Chromotomography for a rotating-prism instrument using backprojection, then filtering.

PubMed

Deming, Ross W

2006-08-01

A simple closed-form solution is derived for reconstructing a 3D spatial-chromatic image cube from a set of chromatically dispersed 2D image frames. The algorithm is tailored for a particular instrument in which the dispersion element is a matching set of mechanically rotated direct vision prisms positioned between a lens and a focal plane array. By using a linear operator formalism to derive the Tikhonov-regularized pseudoinverse operator, it is found that the unique minimum-norm solution is obtained by applying the adjoint operator, followed by 1D filtering with respect to the chromatic variable. Thus the filtering and backprojection (adjoint) steps are applied in reverse order relative to an existing method. Computational efficiency is provided by use of the fast Fourier transform in the filtering step.

Recent Improvements in Aerodynamic Design Optimization on Unstructured Meshes

NASA Technical Reports Server (NTRS)

Nielsen, Eric J.; Anderson, W. Kyle

2000-01-01

Recent improvements in an unstructured-grid method for large-scale aerodynamic design are presented. Previous work had shown such computations to be prohibitively long in a sequential processing environment. Also, robust adjoint solutions and mesh movement procedures were difficult to realize, particularly for viscous flows. To overcome these limiting factors, a set of design codes based on a discrete adjoint method is extended to a multiprocessor environment using a shared memory approach. A nearly linear speedup is demonstrated, and the consistency of the linearizations is shown to remain valid. The full linearization of the residual is used to precondition the adjoint system, and a significantly improved convergence rate is obtained. A new mesh movement algorithm is implemented and several advantages over an existing technique are presented. Several design cases are shown for turbulent flows in two and three dimensions.

Adjoint-Based Algorithms for Adaptation and Design Optimizations on Unstructured Grids

NASA Technical Reports Server (NTRS)

Nielsen, Eric J.

2006-01-01

Schemes based on discrete adjoint algorithms present several exciting opportunities for significantly advancing the current state of the art in computational fluid dynamics. Such methods provide an extremely efficient means for obtaining discretely consistent sensitivity information for hundreds of design variables, opening the door to rigorous, automated design optimization of complex aerospace configuration using the Navier-Stokes equation. Moreover, the discrete adjoint formulation provides a mathematically rigorous foundation for mesh adaptation and systematic reduction of spatial discretization error. Error estimates are also an inherent by-product of an adjoint-based approach, valuable information that is virtually non-existent in today's large-scale CFD simulations. An overview of the adjoint-based algorithm work at NASA Langley Research Center is presented, with examples demonstrating the potential impact on complex computational problems related to design optimization as well as mesh adaptation.

Aerodynamic Design on Unstructured Grids for Turbulent Flows

NASA Technical Reports Server (NTRS)

Anderson, W. Kyle; Bonhaus, Daryl L.

1997-01-01

An aerodynamic design algorithm for turbulent flows using unstructured grids is described. The current approach uses adjoint (costate) variables for obtaining derivatives of the cost function. The solution of the adjoint equations is obtained using an implicit formulation in which the turbulence model is fully coupled with the flow equations when solving for the costate variables. The accuracy of the derivatives is demonstrated by comparison with finite-difference gradients and a few example computations are shown. In addition, a user interface is described which significantly reduces the time required for setting up the design problems. Recommendations on directions of further research into the Navier Stokes design process are made.

Unsteady adjoint for large eddy simulation of a coupled turbine stator-rotor system

NASA Astrophysics Data System (ADS)

Talnikar, Chaitanya; Wang, Qiqi; Laskowski, Gregory

2016-11-01

Unsteady fluid flow simulations like large eddy simulation are crucial in capturing key physics in turbomachinery applications like separation and wake formation in flow over a turbine vane with a downstream blade. To determine how sensitive the design objectives of the coupled system are to control parameters, an unsteady adjoint is needed. It enables the computation of the gradient of an objective with respect to a large number of inputs in a computationally efficient manner. In this paper we present unsteady adjoint solutions for a coupled turbine stator-rotor system. As the transonic fluid flows over the stator vane, the boundary layer transitions to turbulence. The turbulent wake then impinges on the rotor blades, causing early separation. This coupled system exhibits chaotic dynamics which causes conventional adjoint solutions to diverge exponentially, resulting in the corruption of the sensitivities obtained from the adjoint solutions for long-time simulations. In this presentation, adjoint solutions for aerothermal objectives are obtained through a localized adjoint viscosity injection method which aims to stabilize the adjoint solution and maintain accurate sensitivities. Preliminary results obtained from the supercomputer Mira will be shown in the presentation.

Global Instability on Laminar Separation Bubbles-Revisited

NASA Technical Reports Server (NTRS)

Theofilis, Vassilis; Rodriquez, Daniel; Smith, Douglas

2010-01-01

In the last 3 years, global linear instability of LSB has been revisited, using state-of-the-art hardware and algorithms. Eigenspectra of LSB flows have been understood and classified in branches of known and newly-discovered eigenmodes. Major achievements: World-largest numerical solutions of global eigenvalue problems are routinely performed. Key aerodynamic phenomena have been explained via critical point theory, applied to our global mode results. Theoretical foundation for control of LSB flows has been laid. Global mode of LSB at the origin of observable phenomena. U-separation on semi-infinite plate. Stall cells on (stalled) airfoil. Receptivity/Sensitivity/AFC feasible (practical?) via: Adjoint EVP solution. Direct/adjoint coupling (the Crete connection). Minor effect of compressibility on global instability in the subsonic compressible regime. Global instability analysis of LSB in realistic supersonic flows apparently quite some way down the horizon.

New recursive-least-squares algorithms for nonlinear active control of sound and vibration using neural networks.

PubMed

Bouchard, M

2001-01-01

In recent years, a few articles describing the use of neural networks for nonlinear active control of sound and vibration were published. Using a control structure with two multilayer feedforward neural networks (one as a nonlinear controller and one as a nonlinear plant model), steepest descent algorithms based on two distinct gradient approaches were introduced for the training of the controller network. The two gradient approaches were sometimes called the filtered-x approach and the adjoint approach. Some recursive-least-squares algorithms were also introduced, using the adjoint approach. In this paper, an heuristic procedure is introduced for the development of recursive-least-squares algorithms based on the filtered-x and the adjoint gradient approaches. This leads to the development of new recursive-least-squares algorithms for the training of the controller neural network in the two networks structure. These new algorithms produce a better convergence performance than previously published algorithms. Differences in the performance of algorithms using the filtered-x and the adjoint gradient approaches are discussed in the paper. The computational load of the algorithms discussed in the paper is evaluated for multichannel systems of nonlinear active control. Simulation results are presented to compare the convergence performance of the algorithms, showing the convergence gain provided by the new algorithms.

Continuous energy adjoint transport for photons in PHITS

NASA Astrophysics Data System (ADS)

Malins, Alex; Machida, Masahiko; Niita, Koji

2017-09-01

Adjoint Monte Carlo can be an effcient algorithm for solving photon transport problems where the size of the tally is relatively small compared to the source. Such problems are typical in environmental radioactivity calculations, where natural or fallout radionuclides spread over a large area contribute to the air dose rate at a particular location. Moreover photon transport with continuous energy representation is vital for accurately calculating radiation protection quantities. Here we describe the incorporation of an adjoint Monte Carlo capability for continuous energy photon transport into the Particle and Heavy Ion Transport code System (PHITS). An adjoint cross section library for photon interactions was developed based on the JENDL- 4.0 library, by adding cross sections for adjoint incoherent scattering and pair production. PHITS reads in the library and implements the adjoint transport algorithm by Hoogenboom. Adjoint pseudo-photons are spawned within the forward tally volume and transported through space. Currently pseudo-photons can undergo coherent and incoherent scattering within the PHITS adjoint function. Photoelectric absorption is treated implicitly. The calculation result is recovered from the pseudo-photon flux calculated over the true source volume. A new adjoint tally function facilitates this conversion. This paper gives an overview of the new function and discusses potential future developments.

Full-order optimal compensators for flow control: the multiple inputs case

NASA Astrophysics Data System (ADS)

Semeraro, Onofrio; Pralits, Jan O.

2018-03-01

Flow control has been the subject of numerous experimental and theoretical works. We analyze full-order, optimal controllers for large dynamical systems in the presence of multiple actuators and sensors. The full-order controllers do not require any preliminary model reduction or low-order approximation: this feature allows us to assess the optimal performance of an actuated flow without relying on any estimation process or further hypothesis on the disturbances. We start from the original technique proposed by Bewley et al. (Meccanica 51(12):2997-3014, 2016. https://doi.org/10.1007/s11012-016-0547-3), the adjoint of the direct-adjoint (ADA) algorithm. The algorithm is iterative and allows bypassing the solution of the algebraic Riccati equation associated with the optimal control problem, typically infeasible for large systems. In this numerical work, we extend the ADA iteration into a more general framework that includes the design of controllers with multiple, coupled inputs and robust controllers (H_{∞} methods). First, we demonstrate our results by showing the analytical equivalence between the full Riccati solutions and the ADA approximations in the multiple inputs case. In the second part of the article, we analyze the performance of the algorithm in terms of convergence of the solution, by comparing it with analogous techniques. We find an excellent scalability with the number of inputs (actuators), making the method a viable way for full-order control design in complex settings. Finally, the applicability of the algorithm to fluid mechanics problems is shown using the linearized Kuramoto-Sivashinsky equation and the Kármán vortex street past a two-dimensional cylinder.

Receptivity of the compressible mixing layer

NASA Astrophysics Data System (ADS)

Barone, Matthew F.; Lele, Sanjiva K.

2005-09-01

Receptivity of compressible mixing layers to general source distributions is examined by a combined theoretical/computational approach. The properties of solutions to the adjoint Navier Stokes equations are exploited to derive expressions for receptivity in terms of the local value of the adjoint solution. The result is a description of receptivity for arbitrary small-amplitude mass, momentum, and heat sources in the vicinity of a mixing-layer flow, including the edge-scattering effects due to the presence of a splitter plate of finite width. The adjoint solutions are examined in detail for a Mach 1.2 mixing-layer flow. The near field of the adjoint solution reveals regions of relatively high receptivity to direct forcing within the mixing layer, with receptivity to nearby acoustic sources depending on the source type and position. Receptivity ‘nodes’ are present at certain locations near the splitter plate edge where the flow is not sensitive to forcing. The presence of the nodes is explained by interpretation of the adjoint solution as the superposition of incident and scattered fields. The adjoint solution within the boundary layer upstream of the splitter-plate trailing edge reveals a mechanism for transfer of energy from boundary-layer stability modes to Kelvin Helmholtz modes. Extension of the adjoint solution to the far field using a Kirchhoff surface gives the receptivity of the mixing layer to incident sound from distant sources.

Multigrid methods for bifurcation problems: The self adjoint case

NASA Technical Reports Server (NTRS)

Taasan, Shlomo

1987-01-01

This paper deals with multigrid methods for computational problems that arise in the theory of bifurcation and is restricted to the self adjoint case. The basic problem is to solve for arcs of solutions, a task that is done successfully with an arc length continuation method. Other important issues are, for example, detecting and locating singular points as part of the continuation process, switching branches at bifurcation points, etc. Multigrid methods have been applied to continuation problems. These methods work well at regular points and at limit points, while they may encounter difficulties in the vicinity of bifurcation points. A new continuation method that is very efficient also near bifurcation points is presented here. The other issues mentioned above are also treated very efficiently with appropriate multigrid algorithms. For example, it is shown that limit points and bifurcation points can be solved for directly by a multigrid algorithm. Moreover, the algorithms presented here solve the corresponding problems in just a few work units (about 10 or less), where a work unit is the work involved in one local relaxation on the finest grid.

Jointly reconstructing ground motion and resistivity for ERT-based slope stability monitoring

NASA Astrophysics Data System (ADS)

Boyle, Alistair; Wilkinson, Paul B.; Chambers, Jonathan E.; Meldrum, Philip I.; Uhlemann, Sebastian; Adler, Andy

2018-02-01

Electrical resistivity tomography (ERT) is increasingly being used to investigate unstable slopes and monitor the hydrogeological processes within. But movement of electrodes or incorrect placement of electrodes with respect to an assumed model can introduce significant resistivity artefacts into the reconstruction. In this work, we demonstrate a joint resistivity and electrode movement reconstruction algorithm within an iterative Gauss-Newton framework. We apply this to ERT monitoring data from an active slow-moving landslide in the UK. Results show fewer resistivity artefacts and suggest that electrode movement and resistivity can be reconstructed at the same time under certain conditions. A new 2.5-D formulation for the electrode position Jacobian is developed and is shown to give accurate numerical solutions when compared to the adjoint method on 3-D models. On large finite element meshes, the calculation time of the newly developed approach was also proven to be orders of magnitude faster than the 3-D adjoint method and addressed modelling errors in the 2-D perturbation and adjoint electrode position Jacobian.

Adjoint Sensitivity Analysis of Orbital Mechanics: Application to Computations of Observables' Partials with Respect to Harmonics of the Planetary Gravity Fields

NASA Technical Reports Server (NTRS)

Ustinov, Eugene A.; Sunseri, Richard F.

2005-01-01

An approach is presented to the inversion of gravity fields based on evaluation of partials of observables with respect to gravity harmonics using the solution of adjoint problem of orbital dynamics of the spacecraft. Corresponding adjoint operator is derived directly from the linear operator of the linearized forward problem of orbital dynamics. The resulting adjoint problem is similar to the forward problem and can be solved by the same methods. For given highest degree N of gravity harmonics desired, this method involves integration of N adjoint solutions as compared to integration of N2 partials of the forward solution with respect to gravity harmonics in the conventional approach. Thus, for higher resolution gravity models, this approach becomes increasingly more effective in terms of computer resources as compared to the approach based on the solution of the forward problem of orbital dynamics.

Control Theory based Shape Design for the Incompressible Navier-Stokes Equations

NASA Astrophysics Data System (ADS)

Cowles, G.; Martinelli, L.

2003-12-01

A design method for shape optimization in incompressible turbulent viscous flow has been developed and validated for inverse design. The gradient information is determined using a control theory based algorithm. With such an approach, the cost of computing the gradient is negligible. An additional adjoint system must be solved which requires the cost of a single steady state flow solution. Thus, this method has an enormous advantage over traditional finite-difference based algorithms. The method of artificial compressibility is utilized to solve both the flow and adjoint systems. An algebraic turbulence model is used to compute the eddy viscosity. The method is validated using several inverse wing design test cases. In each case, the program must modify the shape of the initial wing such that its pressure distribution matches that of the target wing. Results are shown for the inversion of both finite thickness wings as well as zero thickness wings which can be considered a model of yacht sails.

An optimized treatment for algorithmic differentiation of an important glaciological fixed-point problem

DOE PAGES

Goldberg, Daniel N.; Narayanan, Sri Hari Krishna; Hascoet, Laurent; ...

2016-05-20

We apply an optimized method to the adjoint generation of a time-evolving land ice model through algorithmic differentiation (AD). The optimization involves a special treatment of the fixed-point iteration required to solve the nonlinear stress balance, which differs from a straightforward application of AD software, and leads to smaller memory requirements and in some cases shorter computation times of the adjoint. The optimization is done via implementation of the algorithm of Christianson (1994) for reverse accumulation of fixed-point problems, with the AD tool OpenAD. For test problems, the optimized adjoint is shown to have far lower memory requirements, potentially enablingmore » larger problem sizes on memory-limited machines. In the case of the land ice model, implementation of the algorithm allows further optimization by having the adjoint model solve a sequence of linear systems with identical (as opposed to varying) matrices, greatly improving performance. Finally, the methods introduced here will be of value to other efforts applying AD tools to ice models, particularly ones which solve a hybrid shallow ice/shallow shelf approximation to the Stokes equations.« less

Adjoint Sensitivity Computations for an Embedded-Boundary Cartesian Mesh Method and CAD Geometry

NASA Technical Reports Server (NTRS)

Nemec, Marian; Aftosmis,Michael J.

2006-01-01

Cartesian-mesh methods are perhaps the most promising approach for addressing the issues of flow solution automation for aerodynamic design problems. In these methods, the discretization of the wetted surface is decoupled from that of the volume mesh. This not only enables fast and robust mesh generation for geometry of arbitrary complexity, but also facilitates access to geometry modeling and manipulation using parametric Computer-Aided Design (CAD) tools. Our goal is to combine the automation capabilities of Cartesian methods with an eficient computation of design sensitivities. We address this issue using the adjoint method, where the computational cost of the design sensitivities, or objective function gradients, is esseutially indepeudent of the number of design variables. In previous work, we presented an accurate and efficient algorithm for the solution of the adjoint Euler equations discretized on Cartesian meshes with embedded, cut-cell boundaries. Novel aspects of the algorithm included the computation of surface shape sensitivities for triangulations based on parametric-CAD models and the linearization of the coupling between the surface triangulation and the cut-cells. The objective of the present work is to extend our adjoint formulation to problems involving general shape changes. Central to this development is the computation of volume-mesh sensitivities to obtain a reliable approximation of the objective finction gradient. Motivated by the success of mesh-perturbation schemes commonly used in body-fitted unstructured formulations, we propose an approach based on a local linearization of a mesh-perturbation scheme similar to the spring analogy. This approach circumvents most of the difficulties that arise due to non-smooth changes in the cut-cell layer as the boundary shape evolves and provides a consistent approximation tot he exact gradient of the discretized abjective function. A detailed gradient accurace study is presented to verify our approach. Thereafter, we focus on a shape optimization problem for an Apollo-like reentry capsule. The optimization seeks to enhance the lift-to-drag ratio of the capsule by modifyjing the shape of its heat-shield in conjunction with a center-of-gravity (c.g.) offset. This multipoint and multi-objective optimization problem is used to demonstrate the overall effectiveness of the Cartesian adjoint method for addressing the issues of complex aerodynamic design. This abstract presents only a brief outline of the numerical method and results; full details will be given in the final paper.

An abstract approach to evaporation models in rarefied gas dynamics

NASA Astrophysics Data System (ADS)

Greenberg, W.; van der Mee, C. V. M.

1984-03-01

Strong evaporation models involving 1D stationary problems with linear self-adjoint collision operators and solutions in abstract Hilbert spaces are investigated analytically. An efficient algorithm for locating the transition from existence to nonexistence of solutions is developed and applied to the 1D and 3D BGK model equations and the 3D BGK model in moment form, demonstrating the nonexistence of stationary evaporation states with supersonic drift velocities. Applications to similar models in electron and phonon transport, radiative transfer, and neutron transport are suggested.

Exploratory High-Fidelity Aerostructural Optimization Using an Efficient Monolithic Solution Method

NASA Astrophysics Data System (ADS)

Zhang, Jenmy Zimi

This thesis is motivated by the desire to discover fuel efficient aircraft concepts through exploratory design. An optimization methodology based on tightly integrated high-fidelity aerostructural analysis is proposed, which has the flexibility, robustness, and efficiency to contribute to this goal. The present aerostructural optimization methodology uses an integrated geometry parameterization and mesh movement strategy, which was initially proposed for aerodynamic shape optimization. This integrated approach provides the optimizer with a large amount of geometric freedom for conducting exploratory design, while allowing for efficient and robust mesh movement in the presence of substantial shape changes. In extending this approach to aerostructural optimization, this thesis has addressed a number of important challenges. A structural mesh deformation strategy has been introduced to translate consistently the shape changes described by the geometry parameterization to the structural model. A three-field formulation of the discrete steady aerostructural residual couples the mesh movement equations with the three-dimensional Euler equations and a linear structural analysis. Gradients needed for optimization are computed with a three-field coupled adjoint approach. A number of investigations have been conducted to demonstrate the suitability and accuracy of the present methodology for use in aerostructural optimization involving substantial shape changes. Robustness and efficiency in the coupled solution algorithms is crucial to the success of an exploratory optimization. This thesis therefore also focuses on the design of an effective monolithic solution algorithm for the proposed methodology. This involves using a Newton-Krylov method for the aerostructural analysis and a preconditioned Krylov subspace method for the coupled adjoint solution. Several aspects of the monolithic solution method have been investigated. These include appropriate strategies for scaling and matrix-vector product evaluation, as well as block preconditioning techniques that preserve the modularity between subproblems. The monolithic solution method is applied to problems with varying degrees of fluid-structural coupling, as well as a wing span optimization study. The monolithic solution algorithm typically requires 20%-70% less computing time than its partitioned counterpart. This advantage increases with increasing wing flexibility. The performance of the monolithic solution method is also much less sensitive to the choice of the solution parameter.

Shape determination and control for large space structures

NASA Technical Reports Server (NTRS)

Weeks, C. J.

1981-01-01

An integral operator approach is used to derive solutions to static shape determination and control problems associated with large space structures. Problem assumptions include a linear self-adjoint system model, observations and control forces at discrete points, and performance criteria for the comparison of estimates or control forms. Results are illustrated by simulations in the one dimensional case with a flexible beam model, and in the multidimensional case with a finite model of a large space antenna. Modal expansions for terms in the solution algorithms are presented, using modes from the static or associated dynamic mode. These expansions provide approximated solutions in the event that a used form analytical solution to the system boundary value problem is not available.

Adjoint-Based Aerodynamic Design of Complex Aerospace Configurations

NASA Technical Reports Server (NTRS)

Nielsen, Eric J.

2016-01-01

An overview of twenty years of adjoint-based aerodynamic design research at NASA Langley Research Center is presented. Adjoint-based algorithms provide a powerful tool for efficient sensitivity analysis of complex large-scale computational fluid dynamics (CFD) simulations. Unlike alternative approaches for which computational expense generally scales with the number of design parameters, adjoint techniques yield sensitivity derivatives of a simulation output with respect to all input parameters at the cost of a single additional simulation. With modern large-scale CFD applications often requiring millions of compute hours for a single analysis, the efficiency afforded by adjoint methods is critical in realizing a computationally tractable design optimization capability for such applications.

Trajectory Optimization Using Adjoint Method and Chebyshev Polynomial Approximation for Minimizing Fuel Consumption During Climb

NASA Technical Reports Server (NTRS)

Nguyen, Nhan T.; Hornby, Gregory; Ishihara, Abe

2013-01-01

This paper describes two methods of trajectory optimization to obtain an optimal trajectory of minimum-fuel- to-climb for an aircraft. The first method is based on the adjoint method, and the second method is based on a direct trajectory optimization method using a Chebyshev polynomial approximation and cubic spine approximation. The approximate optimal trajectory will be compared with the adjoint-based optimal trajectory which is considered as the true optimal solution of the trajectory optimization problem. The adjoint-based optimization problem leads to a singular optimal control solution which results in a bang-singular-bang optimal control.

Variational data assimilation for the initial-value dynamo problem.

PubMed

Li, Kuan; Jackson, Andrew; Livermore, Philip W

2011-11-01

The secular variation of the geomagnetic field as observed at the Earth's surface results from the complex magnetohydrodynamics taking place in the fluid core of the Earth. One way to analyze this system is to use the data in concert with an underlying dynamical model of the system through the technique of variational data assimilation, in much the same way as is employed in meteorology and oceanography. The aim is to discover an optimal initial condition that leads to a trajectory of the system in agreement with observations. Taking the Earth's core to be an electrically conducting fluid sphere in which convection takes place, we develop the continuous adjoint forms of the magnetohydrodynamic equations that govern the dynamical system together with the corresponding numerical algorithms appropriate for a fully spectral method. These adjoint equations enable a computationally fast iterative improvement of the initial condition that determines the system evolution. The initial condition depends on the three dimensional form of quantities such as the magnetic field in the entire sphere. For the magnetic field, conservation of the divergence-free condition for the adjoint magnetic field requires the introduction of an adjoint pressure term satisfying a zero boundary condition. We thus find that solving the forward and adjoint dynamo system requires different numerical algorithms. In this paper, an efficient algorithm for numerically solving this problem is developed and tested for two illustrative problems in a whole sphere: one is a kinematic problem with prescribed velocity field, and the second is associated with the Hall-effect dynamo, exhibiting considerable nonlinearity. The algorithm exhibits reliable numerical accuracy and stability. Using both the analytical and the numerical techniques of this paper, the adjoint dynamo system can be solved directly with the same order of computational complexity as that required to solve the forward problem. These numerical techniques form a foundation for ultimate application to observations of the geomagnetic field over the time scale of centuries.

A new mathematical adjoint for the modified SAAF -SN equations

DOE Office of Scientific and Technical Information (OSTI.GOV)

Schunert, Sebastian; Wang, Yaqi; Martineau, Richard

2015-01-01

We present a new adjoint FEM weak form, which can be directly used for evaluating the mathematical adjoint, suitable for perturbation calculations, of the self-adjoint angular flux SN equations (SAAF -SN) without construction and transposition of the underlying coefficient matrix. Stabilization schemes incorporated in the described SAAF -SN method make the mathematical adjoint distinct from the physical adjoint, i.e. the solution of the continuous adjoint equation with SAAF -SN . This weak form is implemented into RattleSnake, the MOOSE (Multiphysics Object-Oriented Simulation Environment) based transport solver. Numerical results verify the correctness of the implementation and show its utility both formore » fixed source and eigenvalue problems.« less

Classical gluon and graviton radiation from the bi-adjoint scalar double copy

NASA Astrophysics Data System (ADS)

Goldberger, Walter D.; Prabhu, Siddharth G.; Thompson, Jedidiah O.

2017-09-01

We find double-copy relations between classical radiating solutions in Yang-Mills theory coupled to dynamical color charges and their counterparts in a cubic bi-adjoint scalar field theory which interacts linearly with particles carrying bi-adjoint charge. The particular color-to-kinematics replacements we employ are motivated by the Bern-Carrasco-Johansson double-copy correspondence for on-shell amplitudes in gauge and gravity theories. They are identical to those recently used to establish relations between classical radiating solutions in gauge theory and in dilaton gravity. Our explicit bi-adjoint solutions are constructed to second order in a perturbative expansion, and map under the double copy onto gauge theory solutions which involve at most cubic gluon self-interactions. If the correspondence is found to persist to higher orders in perturbation theory, our results suggest the possibility of calculating gravitational radiation from colliding compact objects, directly from a scalar field with vastly simpler (purely cubic) Feynman vertices.

The discrete adjoint method for parameter identification in multibody system dynamics.

PubMed

Lauß, Thomas; Oberpeilsteiner, Stefan; Steiner, Wolfgang; Nachbagauer, Karin

2018-01-01

The adjoint method is an elegant approach for the computation of the gradient of a cost function to identify a set of parameters. An additional set of differential equations has to be solved to compute the adjoint variables, which are further used for the gradient computation. However, the accuracy of the numerical solution of the adjoint differential equation has a great impact on the gradient. Hence, an alternative approach is the discrete adjoint method , where the adjoint differential equations are replaced by algebraic equations. Therefore, a finite difference scheme is constructed for the adjoint system directly from the numerical time integration method. The method provides the exact gradient of the discretized cost function subjected to the discretized equations of motion.

Application of Adjoint Methodology in Various Aspects of Sonic Boom Design

NASA Technical Reports Server (NTRS)

Rallabhandi, Sriram K.

2014-01-01

One of the advances in computational design has been the development of adjoint methods allowing efficient calculation of sensitivities in gradient-based shape optimization. This paper discusses two new applications of adjoint methodology that have been developed to aid in sonic boom mitigation exercises. In the first, equivalent area targets are generated using adjoint sensitivities of selected boom metrics. These targets may then be used to drive the vehicle shape during optimization. The second application is the computation of adjoint sensitivities of boom metrics on the ground with respect to parameters such as flight conditions, propagation sampling rate, and selected inputs to the propagation algorithms. These sensitivities enable the designer to make more informed selections of flight conditions at which the chosen cost functionals are less sensitive.

Fully automatic adjoints: a robust and efficient mechanism for generating adjoint ocean models

NASA Astrophysics Data System (ADS)

Ham, D. A.; Farrell, P. E.; Funke, S. W.; Rognes, M. E.

2012-04-01

The problem of generating and maintaining adjoint models is sufficiently difficult that typically only the most advanced and well-resourced community ocean models achieve it. There are two current technologies which each suffer from their own limitations. Algorithmic differentiation, also called automatic differentiation, is employed by models such as the MITGCM [2] and the Alfred Wegener Institute model FESOM [3]. This technique is very difficult to apply to existing code, and requires a major initial investment to prepare the code for automatic adjoint generation. AD tools may also have difficulty with code employing modern software constructs such as derived data types. An alternative is to formulate the adjoint differential equation and to discretise this separately. This approach, known as the continuous adjoint and employed in ROMS [4], has the disadvantage that two different model code bases must be maintained and manually kept synchronised as the model develops. The discretisation of the continuous adjoint is not automatically consistent with that of the forward model, producing an additional source of error. The alternative presented here is to formulate the flow model in the high level language UFL (Unified Form Language) and to automatically generate the model using the software of the FEniCS project. In this approach it is the high level code specification which is differentiated, a task very similar to the formulation of the continuous adjoint [5]. However since the forward and adjoint models are generated automatically, the difficulty of maintaining them vanishes and the software engineering process is therefore robust. The scheduling and execution of the adjoint model, including the application of an appropriate checkpointing strategy is managed by libadjoint [1]. In contrast to the conventional algorithmic differentiation description of a model as a series of primitive mathematical operations, libadjoint employs a new abstraction of the simulation process as a sequence of discrete equations which are assembled and solved. It is the coupling of the respective abstractions employed by libadjoint and the FEniCS project which produces the adjoint model automatically, without further intervention from the model developer. This presentation will demonstrate this new technology through linear and non-linear shallow water test cases. The exceptionally simple model syntax will be highlighted and the correctness of the resulting adjoint simulations will be demonstrated using rigorous convergence tests.

A finite-volume Eulerian-Lagrangian Localized Adjoint Method for solution of the advection-dispersion equation

USGS Publications Warehouse

Healy, R.W.; Russell, T.F.

1993-01-01

A new mass-conservative method for solution of the one-dimensional advection-dispersion equation is derived and discussed. Test results demonstrate that the finite-volume Eulerian-Lagrangian localized adjoint method (FVELLAM) outperforms standard finite-difference methods, in terms of accuracy and efficiency, for solute transport problems that are dominated by advection. For dispersion-dominated problems, the performance of the method is similar to that of standard methods. Like previous ELLAM formulations, FVELLAM systematically conserves mass globally with all types of boundary conditions. FVELLAM differs from other ELLAM approaches in that integrated finite differences, instead of finite elements, are used to approximate the governing equation. This approach, in conjunction with a forward tracking scheme, greatly facilitates mass conservation. The mass storage integral is numerically evaluated at the current time level, and quadrature points are then tracked forward in time to the next level. Forward tracking permits straightforward treatment of inflow boundaries, thus avoiding the inherent problem in backtracking, as used by most characteristic methods, of characteristic lines intersecting inflow boundaries. FVELLAM extends previous ELLAM results by obtaining mass conservation locally on Lagrangian space-time elements. Details of the integration, tracking, and boundary algorithms are presented. Test results are given for problems in Cartesian and radial coordinates.

Comparison of a discrete steepest ascent method with the continuous steepest ascent method for optimal programing

NASA Technical Reports Server (NTRS)

Childs, A. G.

1971-01-01

A discrete steepest ascent method which allows controls which are not piecewise constant (for example, it allows all continuous piecewise linear controls) was derived for the solution of optimal programming problems. This method is based on the continuous steepest ascent method of Bryson and Denham and new concepts introduced by Kelley and Denham in their development of compatible adjoints for taking into account the effects of numerical integration. The method is a generalization of the algorithm suggested by Canon, Cullum, and Polak with the details of the gradient computation given. The discrete method was compared with the continuous method for an aerodynamics problem for which an analytic solution is given by Pontryagin's maximum principle, and numerical results are presented. The discrete method converges more rapidly than the continuous method at first, but then for some undetermined reason, loses its exponential convergence rate. A comparsion was also made for the algorithm of Canon, Cullum, and Polak using piecewise constant controls. This algorithm is very competitive with the continuous algorithm.

Comparative Study of Three Data Assimilation Methods for Ice Sheet Model Initialisation

NASA Astrophysics Data System (ADS)

Mosbeux, Cyrille; Gillet-Chaulet, Fabien; Gagliardini, Olivier

2015-04-01

The current global warming has direct consequences on ice-sheet mass loss contributing to sea level rise. This loss is generally driven by an acceleration of some coastal outlet glaciers and reproducing these mechanisms is one of the major issues in ice-sheet and ice flow modelling. The construction of an initial state, as close as possible to current observations, is required as a prerequisite before producing any reliable projection of the evolution of ice-sheets. For this step, inverse methods are often used to infer badly known or unknown parameters. For instance, the adjoint inverse method has been implemented and applied with success by different authors in different ice flow models in order to infer the basal drag [ Schafer et al., 2012; Gillet-chauletet al., 2012; Morlighem et al., 2010]. Others data fields, such as ice surface and bedrock topography, are easily measurable with more or less uncertainty but only locally along tracks and interpolated on finer model grid. All these approximations lead to errors on the data elevation model and give rise to an ill-posed problem inducing non-physical anomalies in flux divergence [Seroussi et al, 2011]. A solution to dissipate these divergences of flux is to conduct a surface relaxation step at the expense of the accuracy of the modelled surface [Gillet-Chaulet et al., 2012]. Other solutions, based on the inversion of ice thickness and basal drag were proposed [Perego et al., 2014; Pralong & Gudmundsson, 2011]. In this study, we create a twin experiment to compare three different assimilation algorithms based on inverse methods and nudging to constrain the bedrock friction and the bedrock elevation: (i) cyclic inversion of friction parameter and bedrock topography using adjoint method, (ii) cycles coupling inversion of friction parameter using adjoint method and nudging of bedrock topography, (iii) one step inversion of both parameters with adjoint method. The three methods show a clear improvement in parameters knowledge leading to a significant reduction of flux divergence of the model before forecasting.

Adjoint eigenfunctions of temporally recurrent single-spiral solutions in a simple model of atrial fibrillation.

PubMed

Marcotte, Christopher D; Grigoriev, Roman O

2016-09-01

This paper introduces a numerical method for computing the spectrum of adjoint (left) eigenfunctions of spiral wave solutions to reaction-diffusion systems in arbitrary geometries. The method is illustrated by computing over a hundred eigenfunctions associated with an unstable time-periodic single-spiral solution of the Karma model on a square domain. We show that all leading adjoint eigenfunctions are exponentially localized in the vicinity of the spiral tip, although the marginal modes (response functions) demonstrate the strongest localization. We also discuss the implications of the localization for the dynamics and control of unstable spiral waves. In particular, the interaction with no-flux boundaries leads to a drift of spiral waves which can be understood with the help of the response functions.

Adjoint eigenfunctions of temporally recurrent single-spiral solutions in a simple model of atrial fibrillation

NASA Astrophysics Data System (ADS)

Marcotte, Christopher D.; Grigoriev, Roman O.

2016-09-01

This paper introduces a numerical method for computing the spectrum of adjoint (left) eigenfunctions of spiral wave solutions to reaction-diffusion systems in arbitrary geometries. The method is illustrated by computing over a hundred eigenfunctions associated with an unstable time-periodic single-spiral solution of the Karma model on a square domain. We show that all leading adjoint eigenfunctions are exponentially localized in the vicinity of the spiral tip, although the marginal modes (response functions) demonstrate the strongest localization. We also discuss the implications of the localization for the dynamics and control of unstable spiral waves. In particular, the interaction with no-flux boundaries leads to a drift of spiral waves which can be understood with the help of the response functions.

An algorithm for variational data assimilation of contact concentration measurements for atmospheric chemistry models

NASA Astrophysics Data System (ADS)

Penenko, Alexey; Penenko, Vladimir

2014-05-01

Contact concentration measurement data assimilation problem is considered for convection-diffusion-reaction models originating from the atmospheric chemistry study. High dimensionality of models imposes strict requirements on the computational efficiency of the algorithms. Data assimilation is carried out within the variation approach on a single time step of the approximated model. A control function is introduced into the source term of the model to provide flexibility for data assimilation. This function is evaluated as the minimum of the target functional that connects its norm to a misfit between measured and model-simulated data. In the case mathematical model acts as a natural Tikhonov regularizer for the ill-posed measurement data inversion problem. This provides flow-dependent and physically-plausible structure of the resulting analysis and reduces a need to calculate model error covariance matrices that are sought within conventional approach to data assimilation. The advantage comes at the cost of the adjoint problem solution. This issue is solved within the frameworks of splitting-based realization of the basic convection-diffusion-reaction model. The model is split with respect to physical processes and spatial variables. A contact measurement data is assimilated on each one-dimensional convection-diffusion splitting stage. In this case a computationally-efficient direct scheme for both direct and adjoint problem solution can be constructed based on the matrix sweep method. Data assimilation (or regularization) parameter that regulates ratio between model and data in the resulting analysis is obtained with Morozov discrepancy principle. For the proper performance the algorithm takes measurement noise estimation. In the case of Gaussian errors the probability that the used Chi-squared-based estimate is the upper one acts as the assimilation parameter. A solution obtained can be used as the initial guess for data assimilation algorithms that assimilate outside the splitting stages and involve iterations. Splitting method stage that is responsible for chemical transformation processes is realized with the explicit discrete-analytical scheme with respect to time. The scheme is based on analytical extraction of the exponential terms from the solution. This provides unconditional positive sign for the evaluated concentrations. Splitting-based structure of the algorithm provides means for efficient parallel realization. The work is partially supported by the Programs No 4 of Presidium RAS and No 3 of Mathematical Department of RAS, by RFBR project 11-01-00187 and Integrating projects of SD RAS No 8 and 35. Our studies are in the line with the goals of COST Action ES1004.

A comparison of discrete versus continuous adjoint states to invert groundwater flow in heterogeneous dual porosity systems

NASA Astrophysics Data System (ADS)

Delay, Frederick; Badri, Hamid; Fahs, Marwan; Ackerer, Philippe

2017-12-01

Dual porosity models become increasingly used for simulating groundwater flow at the large scale in fractured porous media. In this context, model inversions with the aim of retrieving the system heterogeneity are frequently faced with huge parameterizations for which descent methods of inversion with the assistance of adjoint state calculations are well suited. We compare the performance of discrete and continuous forms of adjoint states associated with the flow equations in a dual porosity system. The discrete form inherits from previous works by some of the authors, as the continuous form is completely new and here fully differentiated for handling all types of model parameters. Adjoint states assist descent methods by calculating the gradient components of the objective function, these being a key to good convergence of inverse solutions. Our comparison on the basis of synthetic exercises show that both discrete and continuous adjoint states can provide very similar solutions close to reference. For highly heterogeneous systems, the calculation grid of the continuous form cannot be too coarse, otherwise the method may show lack of convergence. This notwithstanding, the continuous adjoint state is the most versatile form as its non-intrusive character allows for plugging an inversion toolbox quasi-independent from the code employed for solving the forward problem.

A new approach for developing adjoint models

NASA Astrophysics Data System (ADS)

Farrell, P. E.; Funke, S. W.

2011-12-01

Many data assimilation algorithms rely on the availability of gradients of misfit functionals, which can be efficiently computed with adjoint models. However, the development of an adjoint model for a complex geophysical code is generally very difficult. Algorithmic differentiation (AD, also called automatic differentiation) offers one strategy for simplifying this task: it takes the abstraction that a model is a sequence of primitive instructions, each of which may be differentiated in turn. While extremely successful, this low-level abstraction runs into time-consuming difficulties when applied to the whole codebase of a model, such as differentiating through linear solves, model I/O, calls to external libraries, language features that are unsupported by the AD tool, and the use of multiple programming languages. While these difficulties can be overcome, it requires a large amount of technical expertise and an intimate familiarity with both the AD tool and the model. An alternative to applying the AD tool to the whole codebase is to assemble the discrete adjoint equations and use these to compute the necessary gradients. With this approach, the AD tool must be applied to the nonlinear assembly operators, which are typically small, self-contained units of the codebase. The disadvantage of this approach is that the assembly of the discrete adjoint equations is still very difficult to perform correctly, especially for complex multiphysics models that perform temporal integration; as it stands, this approach is as difficult and time-consuming as applying AD to the whole model. In this work, we have developed a library which greatly simplifies and automates the alternate approach of assembling the discrete adjoint equations. We propose a complementary, higher-level abstraction to that of AD: that a model is a sequence of linear solves. The developer annotates model source code with library calls that build a 'tape' of the operators involved and their dependencies, and supplies callbacks to compute the action of these operators. The library, called libadjoint, is then capable of symbolically manipulating the forward annotation to automatically assemble the adjoint equations. Libadjoint is open source, and is explicitly designed to be bolted-on to an existing discrete model. It can be applied to any discretisation, steady or time-dependent problems, and both linear and nonlinear systems. Using libadjoint has several advantages. It requires the application of an AD tool only to small pieces of code, making the use of AD far more tractable. As libadjoint derives the adjoint equations, the expertise required to develop an adjoint model is greatly diminished. One major advantage of this approach is that the model developer is freed from implementing complex checkpointing strategies for the adjoint model: libadjoint has sufficient information about the forward model to re-play the entire forward solve when necessary, and thus the checkpointing algorithm can be implemented entirely within the library itself. Examples are shown using the Fluidity/ICOM framework, a complex ocean model under development at Imperial College London.

Practical Aerodynamic Design Optimization Based on the Navier-Stokes Equations and a Discrete Adjoint Method

NASA Technical Reports Server (NTRS)

Grossman, Bernard

1999-01-01

The technical details are summarized below: Compressible and incompressible versions of a three-dimensional unstructured mesh Reynolds-averaged Navier-Stokes flow solver have been differentiated and resulting derivatives have been verified by comparisons with finite differences and a complex-variable approach. In this implementation, the turbulence model is fully coupled with the flow equations in order to achieve this consistency. The accuracy demonstrated in the current work represents the first time that such an approach has been successfully implemented. The accuracy of a number of simplifying approximations to the linearizations of the residual have been examined. A first-order approximation to the dependent variables in both the adjoint and design equations has been investigated. The effects of a "frozen" eddy viscosity and the ramifications of neglecting some mesh sensitivity terms were also examined. It has been found that none of the approximations yielded derivatives of acceptable accuracy and were often of incorrect sign. However, numerical experiments indicate that an incomplete convergence of the adjoint system often yield sufficiently accurate derivatives, thereby significantly lowering the time required for computing sensitivity information. The convergence rate of the adjoint solver relative to the flow solver has been examined. Inviscid adjoint solutions typically require one to four times the cost of a flow solution, while for turbulent adjoint computations, this ratio can reach as high as eight to ten. Numerical experiments have shown that the adjoint solver can stall before converging the solution to machine accuracy, particularly for viscous cases. A possible remedy for this phenomenon would be to include the complete higher-order linearization in the preconditioning step, or to employ a simple form of mesh sequencing to obtain better approximations to the solution through the use of coarser meshes. . An efficient surface parameterization based on a free-form deformation technique has been utilized and the resulting codes have been integrated with an optimization package. Lastly, sample optimizations have been shown for inviscid and turbulent flow over an ONERA M6 wing. Drag reductions have been demonstrated by reducing shock strengths across the span of the wing.

Finite-frequency tomography using adjoint methods-Methodology and examples using membrane surface waves

NASA Astrophysics Data System (ADS)

Tape, Carl; Liu, Qinya; Tromp, Jeroen

2007-03-01

We employ adjoint methods in a series of synthetic seismic tomography experiments to recover surface wave phase-speed models of southern California. Our approach involves computing the Fréchet derivative for tomographic inversions via the interaction between a forward wavefield, propagating from the source to the receivers, and an `adjoint' wavefield, propagating from the receivers back to the source. The forward wavefield is computed using a 2-D spectral-element method (SEM) and a phase-speed model for southern California. A `target' phase-speed model is used to generate the `data' at the receivers. We specify an objective or misfit function that defines a measure of misfit between data and synthetics. For a given receiver, the remaining differences between data and synthetics are time-reversed and used as the source of the adjoint wavefield. For each earthquake, the interaction between the regular and adjoint wavefields is used to construct finite-frequency sensitivity kernels, which we call event kernels. An event kernel may be thought of as a weighted sum of phase-specific (e.g. P) banana-doughnut kernels, with weights determined by the measurements. The overall sensitivity is simply the sum of event kernels, which defines the misfit kernel. The misfit kernel is multiplied by convenient orthonormal basis functions that are embedded in the SEM code, resulting in the gradient of the misfit function, that is, the Fréchet derivative. A non-linear conjugate gradient algorithm is used to iteratively improve the model while reducing the misfit function. We illustrate the construction of the gradient and the minimization algorithm, and consider various tomographic experiments, including source inversions, structural inversions and joint source-structure inversions. Finally, we draw connections between classical Hessian-based tomography and gradient-based adjoint tomography.

Efficient block preconditioned eigensolvers for linear response time-dependent density functional theory

NASA Astrophysics Data System (ADS)

Vecharynski, Eugene; Brabec, Jiri; Shao, Meiyue; Govind, Niranjan; Yang, Chao

2017-12-01

We present two efficient iterative algorithms for solving the linear response eigenvalue problem arising from the time dependent density functional theory. Although the matrix to be diagonalized is nonsymmetric, it has a special structure that can be exploited to save both memory and floating point operations. In particular, the nonsymmetric eigenvalue problem can be transformed into an eigenvalue problem that involves the product of two matrices M and K. We show that, because MK is self-adjoint with respect to the inner product induced by the matrix K, this product eigenvalue problem can be solved efficiently by a modified Davidson algorithm and a modified locally optimal block preconditioned conjugate gradient (LOBPCG) algorithm that make use of the K-inner product. The solution of the product eigenvalue problem yields one component of the eigenvector associated with the original eigenvalue problem. We show that the other component of the eigenvector can be easily recovered in an inexpensive postprocessing procedure. As a result, the algorithms we present here become more efficient than existing methods that try to approximate both components of the eigenvectors simultaneously. In particular, our numerical experiments demonstrate that the new algorithms presented here consistently outperform the existing state-of-the-art Davidson type solvers by a factor of two in both solution time and storage.

A New Class of Highly Accurate Differentiation Schemes Based on the Prolate Spheroidal Wave Functions

DTIC Science & Technology

2011-04-07

predictor - corrector scheme. Such an approach for the solution of time-dependent PDEs, which is some- times referred to as the “method of lines,” is studied...particular, λj = i j |λj |. We define the self -adjoint operator Qc : L 2([−1, 1]) → L2([−1, 1]) by the formula Qc(φ) = 1 π ∫ 1 −1 sin( c (x− t)) x− t φ...Gaussian quadratures for bandlimited functions is to use the Newton-type nonlinear optimization algorithm of [14]. Specifically, for bandlimit c and

Efficient block preconditioned eigensolvers for linear response time-dependent density functional theory

DOE Office of Scientific and Technical Information (OSTI.GOV)

Vecharynski, Eugene; Brabec, Jiri; Shao, Meiyue

We present two efficient iterative algorithms for solving the linear response eigen- value problem arising from the time dependent density functional theory. Although the matrix to be diagonalized is nonsymmetric, it has a special structure that can be exploited to save both memory and floating point operations. In particular, the nonsymmetric eigenvalue problem can be transformed into a product eigenvalue problem that is self-adjoint with respect to a K-inner product. This product eigenvalue problem can be solved efficiently by a modified Davidson algorithm and a modified locally optimal block preconditioned conjugate gradient (LOBPCG) algorithm that make use of the K-innermore » product. The solution of the product eigenvalue problem yields one component of the eigenvector associated with the original eigenvalue problem. However, the other component of the eigenvector can be easily recovered in a postprocessing procedure. Therefore, the algorithms we present here are more efficient than existing algorithms that try to approximate both components of the eigenvectors simultaneously. The efficiency of the new algorithms is demonstrated by numerical examples.« less

NEMOTAM: tangent and adjoint models for the ocean modelling platform NEMO

NASA Astrophysics Data System (ADS)

Vidard, A.; Bouttier, P.-A.; Vigilant, F.

2015-04-01

Tangent linear and adjoint models (TAMs) are efficient tools to analyse and to control dynamical systems such as NEMO. They can be involved in a large range of applications such as sensitivity analysis, parameter estimation or the computation of characteristic vectors. A TAM is also required by the 4D-Var algorithm, which is one of the major methods in data assimilation. This paper describes the development and the validation of the tangent linear and adjoint model for the NEMO ocean modelling platform (NEMOTAM). The diagnostic tools that are available alongside NEMOTAM are detailed and discussed, and several applications are also presented.

NEMOTAM: tangent and adjoint models for the ocean modelling platform NEMO

NASA Astrophysics Data System (ADS)

Vidard, A.; Bouttier, P.-A.; Vigilant, F.

2014-10-01

The tangent linear and adjoint model (TAM) are efficient tools to analyse and to control dynamical systems such as NEMO. They can be involved in a large range of applications such as sensitivity analysis, parameter estimation or the computation of characteristics vectors. TAM is also required by the 4-D-VAR algorithm which is one of the major method in Data Assimilation. This paper describes the development and the validation of the Tangent linear and Adjoint Model for the NEMO ocean modelling platform (NEMOTAM). The diagnostic tools that are available alongside NEMOTAM are detailed and discussed and several applications are also presented.

Local-in-Time Adjoint-Based Method for Optimal Control/Design Optimization of Unsteady Compressible Flows

NASA Technical Reports Server (NTRS)

Yamaleev, N. K.; Diskin, B.; Nielsen, E. J.

2009-01-01

.We study local-in-time adjoint-based methods for minimization of ow matching functionals subject to the 2-D unsteady compressible Euler equations. The key idea of the local-in-time method is to construct a very accurate approximation of the global-in-time adjoint equations and the corresponding sensitivity derivative by using only local information available on each time subinterval. In contrast to conventional time-dependent adjoint-based optimization methods which require backward-in-time integration of the adjoint equations over the entire time interval, the local-in-time method solves local adjoint equations sequentially over each time subinterval. Since each subinterval contains relatively few time steps, the storage cost of the local-in-time method is much lower than that of the global adjoint formulation, thus making the time-dependent optimization feasible for practical applications. The paper presents a detailed comparison of the local- and global-in-time adjoint-based methods for minimization of a tracking functional governed by the Euler equations describing the ow around a circular bump. Our numerical results show that the local-in-time method converges to the same optimal solution obtained with the global counterpart, while drastically reducing the memory cost as compared to the global-in-time adjoint formulation.

A hybrid (Monte Carlo/deterministic) approach for multi-dimensional radiation transport

DOE Office of Scientific and Technical Information (OSTI.GOV)

Bal, Guillaume, E-mail: gb2030@columbia.edu; Davis, Anthony B., E-mail: Anthony.B.Davis@jpl.nasa.gov; Kavli Institute for Theoretical Physics, Kohn Hall, University of California, Santa Barbara, CA 93106-4030

2011-08-20

Highlights: {yields} We introduce a variance reduction scheme for Monte Carlo (MC) transport. {yields} The primary application is atmospheric remote sensing. {yields} The technique first solves the adjoint problem using a deterministic solver. {yields} Next, the adjoint solution is used as an importance function for the MC solver. {yields} The adjoint problem is solved quickly since it ignores the volume. - Abstract: A novel hybrid Monte Carlo transport scheme is demonstrated in a scene with solar illumination, scattering and absorbing 2D atmosphere, a textured reflecting mountain, and a small detector located in the sky (mounted on a satellite or amore » airplane). It uses a deterministic approximation of an adjoint transport solution to reduce variance, computed quickly by ignoring atmospheric interactions. This allows significant variance and computational cost reductions when the atmospheric scattering and absorption coefficient are small. When combined with an atmospheric photon-redirection scheme, significant variance reduction (equivalently acceleration) is achieved in the presence of atmospheric interactions.« less

Towards Seismic Tomography Based Upon Adjoint Methods

NASA Astrophysics Data System (ADS)

Tromp, J.; Liu, Q.; Tape, C.; Maggi, A.

2006-12-01

We outline the theory behind tomographic inversions based on 3D reference models, fully numerical 3D wave propagation, and adjoint methods. Our approach involves computing the Fréchet derivatives for tomographic inversions via the interaction between a forward wavefield, propagating from the source to the receivers, and an `adjoint' wavefield, propagating from the receivers back to the source. The forward wavefield is computed using a spectral-element method (SEM) and a heterogeneous wave-speed model, and stored as synthetic seismograms at particular receivers for which there is data. We specify an objective or misfit function that defines a measure of misfit between data and synthetics. For a given receiver, the differences between the data and the synthetics are time reversed and used as the source of the adjoint wavefield. For each earthquake, the interaction between the regular and adjoint wavefields is used to construct finite-frequency sensitivity kernels, which we call event kernel. These kernels may be thought of as weighted sums of measurement-specific banana-donut kernels, with weights determined by the measurements. The overall sensitivity is simply the sum of event kernels, which defines the misfit kernel. The misfit kernel is multiplied by convenient orthonormal basis functions that are embedded in the SEM code, resulting in the gradient of the misfit function, i.e., the Fréchet derivatives. The misfit kernel is multiplied by convenient orthonormal basis functions that are embedded in the SEM code, resulting in the gradient of the misfit function, i.e., the Fréchet derivatives. A conjugate gradient algorithm is used to iteratively improve the model while reducing the misfit function. Using 2D examples for Rayleigh wave phase-speed maps of southern California, we illustrate the construction of the gradient and the minimization algorithm, and consider various tomographic experiments, including source inversions, structural inversions, and joint source-structure inversions. We also illustrate the characteristics of these 3D finite-frequency kernels based upon adjoint simulations for a variety of global arrivals, e.g., Pdiff, P'P', and SKS, and we illustrate how the approach may be used to investigate body- and surface-wave anisotropy. In adjoint tomography any time segment in which the data and synthetics match reasonably well is suitable for measurement, and this implies a much greater number of phases per seismogram can be used compared to classical tomography in which the sensitivity of the measurements is determined analytically for specific arrivals, e.g., P. We use an automated picking algorithm based upon short-term/long-term averages and strict phase and amplitude anomaly criteria to determine arrivals and time windows suitable for measurement. For shallow global events the algorithm typically identifies of the order of 1000~windows suitable for measurement, whereas for a deep event the number can reach 4000. For southern California earthquakes the number of phases is of the order of 100 for a magnitude 4.0 event and up to 450 for a magnitude 5.0 event. We will show examples of event kernels for both global and regional earthquakes. These event kernels form the basis of adjoint tomography.

Covering Resilience: A Recent Development for Binomial Checkpointing

DOE Office of Scientific and Technical Information (OSTI.GOV)

Walther, Andrea; Narayanan, Sri Hari Krishna

In terms of computing time, adjoint methods offer a very attractive alternative to compute gradient information, required, e.g., for optimization purposes. However, together with this very favorable temporal complexity result comes a memory requirement that is in essence proportional with the operation count of the underlying function, e.g., if algorithmic differentiation is used to provide the adjoints. For this reason, checkpointing approaches in many variants have become popular. This paper analyzes an extension of the so-called binomial approach to cover also possible failures of the computing systems. Such a measure of precaution is of special interest for massive parallel simulationsmore » and adjoint calculations where the mean time between failure of the large scale computing system is smaller than the time needed to complete the calculation of the adjoint information. We describe the extensions of standard checkpointing approaches required for such resilience, provide a corresponding implementation and discuss first numerical results.« less

Learning a trajectory using adjoint functions and teacher forcing

NASA Technical Reports Server (NTRS)

Toomarian, Nikzad B.; Barhen, Jacob

1992-01-01

A new methodology for faster supervised temporal learning in nonlinear neural networks is presented which builds upon the concept of adjoint operators to allow fast computation of the gradients of an error functional with respect to all parameters of the neural architecture, and exploits the concept of teacher forcing to incorporate information on the desired output into the activation dynamics. The importance of the initial or final time conditions for the adjoint equations is discussed. A new algorithm is presented in which the adjoint equations are solved simultaneously (i.e., forward in time) with the activation dynamics of the neural network. We also indicate how teacher forcing can be modulated in time as learning proceeds. The results obtained show that the learning time is reduced by one to two orders of magnitude with respect to previously published results, while trajectory tracking is significantly improved. The proposed methodology makes hardware implementation of temporal learning attractive for real-time applications.

Efficient block preconditioned eigensolvers for linear response time-dependent density functional theory

DOE Office of Scientific and Technical Information (OSTI.GOV)

Vecharynski, Eugene; Brabec, Jiri; Shao, Meiyue

Within this paper, we present two efficient iterative algorithms for solving the linear response eigenvalue problem arising from the time dependent density functional theory. Although the matrix to be diagonalized is nonsymmetric, it has a special structure that can be exploited to save both memory and floating point operations. In particular, the nonsymmetric eigenvalue problem can be transformed into an eigenvalue problem that involves the product of two matrices M and K. We show that, because MK is self-adjoint with respect to the inner product induced by the matrix K, this product eigenvalue problem can be solved efficiently by amore » modified Davidson algorithm and a modified locally optimal block preconditioned conjugate gradient (LOBPCG) algorithm that make use of the K-inner product. Additionally, the solution of the product eigenvalue problem yields one component of the eigenvector associated with the original eigenvalue problem. We show that the other component of the eigenvector can be easily recovered in an inexpensive postprocessing procedure. As a result, the algorithms we present here become more efficient than existing methods that try to approximate both components of the eigenvectors simultaneously. In particular, our numerical experiments demonstrate that the new algorithms presented here consistently outperform the existing state-of-the-art Davidson type solvers by a factor of two in both solution time and storage.« less

Efficient block preconditioned eigensolvers for linear response time-dependent density functional theory

DOE Office of Scientific and Technical Information (OSTI.GOV)

Vecharynski, Eugene; Brabec, Jiri; Shao, Meiyue

In this article, we present two efficient iterative algorithms for solving the linear response eigenvalue problem arising from the time dependent density functional theory. Although the matrix to be diagonalized is nonsymmetric, it has a special structure that can be exploited to save both memory and floating point operations. In particular, the nonsymmetric eigenvalue problem can be transformed into an eigenvalue problem that involves the product of two matrices M and K. We show that, because MK is self-adjoint with respect to the inner product induced by the matrix K, this product eigenvalue problem can be solved efficiently by amore » modified Davidson algorithm and a modified locally optimal block preconditioned conjugate gradient (LOBPCG) algorithm that make use of the K-inner product. The solution of the product eigenvalue problem yields one component of the eigenvector associated with the original eigenvalue problem. We show that the other component of the eigenvector can be easily recovered in an inexpensive postprocessing procedure. As a result, the algorithms we present here become more efficient than existing methods that try to approximate both components of the eigenvectors simultaneously. In particular, our numerical experiments demonstrate that the new algorithms presented here consistently outperform the existing state-of-the-art Davidson type solvers by a factor of two in both solution time and storage.« less

Efficient block preconditioned eigensolvers for linear response time-dependent density functional theory

DOE PAGES

Vecharynski, Eugene; Brabec, Jiri; Shao, Meiyue; ...

2017-12-01

In this article, we present two efficient iterative algorithms for solving the linear response eigenvalue problem arising from the time dependent density functional theory. Although the matrix to be diagonalized is nonsymmetric, it has a special structure that can be exploited to save both memory and floating point operations. In particular, the nonsymmetric eigenvalue problem can be transformed into an eigenvalue problem that involves the product of two matrices M and K. We show that, because MK is self-adjoint with respect to the inner product induced by the matrix K, this product eigenvalue problem can be solved efficiently by amore » modified Davidson algorithm and a modified locally optimal block preconditioned conjugate gradient (LOBPCG) algorithm that make use of the K-inner product. The solution of the product eigenvalue problem yields one component of the eigenvector associated with the original eigenvalue problem. We show that the other component of the eigenvector can be easily recovered in an inexpensive postprocessing procedure. As a result, the algorithms we present here become more efficient than existing methods that try to approximate both components of the eigenvectors simultaneously. In particular, our numerical experiments demonstrate that the new algorithms presented here consistently outperform the existing state-of-the-art Davidson type solvers by a factor of two in both solution time and storage.« less

Efficient block preconditioned eigensolvers for linear response time-dependent density functional theory

DOE PAGES

Vecharynski, Eugene; Brabec, Jiri; Shao, Meiyue; ...

2017-08-24

Within this paper, we present two efficient iterative algorithms for solving the linear response eigenvalue problem arising from the time dependent density functional theory. Although the matrix to be diagonalized is nonsymmetric, it has a special structure that can be exploited to save both memory and floating point operations. In particular, the nonsymmetric eigenvalue problem can be transformed into an eigenvalue problem that involves the product of two matrices M and K. We show that, because MK is self-adjoint with respect to the inner product induced by the matrix K, this product eigenvalue problem can be solved efficiently by amore » modified Davidson algorithm and a modified locally optimal block preconditioned conjugate gradient (LOBPCG) algorithm that make use of the K-inner product. Additionally, the solution of the product eigenvalue problem yields one component of the eigenvector associated with the original eigenvalue problem. We show that the other component of the eigenvector can be easily recovered in an inexpensive postprocessing procedure. As a result, the algorithms we present here become more efficient than existing methods that try to approximate both components of the eigenvectors simultaneously. In particular, our numerical experiments demonstrate that the new algorithms presented here consistently outperform the existing state-of-the-art Davidson type solvers by a factor of two in both solution time and storage.« less

A new zonation algorithm with parameter estimation using hydraulic head and subsidence observations.

PubMed

Zhang, Meijing; Burbey, Thomas J; Nunes, Vitor Dos Santos; Borggaard, Jeff

2014-01-01

Parameter estimation codes such as UCODE_2005 are becoming well-known tools in groundwater modeling investigations. These programs estimate important parameter values such as transmissivity (T) and aquifer storage values (Sa ) from known observations of hydraulic head, flow, or other physical quantities. One drawback inherent in these codes is that the parameter zones must be specified by the user. However, such knowledge is often unknown even if a detailed hydrogeological description is available. To overcome this deficiency, we present a discrete adjoint algorithm for identifying suitable zonations from hydraulic head and subsidence measurements, which are highly sensitive to both elastic (Sske) and inelastic (Sskv) skeletal specific storage coefficients. With the advent of interferometric synthetic aperture radar (InSAR), distributed spatial and temporal subsidence measurements can be obtained. A synthetic conceptual model containing seven transmissivity zones, one aquifer storage zone and three interbed zones for elastic and inelastic storage coefficients were developed to simulate drawdown and subsidence in an aquifer interbedded with clay that exhibits delayed drainage. Simulated delayed land subsidence and groundwater head data are assumed to be the observed measurements, to which the discrete adjoint algorithm is called to create approximate spatial zonations of T, Sske , and Sskv . UCODE-2005 is then used to obtain the final optimal parameter values. Calibration results indicate that the estimated zonations calculated from the discrete adjoint algorithm closely approximate the true parameter zonations. This automation algorithm reduces the bias established by the initial distribution of zones and provides a robust parameter zonation distribution. © 2013, National Ground Water Association.

The Gauss-Bonnet operator of an infinite graph

NASA Astrophysics Data System (ADS)

Anné, Colette; Torki-Hamza, Nabila

2015-06-01

We propose a general condition, to ensure essential self-adjointness for the Gauss-Bonnet operator , based on a notion of completeness as Chernoff. This gives essential self-adjointness of the Laplace operator both for functions and 1-forms on infinite graphs. This is used to extend Flanders result concerning solutions of Kirchhoff's laws.

Regularized wave equation migration for imaging and data reconstruction

NASA Astrophysics Data System (ADS)

Kaplan, Sam T.

The reflection seismic experiment results in a measurement (reflection seismic data) of the seismic wavefield. The linear Born approximation to the seismic wavefield leads to a forward modelling operator that we use to approximate reflection seismic data in terms of a scattering potential. We consider approximations to the scattering potential using two methods: the adjoint of the forward modelling operator (migration), and regularized numerical inversion using the forward and adjoint operators. We implement two parameterizations of the forward modelling and migration operators: source-receiver and shot-profile. For both parameterizations, we find requisite Green's function using the split-step approximation. We first develop the forward modelling operator, and then find the adjoint (migration) operator by recognizing a Fredholm integral equation of the first kind. The resulting numerical system is generally under-determined, requiring prior information to find a solution. In source-receiver migration, the parameterization of the scattering potential is understood using the migration imaging condition, and this encourages us to apply sparse prior models to the scattering potential. To that end, we use both a Cauchy prior and a mixed Cauchy-Gaussian prior, finding better resolved estimates of the scattering potential than are given by the adjoint. In shot-profile migration, the parameterization of the scattering potential has its redundancy in multiple active energy sources (i.e. shots). We find that a smallest model regularized inverse representation of the scattering potential gives a more resolved picture of the earth, as compared to the simpler adjoint representation. The shot-profile parameterization allows us to introduce a joint inversion to further improve the estimate of the scattering potential. Moreover, it allows us to introduce a novel data reconstruction algorithm so that limited data can be interpolated/extrapolated. The linearized operators are expensive, encouraging their parallel implementation. For the source-receiver parameterization of the scattering potential this parallelization is non-trivial. Seismic data is typically corrupted by various types of noise. Sparse coding can be used to suppress noise prior to migration. It is a method that stems from information theory and that we apply to noise suppression in seismic data.

Direct Linearization and Adjoint Approaches to Evaluation of Atmospheric Weighting Functions and Surface Partial Derivatives: General Principles, Synergy and Areas of Application

NASA Technical Reports Server (NTRS)

Ustino, Eugene A.

2006-01-01

This slide presentation reviews the observable radiances as functions of atmospheric parameters and of surface parameters; the mathematics of atmospheric weighting functions (WFs) and surface partial derivatives (PDs) are presented; and the equation of the forward radiative transfer (RT) problem is presented. For non-scattering atmospheres this can be done analytically, and all WFs and PDs can be computed analytically using the direct linearization approach. For scattering atmospheres, in general case, the solution of the forward RT problem can be obtained only numerically, but we need only two numerical solutions: one of the forward RT problem and one of the adjoint RT problem to compute all WFs and PDs we can think of. In this presentation we discuss applications of both the linearization and adjoint approaches

Comparison of adjoint and analytical Bayesian inversion methods for constraining Asian sources of carbon monoxide using satellite (MOPITT) measurements of CO columns

NASA Astrophysics Data System (ADS)

Kopacz, Monika; Jacob, Daniel J.; Henze, Daven K.; Heald, Colette L.; Streets, David G.; Zhang, Qiang

2009-02-01

We apply the adjoint of an atmospheric chemical transport model (GEOS-Chem CTM) to constrain Asian sources of carbon monoxide (CO) with 2° × 2.5° spatial resolution using Measurement of Pollution in the Troposphere (MOPITT) satellite observations of CO columns in February-April 2001. Results are compared to the more common analytical method for solving the same Bayesian inverse problem and applied to the same data set. The analytical method is more exact but because of computational limitations it can only constrain emissions over coarse regions. We find that the correction factors to the a priori CO emission inventory from the adjoint inversion are generally consistent with those of the analytical inversion when averaged over the large regions of the latter. The adjoint solution reveals fine-scale variability (cities, political boundaries) that the analytical inversion cannot resolve, for example, in the Indian subcontinent or between Korea and Japan, and some of that variability is of opposite sign which points to large aggregation errors in the analytical solution. Upward correction factors to Chinese emissions from the prior inventory are largest in central and eastern China, consistent with a recent bottom-up revision of that inventory, although the revised inventory also sees the need for upward corrections in southern China where the adjoint and analytical inversions call for downward correction. Correction factors for biomass burning emissions derived from the adjoint and analytical inversions are consistent with a recent bottom-up inventory on the basis of MODIS satellite fire data.

Comparison of Evolutionary (Genetic) Algorithm and Adjoint Methods for Multi-Objective Viscous Airfoil Optimizations

NASA Technical Reports Server (NTRS)

Pulliam, T. H.; Nemec, M.; Holst, T.; Zingg, D. W.; Kwak, Dochan (Technical Monitor)

2002-01-01

A comparison between an Evolutionary Algorithm (EA) and an Adjoint-Gradient (AG) Method applied to a two-dimensional Navier-Stokes code for airfoil design is presented. Both approaches use a common function evaluation code, the steady-state explicit part of the code,ARC2D. The parameterization of the design space is a common B-spline approach for an airfoil surface, which together with a common griding approach, restricts the AG and EA to the same design space. Results are presented for a class of viscous transonic airfoils in which the optimization tradeoff between drag minimization as one objective and lift maximization as another, produces the multi-objective design space. Comparisons are made for efficiency, accuracy and design consistency.

Development of an adjoint sensitivity field-based treatment-planning technique for the use of newly designed directional LDR sources in brachytherapy.

PubMed

Chaswal, V; Thomadsen, B R; Henderson, D L

2012-02-21

The development and application of an automated 3D greedy heuristic (GH) optimization algorithm utilizing the adjoint sensitivity fields for treatment planning to assess the advantage of directional interstitial prostate brachytherapy is presented. Directional and isotropic dose kernels generated using Monte Carlo simulations based on Best Industries model 2301 I-125 source are utilized for treatment planning. The newly developed GH algorithm is employed for optimization of the treatment plans for seven interstitial prostate brachytherapy cases using mixed sources (directional brachytherapy) and using only isotropic sources (conventional brachytherapy). All treatment plans resulted in V100 > 98% and D90 > 45 Gy for the target prostate region. For the urethra region, the D10(Ur), D90(Ur) and V150(Ur) and for the rectum region the V100cc, D2cc, D90(Re) and V90(Re) all are reduced significantly when mixed sources brachytherapy is used employing directional sources. The simulations demonstrated that the use of directional sources in the low dose-rate (LDR) brachytherapy of the prostate clearly benefits in sparing the urethra and the rectum sensitive structures from overdose. The time taken for a conventional treatment plan is less than three seconds, while the time taken for a mixed source treatment plan is less than nine seconds, as tested on an Intel Core2 Duo 2.2 GHz processor with 1GB RAM. The new 3D GH algorithm is successful in generating a feasible LDR brachytherapy treatment planning solution with an extra degree of freedom, i.e. directionality in very little time.

Development of an adjoint sensitivity field-based treatment-planning technique for the use of newly designed directional LDR sources in brachytherapy

NASA Astrophysics Data System (ADS)

Chaswal, V.; Thomadsen, B. R.; Henderson, D. L.

2012-02-01

The development and application of an automated 3D greedy heuristic (GH) optimization algorithm utilizing the adjoint sensitivity fields for treatment planning to assess the advantage of directional interstitial prostate brachytherapy is presented. Directional and isotropic dose kernels generated using Monte Carlo simulations based on Best Industries model 2301 I-125 source are utilized for treatment planning. The newly developed GH algorithm is employed for optimization of the treatment plans for seven interstitial prostate brachytherapy cases using mixed sources (directional brachytherapy) and using only isotropic sources (conventional brachytherapy). All treatment plans resulted in V100 > 98% and D90 > 45 Gy for the target prostate region. For the urethra region, the D10Ur, D90Ur and V150Ur and for the rectum region the V100cc, D2cc, D90Re and V90Re all are reduced significantly when mixed sources brachytherapy is used employing directional sources. The simulations demonstrated that the use of directional sources in the low dose-rate (LDR) brachytherapy of the prostate clearly benefits in sparing the urethra and the rectum sensitive structures from overdose. The time taken for a conventional treatment plan is less than three seconds, while the time taken for a mixed source treatment plan is less than nine seconds, as tested on an Intel Core2 Duo 2.2 GHz processor with 1GB RAM. The new 3D GH algorithm is successful in generating a feasible LDR brachytherapy treatment planning solution with an extra degree of freedom, i.e. directionality in very little time.

An inverse problem strategy based on forward model evaluations: Gradient-based optimization without adjoint solves

DOE Office of Scientific and Technical Information (OSTI.GOV)

Aguilo Valentin, Miguel Alejandro

2016-07-01

This study presents a new nonlinear programming formulation for the solution of inverse problems. First, a general inverse problem formulation based on the compliance error functional is presented. The proposed error functional enables the computation of the Lagrange multipliers, and thus the first order derivative information, at the expense of just one model evaluation. Therefore, the calculation of the Lagrange multipliers does not require the solution of the computationally intensive adjoint problem. This leads to significant speedups for large-scale, gradient-based inverse problems.

Practical Aerodynamic Design Optimization Based on the Navier-Stokes Equations and a Discrete Adjoint Method

NASA Technical Reports Server (NTRS)

Grossman, Bernard

1999-01-01

Compressible and incompressible versions of a three-dimensional unstructured mesh Reynolds-averaged Navier-Stokes flow solver have been differentiated and resulting derivatives have been verified by comparisons with finite differences and a complex-variable approach. In this implementation, the turbulence model is fully coupled with the flow equations in order to achieve this consistency. The accuracy demonstrated in the current work represents the first time that such an approach has been successfully implemented. The accuracy of a number of simplifying approximations to the linearizations of the residual have been examined. A first-order approximation to the dependent variables in both the adjoint and design equations has been investigated. The effects of a "frozen" eddy viscosity and the ramifications of neglecting some mesh sensitivity terms were also examined. It has been found that none of the approximations yielded derivatives of acceptable accuracy and were often of incorrect sign. However, numerical experiments indicate that an incomplete convergence of the adjoint system often yield sufficiently accurate derivatives, thereby significantly lowering the time required for computing sensitivity information. The convergence rate of the adjoint solver relative to the flow solver has been examined. Inviscid adjoint solutions typically require one to four times the cost of a flow solution, while for turbulent adjoint computations, this ratio can reach as high as eight to ten. Numerical experiments have shown that the adjoint solver can stall before converging the solution to machine accuracy, particularly for viscous cases. A possible remedy for this phenomenon would be to include the complete higher-order linearization in the preconditioning step, or to employ a simple form of mesh sequencing to obtain better approximations to the solution through the use of coarser meshes. An efficient surface parameterization based on a free-form deformation technique has been utilized and the resulting codes have been integrated with an optimization package. Lastly, sample optimizations have been shown for inviscid and turbulent flow over an ONERA M6 wing. Drag reductions have been demonstrated by reducing shock strengths across the span of the wing. In order for large scale optimization to become routine, the benefits of parallel architectures should be exploited. Although the flow solver has been parallelized using compiler directives. The parallel efficiency is under 50 percent. Clearly, parallel versions of the codes will have an immediate impact on the ability to design realistic configurations on fine meshes, and this effort is currently underway.

A Computational Study of Shear Layer Receptivity

NASA Astrophysics Data System (ADS)

Barone, Matthew; Lele, Sanjiva

2002-11-01

The receptivity of two-dimensional, compressible shear layers to local and external excitation sources is examined using a computational approach. The family of base flows considered consists of a laminar supersonic stream separated from nearly quiescent fluid by a thin, rigid splitter plate with a rounded trailing edge. The linearized Euler and linearized Navier-Stokes equations are solved numerically in the frequency domain. The flow solver is based on a high order finite difference scheme, coupled with an overset mesh technique developed for computational aeroacoustics applications. Solutions are obtained for acoustic plane wave forcing near the most unstable shear layer frequency, and are compared to the existing low frequency theory. An adjoint formulation to the present problem is developed, and adjoint equation calculations are performed using the same numerical methods as for the regular equation sets. Solutions to the adjoint equations are used to shed light on the mechanisms which control the receptivity of finite-width compressible shear layers.

Constrained Multipoint Aerodynamic Shape Optimization Using an Adjoint Formulation and Parallel Computers

NASA Technical Reports Server (NTRS)

Reuther, James; Jameson, Antony; Alonso, Juan Jose; Rimlinger, Mark J.; Saunders, David

1997-01-01

An aerodynamic shape optimization method that treats the design of complex aircraft configurations subject to high fidelity computational fluid dynamics (CFD), geometric constraints and multiple design points is described. The design process will be greatly accelerated through the use of both control theory and distributed memory computer architectures. Control theory is employed to derive the adjoint differential equations whose solution allows for the evaluation of design gradient information at a fraction of the computational cost required by previous design methods. The resulting problem is implemented on parallel distributed memory architectures using a domain decomposition approach, an optimized communication schedule, and the MPI (Message Passing Interface) standard for portability and efficiency. The final result achieves very rapid aerodynamic design based on a higher order CFD method. In order to facilitate the integration of these high fidelity CFD approaches into future multi-disciplinary optimization (NW) applications, new methods must be developed which are capable of simultaneously addressing complex geometries, multiple objective functions, and geometric design constraints. In our earlier studies, we coupled the adjoint based design formulations with unconstrained optimization algorithms and showed that the approach was effective for the aerodynamic design of airfoils, wings, wing-bodies, and complex aircraft configurations. In many of the results presented in these earlier works, geometric constraints were satisfied either by a projection into feasible space or by posing the design space parameterization such that it automatically satisfied constraints. Furthermore, with the exception of reference 9 where the second author initially explored the use of multipoint design in conjunction with adjoint formulations, our earlier works have focused on single point design efforts. Here we demonstrate that the same methodology may be extended to treat complete configuration designs subject to multiple design points and geometric constraints. Examples are presented for both transonic and supersonic configurations ranging from wing alone designs to complex configuration designs involving wing, fuselage, nacelles and pylons.

Development of CO2 inversion system based on the adjoint of the global coupled transport model

NASA Astrophysics Data System (ADS)

Belikov, Dmitry; Maksyutov, Shamil; Chevallier, Frederic; Kaminski, Thomas; Ganshin, Alexander; Blessing, Simon

2014-05-01

We present the development of an inverse modeling system employing an adjoint of the global coupled transport model consisting of the National Institute for Environmental Studies (NIES) Eulerian transport model (TM) and the Lagrangian plume diffusion model (LPDM) FLEXPART. NIES TM is a three-dimensional atmospheric transport model, which solves the continuity equation for a number of atmospheric tracers on a grid spanning the entire globe. Spatial discretization is based on a reduced latitude-longitude grid and a hybrid sigma-isentropic coordinate in the vertical. NIES TM uses a horizontal resolution of 2.5°×2.5°. However, to resolve synoptic-scale tracer distributions and to have the ability to optimize fluxes at resolutions of 0.5° and higher we coupled NIES TM with the Lagrangian model FLEXPART. The Lagrangian component of the forward and adjoint models uses precalculated responses of the observed concentration to the surface fluxes and 3-D concentrations field simulated with the FLEXPART model. NIES TM and FLEXPART are driven by JRA-25/JCDAS reanalysis dataset. Construction of the adjoint of the Lagrangian part is less complicated, as LPDMs calculate the sensitivity of measurements to the surrounding emissions field by tracking a large number of "particles" backwards in time. Developing of the adjoint to Eulerian part was performed with automatic differentiation tool the Transformation of Algorithms in Fortran (TAF) software (http://www.FastOpt.com). This method leads to the discrete adjoint of NIES TM. The main advantage of the discrete adjoint is that the resulting gradients of the numerical cost function are exact, even for nonlinear algorithms. The overall advantages of our method are that: 1. No code modification of Lagrangian model is required, making it applicable to combination of global NIES TM and any Lagrangian model; 2. Once run, the Lagrangian output can be applied to any chemically neutral gas; 3. High-resolution results can be obtained over limited regions close to the monitoring sites (using the LPDM part), and at coarse resolution for the rest of the globe (using the Eulerian part), minimizing aggregation errors and computation cost. The adjoint of the coupled high-resolution Eulerian-Lagrangian model will be incorporated into the PYVAR CO2 variational inverse system (Chevallier et al., 2005). Chevallier, F., Fisher, M., Peylin, P., Serrar, S., Bousquet, P., Bréon, F.-M., Chédin, A., and Ciais, P.: Inferring CO2 sources and sinks from satellite observations: method and application to TOVS data, J. Geophys. Res., 110, D24309, doi:10.1029/2005JD006390, 2005.

Variational approach to stability boundary for the Taylor-Goldstein equation

NASA Astrophysics Data System (ADS)

Hirota, Makoto; Morrison, Philip J.

2015-11-01

Linear stability of inviscid stratified shear flow is studied by developing an efficient method for finding neutral (i.e., marginally stable) solutions of the Taylor-Goldstein equation. The classical Miles-Howard criterion states that stratified shear flow is stable if the local Richardson number JR is greater than 1/4 everywhere. In this work, the case of JR > 0 everywhere is considered by assuming strictly monotonic and smooth profiles of the ambient shear flow and density. It is shown that singular neutral modes that are embedded in the continuous spectrum can be found by solving one-parameter families of self-adjoint eigenvalue problems. The unstable ranges of wavenumber are searched for accurately and efficiently by adopting this method in a numerical algorithm. Because the problems are self-adjoint, the variational method can be applied to ascertain the existence of singular neutral modes. For certain shear flow and density profiles, linear stability can be proven by showing the non-existence of a singular neutral mode. New sufficient conditions, extensions of the Rayleigh-Fjortoft stability criterion for unstratified shear flows, are derived in this manner. This work was supported by JSPS Strategic Young Researcher Overseas Visits Program for Accelerating Brain Circulation # 55053270.

Adjoint of the global Eulerian-Lagrangian coupled atmospheric transport model (A-GELCA v1.0): development and validation

NASA Astrophysics Data System (ADS)

Belikov, Dmitry A.; Maksyutov, Shamil; Yaremchuk, Alexey; Ganshin, Alexander; Kaminski, Thomas; Blessing, Simon; Sasakawa, Motoki; Gomez-Pelaez, Angel J.; Starchenko, Alexander

2016-02-01

We present the development of the Adjoint of the Global Eulerian-Lagrangian Coupled Atmospheric (A-GELCA) model that consists of the National Institute for Environmental Studies (NIES) model as an Eulerian three-dimensional transport model (TM), and FLEXPART (FLEXible PARTicle dispersion model) as the Lagrangian Particle Dispersion Model (LPDM). The forward tangent linear and adjoint components of the Eulerian model were constructed directly from the original NIES TM code using an automatic differentiation tool known as TAF (Transformation of Algorithms in Fortran; http://www.FastOpt.com, with additional manual pre- and post-processing aimed at improving transparency and clarity of the code and optimizing the performance of the computing, including MPI (Message Passing Interface). The Lagrangian component did not require any code modification, as LPDMs are self-adjoint and track a significant number of particles backward in time in order to calculate the sensitivity of the observations to the neighboring emission areas. The constructed Eulerian adjoint was coupled with the Lagrangian component at a time boundary in the global domain. The simulations presented in this work were performed using the A-GELCA model in forward and adjoint modes. The forward simulation shows that the coupled model improves reproduction of the seasonal cycle and short-term variability of CO2. Mean bias and standard deviation for five of the six Siberian sites considered decrease roughly by 1 ppm when using the coupled model. The adjoint of the Eulerian model was shown, through several numerical tests, to be very accurate (within machine epsilon with mismatch around to ±6 e-14) compared to direct forward sensitivity calculations. The developed adjoint of the coupled model combines the flux conservation and stability of an Eulerian discrete adjoint formulation with the flexibility, accuracy, and high resolution of a Lagrangian backward trajectory formulation. A-GELCA will be incorporated into a variational inversion system designed to optimize surface fluxes of greenhouse gases.

A variational data assimilation system for the range dependent acoustic model using the representer method: Theoretical derivations.

PubMed

Ngodock, Hans; Carrier, Matthew; Fabre, Josette; Zingarelli, Robert; Souopgui, Innocent

2017-07-01

This study presents the theoretical framework for variational data assimilation of acoustic pressure observations into an acoustic propagation model, namely, the range dependent acoustic model (RAM). RAM uses the split-step Padé algorithm to solve the parabolic equation. The assimilation consists of minimizing a weighted least squares cost function that includes discrepancies between the model solution and the observations. The minimization process, which uses the principle of variations, requires the derivation of the tangent linear and adjoint models of the RAM. The mathematical derivations are presented here, and, for the sake of brevity, a companion study presents the numerical implementation and results from the assimilation simulated acoustic pressure observations.

Airfoil optimization for unsteady flows with application to high-lift noise reduction

NASA Astrophysics Data System (ADS)

Rumpfkeil, Markus Peer

The use of steady-state aerodynamic optimization methods in the computational fluid dynamic (CFD) community is fairly well established. In particular, the use of adjoint methods has proven to be very beneficial because their cost is independent of the number of design variables. The application of numerical optimization to airframe-generated noise, however, has not received as much attention, but with the significant quieting of modern engines, airframe noise now competes with engine noise. Optimal control techniques for unsteady flows are needed in order to be able to reduce airframe-generated noise. In this thesis, a general framework is formulated to calculate the gradient of a cost function in a nonlinear unsteady flow environment via the discrete adjoint method. The unsteady optimization algorithm developed in this work utilizes a Newton-Krylov approach since the gradient-based optimizer uses the quasi-Newton method BFGS, Newton's method is applied to the nonlinear flow problem, GMRES is used to solve the resulting linear problem inexactly, and last but not least the linear adjoint problem is solved using Bi-CGSTAB. The flow is governed by the unsteady two-dimensional compressible Navier-Stokes equations in conjunction with a one-equation turbulence model, which are discretized using structured grids and a finite difference approach. The effectiveness of the unsteady optimization algorithm is demonstrated by applying it to several problems of interest including shocktubes, pulses in converging-diverging nozzles, rotating cylinders, transonic buffeting, and an unsteady trailing-edge flow. In order to address radiated far-field noise, an acoustic wave propagation program based on the Ffowcs Williams and Hawkings (FW-H) formulation is implemented and validated. The general framework is then used to derive the adjoint equations for a novel hybrid URANS/FW-H optimization algorithm in order to be able to optimize the shape of airfoils based on their calculated far-field pressure fluctuations. Validation and application results for this novel hybrid URANS/FW-H optimization algorithm show that it is possible to optimize the shape of an airfoil in an unsteady flow environment to minimize its radiated far-field noise while maintaining good aerodynamic performance.

Quasimodular instanton partition function and the elliptic solution of Korteweg-de Vries equations

NASA Astrophysics Data System (ADS)

He, Wei

2015-02-01

The Gauge/Bethe correspondence relates Omega-deformed N = 2 supersymmetric gauge theories to some quantum integrable models, in simple cases the integrable models can be treated as solvable quantum mechanics models. For SU(2) gauge theory with an adjoint matter, or with 4 fundamental matters, the potential of corresponding quantum model is the elliptic function. If the mass of matter takes special value then the potential is an elliptic solution of KdV hierarchy. We show that the deformed prepotential of gauge theory can be obtained from the average densities of conserved charges of the classical KdV solution, the UV gauge coupling dependence is assembled into the Eisenstein series. The gauge theory with adjoint mass is taken as the example.

Self-consistent adjoint analysis for topology optimization of electromagnetic waves

NASA Astrophysics Data System (ADS)

Deng, Yongbo; Korvink, Jan G.

2018-05-01

In topology optimization of electromagnetic waves, the Gâteaux differentiability of the conjugate operator to the complex field variable results in the complexity of the adjoint sensitivity, which evolves the original real-valued design variable to be complex during the iterative solution procedure. Therefore, the self-inconsistency of the adjoint sensitivity is presented. To enforce the self-consistency, the real part operator has been used to extract the real part of the sensitivity to keep the real-value property of the design variable. However, this enforced self-consistency can cause the problem that the derived structural topology has unreasonable dependence on the phase of the incident wave. To solve this problem, this article focuses on the self-consistent adjoint analysis of the topology optimization problems for electromagnetic waves. This self-consistent adjoint analysis is implemented by splitting the complex variables of the wave equations into the corresponding real parts and imaginary parts, sequentially substituting the split complex variables into the wave equations with deriving the coupled equations equivalent to the original wave equations, where the infinite free space is truncated by the perfectly matched layers. Then, the topology optimization problems of electromagnetic waves are transformed into the forms defined on real functional spaces instead of complex functional spaces; the adjoint analysis of the topology optimization problems is implemented on real functional spaces with removing the variational of the conjugate operator; the self-consistent adjoint sensitivity is derived, and the phase-dependence problem is avoided for the derived structural topology. Several numerical examples are implemented to demonstrate the robustness of the derived self-consistent adjoint analysis.

Optical properties reconstruction using the adjoint method based on the radiative transfer equation

NASA Astrophysics Data System (ADS)

Addoum, Ahmad; Farges, Olivier; Asllanaj, Fatmir

2018-01-01

An efficient algorithm is proposed to reconstruct the spatial distribution of optical properties in heterogeneous media like biological tissues. The light transport through such media is accurately described by the radiative transfer equation in the frequency-domain. The adjoint method is used to efficiently compute the objective function gradient with respect to optical parameters. Numerical tests show that the algorithm is accurate and robust to retrieve simultaneously the absorption μa and scattering μs coefficients for lowly and highly absorbing medium. Moreover, the simultaneous reconstruction of μs and the anisotropy factor g of the Henyey-Greenstein phase function is achieved with a reasonable accuracy. The main novelty in this work is the reconstruction of g which might open the possibility to image this parameter in tissues as an additional contrast agent in optical tomography.

Sensitivity Analysis for Steady State Groundwater Flow Using Adjoint Operators

NASA Astrophysics Data System (ADS)

Sykes, J. F.; Wilson, J. L.; Andrews, R. W.

1985-03-01

Adjoint sensitivity theory is currently being considered as a potential method for calculating the sensitivity of nuclear waste repository performance measures to the parameters of the system. For groundwater flow systems, performance measures of interest include piezometric heads in the vicinity of a waste site, velocities or travel time in aquifers, and mass discharge to biosphere points. The parameters include recharge-discharge rates, prescribed boundary heads or fluxes, formation thicknesses, and hydraulic conductivities. The derivative of a performance measure with respect to the system parameters is usually taken as a measure of sensitivity. To calculate sensitivities, adjoint sensitivity equations are formulated from the equations describing the primary problem. The solution of the primary problem and the adjoint sensitivity problem enables the determination of all of the required derivatives and hence related sensitivity coefficients. In this study, adjoint sensitivity theory is developed for equations of two-dimensional steady state flow in a confined aquifer. Both the primary flow equation and the adjoint sensitivity equation are solved using the Galerkin finite element method. The developed computer code is used to investigate the regional flow parameters of the Leadville Formation of the Paradox Basin in Utah. The results illustrate the sensitivity of calculated local heads to the boundary conditions. Alternatively, local velocity related performance measures are more sensitive to hydraulic conductivities.

Optimization of wind plant layouts using an adjoint approach

DOE PAGES

King, Ryan N.; Dykes, Katherine; Graf, Peter; ...

2017-03-10

Using adjoint optimization and three-dimensional steady-state Reynolds-averaged Navier–Stokes (RANS) simulations, we present a new gradient-based approach for optimally siting wind turbines within utility-scale wind plants. By solving the adjoint equations of the flow model, the gradients needed for optimization are found at a cost that is independent of the number of control variables, thereby permitting optimization of large wind plants with many turbine locations. Moreover, compared to the common approach of superimposing prescribed wake deficits onto linearized flow models, the computational efficiency of the adjoint approach allows the use of higher-fidelity RANS flow models which can capture nonlinear turbulent flowmore » physics within a wind plant. The steady-state RANS flow model is implemented in the Python finite-element package FEniCS and the derivation and solution of the discrete adjoint equations are automated within the dolfin-adjoint framework. Gradient-based optimization of wind turbine locations is demonstrated for idealized test cases that reveal new optimization heuristics such as rotational symmetry, local speedups, and nonlinear wake curvature effects. Layout optimization is also demonstrated on more complex wind rose shapes, including a full annual energy production (AEP) layout optimization over 36 inflow directions and 5 wind speed bins.« less

Optimization of wind plant layouts using an adjoint approach

DOE Office of Scientific and Technical Information (OSTI.GOV)

King, Ryan N.; Dykes, Katherine; Graf, Peter

Using adjoint optimization and three-dimensional steady-state Reynolds-averaged Navier–Stokes (RANS) simulations, we present a new gradient-based approach for optimally siting wind turbines within utility-scale wind plants. By solving the adjoint equations of the flow model, the gradients needed for optimization are found at a cost that is independent of the number of control variables, thereby permitting optimization of large wind plants with many turbine locations. Moreover, compared to the common approach of superimposing prescribed wake deficits onto linearized flow models, the computational efficiency of the adjoint approach allows the use of higher-fidelity RANS flow models which can capture nonlinear turbulent flowmore » physics within a wind plant. The steady-state RANS flow model is implemented in the Python finite-element package FEniCS and the derivation and solution of the discrete adjoint equations are automated within the dolfin-adjoint framework. Gradient-based optimization of wind turbine locations is demonstrated for idealized test cases that reveal new optimization heuristics such as rotational symmetry, local speedups, and nonlinear wake curvature effects. Layout optimization is also demonstrated on more complex wind rose shapes, including a full annual energy production (AEP) layout optimization over 36 inflow directions and 5 wind speed bins.« less

Doppler Temperature Coefficient Calculations Using Adjoint-Weighted Tallies and Continuous Energy Cross Sections in MCNP6

NASA Astrophysics Data System (ADS)

Gonzales, Matthew Alejandro

The calculation of the thermal neutron Doppler temperature reactivity feedback co-efficient, a key parameter in the design and safe operation of advanced reactors, using first order perturbation theory in continuous energy Monte Carlo codes is challenging as the continuous energy adjoint flux is not readily available. Traditional approaches of obtaining the adjoint flux attempt to invert the random walk process as well as require data corresponding to all temperatures and their respective temperature derivatives within the system in order to accurately calculate the Doppler temperature feedback. A new method has been developed using adjoint-weighted tallies and On-The-Fly (OTF) generated continuous energy cross sections within the Monte Carlo N-Particle (MCNP6) transport code. The adjoint-weighted tallies are generated during the continuous energy k-eigenvalue Monte Carlo calculation. The weighting is based upon the iterated fission probability interpretation of the adjoint flux, which is the steady state population in a critical nuclear reactor caused by a neutron introduced at that point in phase space. The adjoint-weighted tallies are produced in a forward calculation and do not require an inversion of the random walk. The OTF cross section database uses a high order functional expansion between points on a user-defined energy-temperature mesh in which the coefficients with respect to a polynomial fitting in temperature are stored. The coefficients of the fits are generated before run- time and called upon during the simulation to produce cross sections at any given energy and temperature. The polynomial form of the OTF cross sections allows the possibility of obtaining temperature derivatives of the cross sections on-the-fly. The use of Monte Carlo sampling of adjoint-weighted tallies and the capability of computing derivatives of continuous energy cross sections with respect to temperature are used to calculate the Doppler temperature coefficient in a research version of MCNP6. Temperature feedback results from the cross sections themselves, changes in the probability density functions, as well as changes in the density of the materials. The focus of this work is specific to the Doppler temperature feedback which result from Doppler broadening of cross sections as well as changes in the probability density function within the scattering kernel. This method is compared against published results using Mosteller's numerical benchmark to show accurate evaluations of the Doppler temperature coefficient, fuel assembly calculations, and a benchmark solution based on the heavy gas model for free-gas elastic scattering. An infinite medium benchmark for neutron free gas elastic scattering for large scattering ratios and constant absorption cross section has been developed using the heavy gas model. An exact closed form solution for the neutron energy spectrum is obtained in terms of the confluent hypergeometric function and compared against spectra for the free gas scattering model in MCNP6. Results show a quick increase in convergence of the analytic energy spectrum to the MCNP6 code with increasing target size, showing absolute relative differences of less than 5% for neutrons scattering with carbon. The analytic solution has been generalized to accommodate piecewise constant in energy absorption cross section to produce temperature feedback. Results reinforce the constraints in which heavy gas theory may be applied resulting in a significant target size to accommodate increasing cross section structure. The energy dependent piecewise constant cross section heavy gas model was used to produce a benchmark calculation of the Doppler temperature coefficient to show accurate calculations when using the adjoint-weighted method. Results show the Doppler temperature coefficient using adjoint weighting and cross section derivatives accurately obtains the correct solution within statistics as well as reduce computer runtimes by a factor of 50.

Electrical Resistivity Tomography using a finite element based BFGS algorithm with algebraic multigrid preconditioning

NASA Astrophysics Data System (ADS)

Codd, A. L.; Gross, L.

2018-03-01

We present a new inversion method for Electrical Resistivity Tomography which, in contrast to established approaches, minimizes the cost function prior to finite element discretization for the unknown electric conductivity and electric potential. Minimization is performed with the Broyden-Fletcher-Goldfarb-Shanno method (BFGS) in an appropriate function space. BFGS is self-preconditioning and avoids construction of the dense Hessian which is the major obstacle to solving large 3-D problems using parallel computers. In addition to the forward problem predicting the measurement from the injected current, the so-called adjoint problem also needs to be solved. For this problem a virtual current is injected through the measurement electrodes and an adjoint electric potential is obtained. The magnitude of the injected virtual current is equal to the misfit at the measurement electrodes. This new approach has the advantage that the solution process of the optimization problem remains independent to the meshes used for discretization and allows for mesh adaptation during inversion. Computation time is reduced by using superposition of pole loads for the forward and adjoint problems. A smoothed aggregation algebraic multigrid (AMG) preconditioned conjugate gradient is applied to construct the potentials for a given electric conductivity estimate and for constructing a first level BFGS preconditioner. Through the additional reuse of AMG operators and coarse grid solvers inversion time for large 3-D problems can be reduced further. We apply our new inversion method to synthetic survey data created by the resistivity profile representing the characteristics of subsurface fluid injection. We further test it on data obtained from a 2-D surface electrode survey on Heron Island, a small tropical island off the east coast of central Queensland, Australia.

Novel numerical techniques for magma dynamics

NASA Astrophysics Data System (ADS)

Rhebergen, S.; Katz, R. F.; Wathen, A.; Alisic, L.; Rudge, J. F.; Wells, G.

2013-12-01

We discuss the development of finite element techniques and solvers for magma dynamics computations. These are implemented within the FEniCS framework. This approach allows for user-friendly, expressive, high-level code development, but also provides access to powerful, scalable numerical solvers and a large family of finite element discretisations. With the recent addition of dolfin-adjoint, FeniCS supports automated adjoint and tangent-linear models, enabling the rapid development of Generalised Stability Analysis. The ability to easily scale codes to three dimensions with large meshes, and/or to apply intricate adjoint calculations means that efficiency of the numerical algorithms is vital. We therefore describe our development and analysis of preconditioners designed specifically for finite element discretizations of equations governing magma dynamics. The preconditioners are based on Elman-Silvester-Wathen methods for the Stokes equation, and we extend these to flows with compaction. Our simulations are validated by comparison of results with laboratory experiments on partially molten aggregates.

An improved algorithm for balanced POD through an analytic treatment of impulse response tails

NASA Astrophysics Data System (ADS)

Tu, Jonathan H.; Rowley, Clarence W.

2012-06-01

We present a modification of the balanced proper orthogonal decomposition (balanced POD) algorithm for systems with simple impulse response tails. In this new method, we use dynamic mode decomposition (DMD) to estimate the slowly decaying eigenvectors that dominate the long-time behavior of the direct and adjoint impulse responses. This is done using a new, low-memory variant of the DMD algorithm, appropriate for large datasets. We then formulate analytic expressions for the contribution of these eigenvectors to the controllability and observability Gramians. These contributions can be accounted for in the balanced POD algorithm by simply appending the impulse response snapshot matrices (direct and adjoint, respectively) with particular linear combinations of the slow eigenvectors. Aside from these additions to the snapshot matrices, the algorithm remains unchanged. By treating the tails analytically, we eliminate the need to run long impulse response simulations, lowering storage requirements and speeding up ensuing computations. To demonstrate its effectiveness, we apply this method to two examples: the linearized, complex Ginzburg-Landau equation, and the two-dimensional fluid flow past a cylinder. As expected, reduced-order models computed using an analytic tail match or exceed the accuracy of those computed using the standard balanced POD procedure, at a fraction of the cost.

Greenland Regional and Ice Sheet-wide Geometry Sensitivity to Boundary and Initial conditions

NASA Astrophysics Data System (ADS)

Logan, L. C.; Narayanan, S. H. K.; Greve, R.; Heimbach, P.

2017-12-01

Ice sheet and glacier model outputs require inputs from uncertainly known initial and boundary conditions, and other parameters. Conservation and constitutive equations formalize the relationship between model inputs and outputs, and the sensitivity of model-derived quantities of interest (e.g., ice sheet volume above floatation) to model variables can be obtained via the adjoint model of an ice sheet. We show how one particular ice sheet model, SICOPOLIS (SImulation COde for POLythermal Ice Sheets), depends on these inputs through comprehensive adjoint-based sensitivity analyses. SICOPOLIS discretizes the shallow-ice and shallow-shelf approximations for ice flow, and is well-suited for paleo-studies of Greenland and Antarctica, among other computational domains. The adjoint model of SICOPOLIS was developed via algorithmic differentiation, facilitated by the source transformation tool OpenAD (developed at Argonne National Lab). While model sensitivity to various inputs can be computed by costly methods involving input perturbation simulations, the time-dependent adjoint model of SICOPOLIS delivers model sensitivities to initial and boundary conditions throughout time at lower cost. Here, we explore both the sensitivities of the Greenland Ice Sheet's entire and regional volumes to: initial ice thickness, precipitation, basal sliding, and geothermal flux over the Holocene epoch. Sensitivity studies such as described here are now accessible to the modeling community, based on the latest version of SICOPOLIS that has been adapted for OpenAD to generate correct and efficient adjoint code.

Fluorescence molecular imaging based on the adjoint radiative transport equation

NASA Astrophysics Data System (ADS)

Asllanaj, Fatmir; Addoum, Ahmad; Rodolphe Roche, Jean

2018-07-01

A new reconstruction algorithm for fluorescence diffuse optical tomography of biological tissues is proposed. The radiative transport equation in the frequency domain is used to model light propagation. The adjoint method studied in this work provides an efficient way for solving the inverse problem. The methodology is applied to a 2D tissue-like phantom subjected to a collimated laser beam. Indocyanine Green is used as fluorophore. Reconstructed images of the spatial fluorophore absorption distribution is assessed taking into account the residual fluorescence in the medium. We show that illuminating the tissue surface from a collimated centered direction near the inclusion gaves a better reconstruction quality. Two closely positioned inclusions can be accurately localized. Additionally, their fluorophore absorption coefficients can be quantified. However, the algorithm failes to reconstruct smaller or deeper inclusions. This is due to light attenuation in the medium. Reconstructions with noisy data are also achieved with a reasonable accuracy.

On the symmetry of the boundary conditions of the volume potential

NASA Astrophysics Data System (ADS)

Kal'menov, Tynysbek Sh.; Arepova, Gaukhar; Suragan, Durvudkhan

2017-09-01

It is well known that the volume potential determines the mass or the charge distributed over the domain with density f. The volume potential is extensively used in function theory and embedding theorems. It is also well known that the volume potential gives a solution to an inhomogeneous equation. And it generates a linear self-adjoint operator. It is known that self-adjoint differential operators are generated by boundary conditions. In our previous papers for an arbitrary domain a boundary condition on the volume potential is given. In the past, it was not possible to prove the self-adjointness of these obtained boundary conditions. In the present paper, we prove the symmetry of boundary condition for the volume potential.

Estimation of River Bathymetry from ATI-SAR Data

NASA Astrophysics Data System (ADS)

Almeida, T. G.; Walker, D. T.; Farquharson, G.

2013-12-01

A framework for estimation of river bathymetry from surface velocity observation data is presented using variational inverse modeling applied to the 2D depth-averaged, shallow-water equations (SWEs) including bottom friction. We start with with a cost function defined by the error between observed and estimated surface velocities, and introduce the SWEs as a constraint on the velocity field. The constrained minimization problem is converted to an unconstrained minimization through the use of Lagrange multipliers, and an adjoint SWE model is developed. The adjoint model solution is used to calculate the gradient of the cost function with respect to river bathymetry. The gradient is used in a descent algorithm to determine the bathymetry that yields a surface velocity field that is a best-fit to the observational data. In applying the algorithm, the 2D depth-averaged flow is computed assuming a known, constant discharge rate and a known, uniform bottom-friction coefficient; a correlation relating surface velocity and depth-averaged velocity is also used. Observation data was collected using a dual beam squinted along-track-interferometric, synthetic-aperture radar (ATI-SAR) system, which provides two independent components of the surface velocity, oriented roughly 30 degrees fore and aft of broadside, offering high-resolution bank-to-bank velocity vector coverage of the river. Data and bathymetry estimation results are presented for two rivers, the Snohomish River near Everett, WA and the upper Sacramento River, north of Colusa, CA. The algorithm results are compared to available measured bathymetry data, with favorable results. General trends show that the water-depth estimates are most accurate in shallow regions, and performance is sensitive to the accuracy of the specified discharge rate and bottom friction coefficient. The results also indicate that, for a given reach, the estimated water depth reaches a maximum that is smaller than the true depth; this apparent maximum depth scales with the true river depth and discharge rate, so that the deepest parts of the river show the largest bathymetry errors.

Variational methods for direct/inverse problems of atmospheric dynamics and chemistry

NASA Astrophysics Data System (ADS)

Penenko, Vladimir; Penenko, Alexey; Tsvetova, Elena

2013-04-01

We present a variational approach for solving direct and inverse problems of atmospheric hydrodynamics and chemistry. It is important that the accurate matching of numerical schemes has to be provided in the chain of objects: direct/adjoint problems - sensitivity relations - inverse problems, including assimilation of all available measurement data. To solve the problems we have developed a new enhanced set of cost-effective algorithms. The matched description of the multi-scale processes is provided by a specific choice of the variational principle functionals for the whole set of integrated models. Then all functionals of variational principle are approximated in space and time by splitting and decomposition methods. Such approach allows us to separately consider, for example, the space-time problems of atmospheric chemistry in the frames of decomposition schemes for the integral identity sum analogs of the variational principle at each time step and in each of 3D finite-volumes. To enhance the realization efficiency, the set of chemical reactions is divided on the subsets related to the operators of production and destruction. Then the idea of the Euler's integrating factors is applied in the frames of the local adjoint problem technique [1]-[3]. The analytical solutions of such adjoint problems play the role of integrating factors for differential equations describing atmospheric chemistry. With their help, the system of differential equations is transformed to the equivalent system of integral equations. As a result we avoid the construction and inversion of preconditioning operators containing the Jacobi matrixes which arise in traditional implicit schemes for ODE solution. This is the main advantage of our schemes. At the same time step but on the different stages of the "global" splitting scheme, the system of atmospheric dynamic equations is solved. For convection - diffusion equations for all state functions in the integrated models we have developed the monotone and stable discrete-analytical numerical schemes [1]-[3] conserving the positivity of the chemical substance concentrations and possessing the properties of energy and mass balance that are postulated in the general variational principle for integrated models. All algorithms for solution of transport, diffusion and transformation problems are direct (without iterations). The work is partially supported by the Programs No 4 of Presidium RAS and No 3 of Mathematical Department of RAS, by RFBR project 11-01-00187 and Integrating projects of SD RAS No 8 and 35. Our studies are in the line with the goals of COST Action ES1004. References Penenko V., Tsvetova E. Discrete-analytical methods for the implementation of variational principles in environmental applications// Journal of computational and applied mathematics, 2009, v. 226, 319-330. Penenko A.V. Discrete-analytic schemes for solving an inverse coefficient heat conduction problem in a layered medium with gradient methods// Numerical Analysis and Applications, 2012, V. 5, pp 326-341. V. Penenko, E. Tsvetova. Variational methods for constructing the monotone approximations for atmospheric chemistry models //Numerical Analysis and Applications, 2013 (in press).

Full moment tensor and source location inversion based on full waveform adjoint method

NASA Astrophysics Data System (ADS)

Morency, C.

2012-12-01

The development of high-performance computing and numerical techniques enabled global and regional tomography to reach high levels of precision, and seismic adjoint tomography has become a state-of-the-art tomographic technique. The method was successfully used for crustal tomography of Southern California (Tape et al., 2009) and Europe (Zhu et al., 2012). Here, I will focus on the determination of source parameters (full moment tensor and location) based on the same approach (Kim et al, 2011). The method relies on full wave simulations and takes advantage of the misfit between observed and synthetic seismograms. An adjoint wavefield is calculated by back-propagating the difference between observed and synthetics from the receivers to the source. The interaction between this adjoint wavefield and the regular forward wavefield helps define Frechet derivatives of the source parameters, that is, the sensitivity of the misfit with respect to the source parameters. Source parameters are then recovered by minimizing the misfit based on a conjugate gradient algorithm using the Frechet derivatives. First, I will demonstrate the method on synthetic cases before tackling events recorded at the Geysers. The velocity model used at the Geysers is based on the USGS 3D velocity model. Waveform datasets come from the Northern California Earthquake Data Center. Finally, I will discuss strategies to ultimately use this method to characterize smaller events for microseismic and induced seismicity monitoring. References: - Tape, C., Q. Liu, A. Maggi, and J. Tromp, 2009, Adjoint tomography of the Southern California crust: Science, 325, 988992. - Zhu, H., Bozdag, E., Peter, D., and Tromp, J., 2012, Structure of the European upper mantle revealed by adjoint method: Nature Geoscience, 5, 493-498. - Kim, Y., Q. Liu, and J. Tromp, 2011, Adjoint centroid-moment tensor inversions: Geophys. J. Int., 186, 264278. Prepared by LLNL under Contract DE-AC52-07NA27344.

A demonstration of adjoint methods for multi-dimensional remote sensing of the atmosphere and surface

NASA Astrophysics Data System (ADS)

Martin, William G. K.; Hasekamp, Otto P.

2018-01-01

In previous work, we derived the adjoint method as a computationally efficient path to three-dimensional (3D) retrievals of clouds and aerosols. In this paper we will demonstrate the use of adjoint methods for retrieving two-dimensional (2D) fields of cloud extinction. The demonstration uses a new 2D radiative transfer solver (FSDOM). This radiation code was augmented with adjoint methods to allow efficient derivative calculations needed to retrieve cloud and surface properties from multi-angle reflectance measurements. The code was then used in three synthetic retrieval studies. Our retrieval algorithm adjusts the cloud extinction field and surface albedo to minimize the measurement misfit function with a gradient-based, quasi-Newton approach. At each step we compute the value of the misfit function and its gradient with two calls to the solver FSDOM. First we solve the forward radiative transfer equation to compute the residual misfit with measurements, and second we solve the adjoint radiative transfer equation to compute the gradient of the misfit function with respect to all unknowns. The synthetic retrieval studies verify that adjoint methods are scalable to retrieval problems with many measurements and unknowns. We can retrieve the vertically-integrated optical depth of moderately thick clouds as a function of the horizontal coordinate. It is also possible to retrieve the vertical profile of clouds that are separated by clear regions. The vertical profile retrievals improve for smaller cloud fractions. This leads to the conclusion that cloud edges actually increase the amount of information that is available for retrieving the vertical profile of clouds. However, to exploit this information one must retrieve the horizontally heterogeneous cloud properties with a 2D (or 3D) model. This prototype shows that adjoint methods can efficiently compute the gradient of the misfit function. This work paves the way for the application of similar methods to 3D remote sensing problems.

A Direct Algorithm Maple Package of One-Dimensional Optimal System for Group Invariant Solutions

NASA Astrophysics Data System (ADS)

Zhang, Lin; Han, Zhong; Chen, Yong

2018-01-01

To construct the one-dimensional optimal system of finite dimensional Lie algebra automatically, we develop a new Maple package One Optimal System. Meanwhile, we propose a new method to calculate the adjoint transformation matrix and find all the invariants of Lie algebra in spite of Killing form checking possible constraints of each classification. Besides, a new conception called invariance set is raised. Moreover, this Maple package is proved to be more efficiency and precise than before by applying it to some classic examples. Supported by the Global Change Research Program of China under Grant No. 2015CB95390, National Natural Science Foundation of China under Grant Nos. 11675054 and 11435005, and Shanghai Collaborative Innovation Center of Trustworthy Software for Internet of Things under Grant No. ZF1213

An adjoint-based framework for maximizing mixing in binary fluids

NASA Astrophysics Data System (ADS)

Eggl, Maximilian; Schmid, Peter

2017-11-01

Mixing in the inertial, but laminar parameter regime is a common application in a wide range of industries. Enhancing the efficiency of mixing processes thus has a fundamental effect on product quality, material homogeneity and, last but not least, production costs. In this project, we address mixing efficiency in the above mentioned regime (Reynolds number Re = 1000 , Peclet number Pe = 1000) by developing and demonstrating an algorithm based on nonlinear adjoint looping that minimizes the variance of a passive scalar field which models our binary Newtonian fluids. The numerical method is based on the FLUSI code (Engels et al. 2016), a Fourier pseudo-spectral code, which we modified and augmented by scalar transport and adjoint equations. Mixing is accomplished by moving stirrers which are numerically modeled using a penalization approach. In our two-dimensional simulations we consider rotating circular and elliptic stirrers and extract optimal mixing strategies from the iterative scheme. The case of optimizing shape and rotational speed of the stirrers will be demonstrated.

Linearized inversion of multiple scattering seismic energy

NASA Astrophysics Data System (ADS)

Aldawood, Ali; Hoteit, Ibrahim; Zuberi, Mohammad

2014-05-01

Internal multiples deteriorate the quality of the migrated image obtained conventionally by imaging single scattering energy. So, imaging seismic data with the single-scattering assumption does not locate multiple bounces events in their actual subsurface positions. However, imaging internal multiples properly has the potential to enhance the migrated image because they illuminate zones in the subsurface that are poorly illuminated by single scattering energy such as nearly vertical faults. Standard migration of these multiples provides subsurface reflectivity distributions with low spatial resolution and migration artifacts due to the limited recording aperture, coarse sources and receivers sampling, and the band-limited nature of the source wavelet. The resultant image obtained by the adjoint operator is a smoothed depiction of the true subsurface reflectivity model and is heavily masked by migration artifacts and the source wavelet fingerprint that needs to be properly deconvolved. Hence, we proposed a linearized least-square inversion scheme to mitigate the effect of the migration artifacts, enhance the spatial resolution, and provide more accurate amplitude information when imaging internal multiples. The proposed algorithm uses the least-square image based on single-scattering assumption as a constraint to invert for the part of the image that is illuminated by internal scattering energy. Then, we posed the problem of imaging double-scattering energy as a least-square minimization problem that requires solving the normal equation of the following form: GTGv = GTd, (1) where G is a linearized forward modeling operator that predicts double-scattered seismic data. Also, GT is a linearized adjoint operator that image double-scattered seismic data. Gradient-based optimization algorithms solve this linear system. Hence, we used a quasi-Newton optimization technique to find the least-square minimizer. In this approach, an estimate of the Hessian matrix that contains curvature information is modified at every iteration by a low-rank update based on gradient changes at every step. At each iteration, the data residual is imaged using GT to determine the model update. Application of the linearized inversion to synthetic data to image a vertical fault plane demonstrate the effectiveness of this methodology to properly delineate the vertical fault plane and give better amplitude information than the standard migrated image using the adjoint operator that takes into account internal multiples. Thus, least-square imaging of multiple scattering enhances the spatial resolution of the events illuminated by internal scattering energy. It also deconvolves the source signature and helps remove the fingerprint of the acquisition geometry. The final image is obtained by the superposition of the least-square solution based on single scattering assumption and the least-square solution based on double scattering assumption.

Axisymmetric problem of fretting wear for a foundation with a nonuniform coating and rough punch

NASA Astrophysics Data System (ADS)

Manzhirov, A. V.; Kazakov, K. E.

2018-05-01

The axisymmetric contact problem with fretting wear for an elastic foundation with a longitudinally nonuniform (surface nonuniform) coating and a rigid punch with a rough foundation has been solved for the first time. The case of linear wear is considered. The nonuniformity of the coating and punch roughness are described by a different rapidly changing functions. This strong nonuniformity arises when coatings are deposited using modern additive manufacturing technologies. The problem is reduced the solution of an integral equation with two different integral operators: a compact self-adjoint positively defined operator with respect to the coordinate and the non-self-adjoint integral Volterra operator with respect to time. The solution is obtained in series using the projection method of the authors. The efficiency of the proposed approach for constructing a high-accuracy approximate solution to the problem (with only a few expansion terms retained) is demonstrated.

Multi-objective Optimization Strategies Using Adjoint Method and Game Theory in Aerodynamics

NASA Astrophysics Data System (ADS)

Tang, Zhili

2006-08-01

There are currently three different game strategies originated in economics: (1) Cooperative games (Pareto front), (2) Competitive games (Nash game) and (3) Hierarchical games (Stackelberg game). Each game achieves different equilibria with different performance, and their players play different roles in the games. Here, we introduced game concept into aerodynamic design, and combined it with adjoint method to solve multi-criteria aerodynamic optimization problems. The performance distinction of the equilibria of these three game strategies was investigated by numerical experiments. We computed Pareto front, Nash and Stackelberg equilibria of the same optimization problem with two conflicting and hierarchical targets under different parameterizations by using the deterministic optimization method. The numerical results show clearly that all the equilibria solutions are inferior to the Pareto front. Non-dominated Pareto front solutions are obtained, however the CPU cost to capture a set of solutions makes the Pareto front an expensive tool to the designer.

[Purifying process of gynostemma pentaphyllum saponins based on "adjoint marker" online control technology and identification of their compositions by UPLC-QTOF-MS].

PubMed

Fan, Dong-Dong; Kuang, Yan-Hui; Dong, Li-Hua; Ye, Xiao; Chen, Liang-Mian; Zhang, Dong; Ma, Zhen-Shan; Wang, Jin-Yu; Zhu, Jing-Jing; Wang, Zhi-Min; Wang, De-Qin; Li, Chu-Yuan

2017-04-01

To optimize the purification process of gynostemma pentaphyllum saponins (GPS) based on "adjoint marker" online control technology with GPS as the testing index. UPLC-QTOF-MS technology was used for qualitative analysis. "Adjoint marker" online control results showed that the end point of load sample was that the UV absorbance of effluent liquid was equal to half of that of load sample solution, and the absorbance was basically stable when the end point was stable. In UPLC-QTOF-MS qualitative analysis, 16 saponins were identified from GPS, including 13 known gynostemma saponins and 3 new saponins. This optimized method was proved to be simple, scientific, reasonable, easy for online determination, real-time record, and can be better applied to the mass production and automation of production. The results of qualitative analysis indicated that the "adjoint marker" online control technology can well retain main efficacy components of medicinal materials, and provide analysis tools for the process control and quality traceability. Copyright© by the Chinese Pharmaceutical Association.

Metabolic Flux Analysis in Isotope Labeling Experiments Using the Adjoint Approach.

PubMed

Mottelet, Stephane; Gaullier, Gil; Sadaka, Georges

2017-01-01

Comprehension of metabolic pathways is considerably enhanced by metabolic flux analysis (MFA-ILE) in isotope labeling experiments. The balance equations are given by hundreds of algebraic (stationary MFA) or ordinary differential equations (nonstationary MFA), and reducing the number of operations is therefore a crucial part of reducing the computation cost. The main bottleneck for deterministic algorithms is the computation of derivatives, particularly for nonstationary MFA. In this article, we explain how the overall identification process may be speeded up by using the adjoint approach to compute the gradient of the residual sum of squares. The proposed approach shows significant improvements in terms of complexity and computation time when it is compared with the usual (direct) approach. Numerical results are obtained for the central metabolic pathways of Escherichia coli and are validated against reference software in the stationary case. The methods and algorithms described in this paper are included in the sysmetab software package distributed under an Open Source license at http://forge.scilab.org/index.php/p/sysmetab/.

Technique for Calculating Solution Derivatives With Respect to Geometry Parameters in a CFD Code

NASA Technical Reports Server (NTRS)

Mathur, Sanjay

2011-01-01

A solution has been developed to the challenges of computation of derivatives with respect to geometry, which is not straightforward because these are not typically direct inputs to the computational fluid dynamics (CFD) solver. To overcome these issues, a procedure has been devised that can be used without having access to the mesh generator, while still being applicable to all types of meshes. The basic approach is inspired by the mesh motion algorithms used to deform the interior mesh nodes in a smooth manner when the surface nodes, for example, are in a fluid structure interaction problem. The general idea is to model the mesh edges and nodes as constituting a spring-mass system. Changes to boundary node locations are propagated to interior nodes by allowing them to assume their new equilibrium positions, for instance, one where the forces on each node are in balance. The main advantage of the technique is that it is independent of the volumetric mesh generator, and can be applied to structured, unstructured, single- and multi-block meshes. It essentially reduces the problem down to defining the surface mesh node derivatives with respect to the geometry parameters of interest. For analytical geometries, this is quite straightforward. In the more general case, one would need to be able to interrogate the underlying parametric CAD (computer aided design) model and to evaluate the derivatives either analytically, or by a finite difference technique. Because the technique is based on a partial differential equation (PDE), it is applicable not only to forward mode problems (where derivatives of all the output quantities are computed with respect to a single input), but it could also be extended to the adjoint problem, either by using an analytical adjoint of the PDE or a discrete analog.

Adjoint equations and analysis of complex systems: Application to virus infection modelling

NASA Astrophysics Data System (ADS)

Marchuk, G. I.; Shutyaev, V.; Bocharov, G.

2005-12-01

Recent development of applied mathematics is characterized by ever increasing attempts to apply the modelling and computational approaches across various areas of the life sciences. The need for a rigorous analysis of the complex system dynamics in immunology has been recognized since more than three decades ago. The aim of the present paper is to draw attention to the method of adjoint equations. The methodology enables to obtain information about physical processes and examine the sensitivity of complex dynamical systems. This provides a basis for a better understanding of the causal relationships between the immune system's performance and its parameters and helps to improve the experimental design in the solution of applied problems. We show how the adjoint equations can be used to explain the changes in hepatitis B virus infection dynamics between individual patients.

Optimal control and optimal trajectories of regional macroeconomic dynamics based on the Pontryagin maximum principle

NASA Astrophysics Data System (ADS)

Bulgakov, V. K.; Strigunov, V. V.

2009-05-01

The Pontryagin maximum principle is used to prove a theorem concerning optimal control in regional macroeconomics. A boundary value problem for optimal trajectories of the state and adjoint variables is formulated, and optimal curves are analyzed. An algorithm is proposed for solving the boundary value problem of optimal control. The performance of the algorithm is demonstrated by computing an optimal control and the corresponding optimal trajectories.

Analytical solution for the advection-dispersion transport equation in layered media

USDA-ARS?s Scientific Manuscript database

The advection-dispersion transport equation with first-order decay was solved analytically for multi-layered media using the classic integral transform technique (CITT). The solution procedure used an associated non-self-adjoint advection-diffusion eigenvalue problem that had the same form and coef...

Formulation for Simultaneous Aerodynamic Analysis and Design Optimization

NASA Technical Reports Server (NTRS)

Hou, G. W.; Taylor, A. C., III; Mani, S. V.; Newman, P. A.

1993-01-01

An efficient approach for simultaneous aerodynamic analysis and design optimization is presented. This approach does not require the performance of many flow analyses at each design optimization step, which can be an expensive procedure. Thus, this approach brings us one step closer to meeting the challenge of incorporating computational fluid dynamic codes into gradient-based optimization techniques for aerodynamic design. An adjoint-variable method is introduced to nullify the effect of the increased number of design variables in the problem formulation. The method has been successfully tested on one-dimensional nozzle flow problems, including a sample problem with a normal shock. Implementations of the above algorithm are also presented that incorporate Newton iterations to secure a high-quality flow solution at the end of the design process. Implementations with iterative flow solvers are possible and will be required for large, multidimensional flow problems.

A Scalable, Parallel Approach for Multi-Point, High-Fidelity Aerostructural Optimization of Aircraft Configurations

NASA Astrophysics Data System (ADS)

Kenway, Gaetan K. W.

This thesis presents new tools and techniques developed to address the challenging problem of high-fidelity aerostructural optimization with respect to large numbers of design variables. A new mesh-movement scheme is developed that is both computationally efficient and sufficiently robust to accommodate large geometric design changes and aerostructural deformations. A fully coupled Newton-Krylov method is presented that accelerates the convergence of aerostructural systems and provides a 20% performance improvement over the traditional nonlinear block Gauss-Seidel approach and can handle more exible structures. A coupled adjoint method is used that efficiently computes derivatives for a gradient-based optimization algorithm. The implementation uses only machine accurate derivative techniques and is verified to yield fully consistent derivatives by comparing against the complex step method. The fully-coupled large-scale coupled adjoint solution method is shown to have 30% better performance than the segregated approach. The parallel scalability of the coupled adjoint technique is demonstrated on an Euler Computational Fluid Dynamics (CFD) model with more than 80 million state variables coupled to a detailed structural finite-element model of the wing with more than 1 million degrees of freedom. Multi-point high-fidelity aerostructural optimizations of a long-range wide-body, transonic transport aircraft configuration are performed using the developed techniques. The aerostructural analysis employs Euler CFD with a 2 million cell mesh and a structural finite element model with 300 000 DOF. Two design optimization problems are solved: one where takeoff gross weight is minimized, and another where fuel burn is minimized. Each optimization uses a multi-point formulation with 5 cruise conditions and 2 maneuver conditions. The optimization problems have 476 design variables are optimal results are obtained within 36 hours of wall time using 435 processors. The TOGW minimization results in a 4.2% reduction in TOGW with a 6.6% fuel burn reduction, while the fuel burn optimization resulted in a 11.2% fuel burn reduction with no change to the takeoff gross weight.

Gradient descent learning algorithm overview: a general dynamical systems perspective.

PubMed

Baldi, P

1995-01-01

Gives a unified treatment of gradient descent learning algorithms for neural networks using a general framework of dynamical systems. This general approach organizes and simplifies all the known algorithms and results which have been originally derived for different problems (fixed point/trajectory learning), for different models (discrete/continuous), for different architectures (forward/recurrent), and using different techniques (backpropagation, variational calculus, adjoint methods, etc.). The general approach can also be applied to derive new algorithms. The author then briefly examines some of the complexity issues and limitations intrinsic to gradient descent learning. Throughout the paper, the author focuses on the problem of trajectory learning.

Optical solitons, nonlinear self-adjointness and conservation laws for the cubic nonlinear Shrödinger's equation with repulsive delta potential

NASA Astrophysics Data System (ADS)

Baleanu, Dumitru; Inc, Mustafa; Aliyu, Aliyu Isa; Yusuf, Abdullahi

2017-11-01

In this paper, the complex envelope function ansatz method is used to acquire the optical solitons to the cubic nonlinear Shrödinger's equation with repulsive delta potential (δ-NLSE). The method reveals dark and bright optical solitons. The necessary constraint conditions which guarantee the existence of the solitons are also presented. We studied the δ-NLSE by analyzing a system of partial differential equations (PDEs) obtained by decomposing the equation into real and imaginary components. We derive the Lie point symmetry generators of the system and prove that the system is nonlinearly self-adjoint with an explicit form of a differential substitution satisfying the nonlinear self-adjoint condition. Then we use these facts to establish a set of conserved vectors for the system using the general Cls theorem presented by Ibragimov. Some interesting figures for the acquired solutions are also presented.

Generalized Ince Gaussian beams

NASA Astrophysics Data System (ADS)

Bandres, Miguel A.; Gutiérrez-Vega, Julio C.

2006-08-01

In this work we present a detailed analysis of the tree families of generalized Gaussian beams, which are the generalized Hermite, Laguerre, and Ince Gaussian beams. The generalized Gaussian beams are not the solution of a Hermitian operator at an arbitrary z plane. We derived the adjoint operator and the adjoint eigenfunctions. Each family of generalized Gaussian beams forms a complete biorthonormal set with their adjoint eigenfunctions, therefore, any paraxial field can be described as a superposition of a generalized family with the appropriate weighting and phase factors. Each family of generalized Gaussian beams includes the standard and elegant corresponding families as particular cases when the parameters of the generalized families are chosen properly. The generalized Hermite Gaussian and Laguerre Gaussian beams correspond to limiting cases of the generalized Ince Gaussian beams when the ellipticity parameter of the latter tends to infinity or to zero, respectively. The expansion formulas among the three generalized families and their Fourier transforms are also presented.

Adjoint Techniques for Topology Optimization of Structures Under Damage Conditions

NASA Technical Reports Server (NTRS)

Akgun, Mehmet A.; Haftka, Raphael T.

2000-01-01

The objective of this cooperative agreement was to seek computationally efficient ways to optimize aerospace structures subject to damage tolerance criteria. Optimization was to involve sizing as well as topology optimization. The work was done in collaboration with Steve Scotti, Chauncey Wu and Joanne Walsh at the NASA Langley Research Center. Computation of constraint sensitivity is normally the most time-consuming step of an optimization procedure. The cooperative work first focused on this issue and implemented the adjoint method of sensitivity computation (Haftka and Gurdal, 1992) in an optimization code (runstream) written in Engineering Analysis Language (EAL). The method was implemented both for bar and plate elements including buckling sensitivity for the latter. Lumping of constraints was investigated as a means to reduce the computational cost. Adjoint sensitivity computation was developed and implemented for lumped stress and buckling constraints. Cost of the direct method and the adjoint method was compared for various structures with and without lumping. The results were reported in two papers (Akgun et al., 1998a and 1999). It is desirable to optimize topology of an aerospace structure subject to a large number of damage scenarios so that a damage tolerant structure is obtained. Including damage scenarios in the design procedure is critical in order to avoid large mass penalties at later stages (Haftka et al., 1983). A common method for topology optimization is that of compliance minimization (Bendsoe, 1995) which has not been used for damage tolerant design. In the present work, topology optimization is treated as a conventional problem aiming to minimize the weight subject to stress constraints. Multiple damage configurations (scenarios) are considered. Each configuration has its own structural stiffness matrix and, normally, requires factoring of the matrix and solution of the system of equations. Damage that is expected to be tolerated is local and represents a small change in the stiffness matrix compared to the baseline (undamaged) structure. The exact solution to a slightly modified set of equations can be obtained from the baseline solution economically without actually solving the modified system.. Shennan-Morrison-Woodbury (SMW) formulas are matrix update formulas that allow this (Akgun et al., 1998b). SMW formulas were therefore used here to compute adjoint displacements for sensitivity computation and structural displacements in damaged configurations.

On the Direct Assimilation of Along-track Sea Surface Height Observations into a Free-surface Ocean Model Using a Weak Constraints Four Dimensional Variational (4dvar) Method

NASA Astrophysics Data System (ADS)

Ngodock, H.; Carrier, M.; Smith, S. R.; Souopgui, I.; Martin, P.; Jacobs, G. A.

2016-02-01

The representer method is adopted for solving a weak constraints 4dvar problem for the assimilation of ocean observations including along-track SSH, using a free surface ocean model. Direct 4dvar assimilation of SSH observations along the satellite tracks requires that the adjoint model be integrated with Dirac impulses on the right hand side of the adjoint equations for the surface elevation equation. The solution of this adjoint model will inevitably include surface gravity waves, and it constitutes the forcing for the tangent linear model (TLM) according to the representer method. This yields an analysis that is contaminated by gravity waves. A method for avoiding the generation of the surface gravity waves in the analysis is proposed in this study; it consists of removing the adjoint of the free surface from the right hand side (rhs) of the free surface mode in the TLM. The information from the SSH observations will still propagate to all other variables via the adjoint of the balance relationship between the barotropic and baroclinic modes, resulting in the correction to the surface elevation. Two assimilation experiments are carried out in the Gulf of Mexico: one with adjoint forcing included on the rhs of the TLM free surface equation, and the other without. Both analyses are evaluated against the assimilated SSH observations, SSH maps from Aviso and independent surface drifters, showing that the analysis that did not include adjoint forcing in the free surface is more accurate. This study shows that when a weak constraint 4dvar approach is considered for the assimilation of along-track SSH observations using a free surface model, with the aim of correcting the mesoscale circulation, an independent model error should not be assigned to the free surface.

Monte Carlo Perturbation Theory Estimates of Sensitivities to System Dimensions

DOE PAGES

Burke, Timothy P.; Kiedrowski, Brian C.

2017-12-11

Here, Monte Carlo methods are developed using adjoint-based perturbation theory and the differential operator method to compute the sensitivities of the k-eigenvalue, linear functions of the flux (reaction rates), and bilinear functions of the forward and adjoint flux (kinetics parameters) to system dimensions for uniform expansions or contractions. The calculation of sensitivities to system dimensions requires computing scattering and fission sources at material interfaces using collisions occurring at the interface—which is a set of events with infinitesimal probability. Kernel density estimators are used to estimate the source at interfaces using collisions occurring near the interface. The methods for computing sensitivitiesmore » of linear and bilinear ratios are derived using the differential operator method and adjoint-based perturbation theory and are shown to be equivalent to methods previously developed using a collision history–based approach. The methods for determining sensitivities to system dimensions are tested on a series of fast, intermediate, and thermal critical benchmarks as well as a pressurized water reactor benchmark problem with iterated fission probability used for adjoint-weighting. The estimators are shown to agree within 5% and 3σ of reference solutions obtained using direct perturbations with central differences for the majority of test problems.« less

Consistent Adjoint Driven Importance Sampling using Space, Energy and Angle

DOE Office of Scientific and Technical Information (OSTI.GOV)

Peplow, Douglas E.; Mosher, Scott W; Evans, Thomas M

2012-08-01

For challenging radiation transport problems, hybrid methods combine the accuracy of Monte Carlo methods with the global information present in deterministic methods. One of the most successful hybrid methods is CADIS Consistent Adjoint Driven Importance Sampling. This method uses a deterministic adjoint solution to construct a biased source distribution and consistent weight windows to optimize a specific tally in a Monte Carlo calculation. The method has been implemented into transport codes using just the spatial and energy information from the deterministic adjoint and has been used in many applications to compute tallies with much higher figures-of-merit than analog calculations. CADISmore » also outperforms user-supplied importance values, which usually take long periods of user time to develop. This work extends CADIS to develop weight windows that are a function of the position, energy, and direction of the Monte Carlo particle. Two types of consistent source biasing are presented: one method that biases the source in space and energy while preserving the original directional distribution and one method that biases the source in space, energy, and direction. Seven simple example problems are presented which compare the use of the standard space/energy CADIS with the new space/energy/angle treatments.« less

2D Inviscid and Viscous Inverse Design Using Continuous Adjoint and Lax-Wendroff Formulation

NASA Astrophysics Data System (ADS)

Proctor, Camron Lisle

The continuous adjoint (CA) technique for optimization and/or inverse-design of aerodynamic components has seen nearly 30 years of documented success in academia. The benefits of using CA versus a direct sensitivity analysis are shown repeatedly in the literature. However, the use of CA in industry is relatively unheard-of. The sparseness of industry contributions to the field may be attributed to the tediousness of the derivation and/or to the difficulties in implementation due to the lack of well-documented adjoint numerical methods. The focus of this work has been to thoroughly document the techniques required to build a two-dimensional CA inverse-design tool. To this end, this work begins with a short background on computational fluid dynamics (CFD) and the use of optimization tools in conjunction with CFD tools to solve aerodynamic optimization problems. A thorough derivation of the continuous adjoint equations and the accompanying gradient calculations for inviscid and viscous constraining equations follows the introduction. Next, the numerical techniques used for solving the partial differential equations (PDEs) governing the flow equations and the adjoint equations are described. Numerical techniques for the supplementary equations are discussed briefly. Subsequently, a verification of the efficacy of the inverse design tool, for the inviscid adjoint equations as well as possible numerical implementation pitfalls are discussed. The NACA0012 airfoil is used as an initial airfoil and surface pressure distribution and the NACA16009 is used as the desired pressure and vice versa. Using a Savitsky-Golay gradient filter, convergence (defined as a cost function<1E-5) is reached in approximately 220 design iteration using 121 design variables. The inverse-design using inviscid adjoint equations results are followed by the discussion of the viscous inverse design results and techniques used to further the convergence of the optimizer. The relationship between limiting step-size and convergence in a line-search optimization is shown to slightly decrease the final cost function at significant computational cost. A gradient damping technique is presented and shown to increase the convergence rate for the optimization in viscous problems, at a negligible increase in computational cost, but is insufficient to converge the solution. Systematically including adjacent surface vertices in the perturbation of a design variable, also a surface vertex, is shown to affect the convergence capability of the viscous optimizer. Finally, a comparison of using inviscid adjoint equations, as opposed to viscous adjoint equations, on viscous flow is presented, and the inviscid adjoint paired with viscous flow is found to reduce the cost function further than the viscous adjoint for the presented problem.

Low-thrust trajectory analysis for the geosynchronous mission

NASA Technical Reports Server (NTRS)

Jasper, T. P.

1973-01-01

Methodology employed in development of a computer program designed to analyze optimal low-thrust trajectories is described, and application of the program to a Solar Electric Propulsion Stage (SEPS) geosynchronous mission is discussed. To avoid the zero inclination and eccentricity singularities which plague many small-force perturbation techniques, a special set of state variables (equinoctial) is used. Adjoint equations are derived for the minimum time problem and are also free from the singularities. Solutions to the state and adjoint equations are obtained by both orbit averaging and precision numerical integration; an evaluation of these approaches is made.

The fast neutron fluence and the activation detector activity calculations using the effective source method and the adjoint function

DOE Office of Scientific and Technical Information (OSTI.GOV)

Hep, J.; Konecna, A.; Krysl, V.

2011-07-01

This paper describes the application of effective source in forward calculations and the adjoint method to the solution of fast neutron fluence and activation detector activities in the reactor pressure vessel (RPV) and RPV cavity of a VVER-440 reactor. Its objective is the demonstration of both methods on a practical task. The effective source method applies the Boltzmann transport operator to time integrated source data in order to obtain neutron fluence and detector activities. By weighting the source data by time dependent decay of the detector activity, the result of the calculation is the detector activity. Alternatively, if the weightingmore » is uniform with respect to time, the result is the fluence. The approach works because of the inherent linearity of radiation transport in non-multiplying time-invariant media. Integrated in this way, the source data are referred to as the effective source. The effective source in the forward calculations method thereby enables the analyst to replace numerous intensive transport calculations with a single transport calculation in which the time dependence and magnitude of the source are correctly represented. In this work, the effective source method has been expanded slightly in the following way: neutron source data were performed with few group method calculation using the active core calculation code MOBY-DICK. The follow-up neutron transport calculation was performed using the neutron transport code TORT to perform multigroup calculations. For comparison, an alternative method of calculation has been used based upon adjoint functions of the Boltzmann transport equation. Calculation of the three-dimensional (3-D) adjoint function for each required computational outcome has been obtained using the deterministic code TORT and the cross section library BGL440. Adjoint functions appropriate to the required fast neutron flux density and neutron reaction rates have been calculated for several significant points within the RPV and RPV cavity of the VVER-440 reacto rand located axially at the position of maximum power and at the position of the weld. Both of these methods (the effective source and the adjoint function) are briefly described in the present paper. The paper also describes their application to the solution of fast neutron fluence and detectors activities for the VVER-440 reactor. (authors)« less

Improvement of basal conditions knowledge in Antarctica using data assimilation methods

NASA Astrophysics Data System (ADS)

Mosbeux, C.; Gillet-Chaulet, F.; Gagliardini, O.

2017-12-01

The current global warming seems to have direct consequences on ice-sheet mass loss. Unfortunately, as highlighted in the last IPCC report, current ice-sheets models face several difficulties in assessing the future evolution of the dynamics of ice sheets for the next century. Indeed, projections are still plagued with high uncertainties partially due to the poor representation of occurring physical processes, but also due to the poor initialisation of ice flow models. More specifically, simulations are very sensitive to initial parameters such as the basal friction between ice-sheet and bedrock and the bedrock topography which are still badly known because of a lack of direct observations or large uncertainty on measurements. Improving the knowledge of these two parameters in Greenland and Antarctica is therefore a prerequisite for making reliable projections. Data assimilation methods have been developed in order to overcome this problem such as the Bayesian approach of Pralong and Gudmundsson (2009) or the adjoint method tested by Goldberg and Heimbach (2013) and Perego et al. (2014). The present work is based on two different assimilation algorithms to better constrain both basal drag and bedrock elevation parameters. The first algorithm is entirely based on the adjoint method while the second one uses an iterative method coupling inversion of basal friction based on an adjoint method and through an inversion of bedrock topography using a nudging method. Both algorithms have been implemented in the finite element ice sheet and ice flow model Elmer/Ice and have been tested in a twin experiment showing a clear improvement of both parameters knowledge (Mosbeux et al., 2016). Here, the methods are applied to a real 3D case in East Antarctica and with an ensemble method approach. The application of both algorithms reduces the uncertainty on basal conditions, for instance by providing more details to the basal geometry when compared to usual DEM. Moreover, as in the previous experiment, the reconstruction of both basal elevation and basal friction significantly decreases ice flux divergence anomalies when compared to classical methods where only the friction is inverted. Finally, we conduct prognostic simulations, allowing to assess the impact of the different initialisations obtained with the ensemble method.

Towards adjoint-based inversion for rheological parameters in nonlinear viscous mantle flow

NASA Astrophysics Data System (ADS)

Worthen, Jennifer; Stadler, Georg; Petra, Noemi; Gurnis, Michael; Ghattas, Omar

2014-09-01

We address the problem of inferring mantle rheological parameter fields from surface velocity observations and instantaneous nonlinear mantle flow models. We formulate this inverse problem as an infinite-dimensional nonlinear least squares optimization problem governed by nonlinear Stokes equations. We provide expressions for the gradient of the cost functional of this optimization problem with respect to two spatially-varying rheological parameter fields: the viscosity prefactor and the exponent of the second invariant of the strain rate tensor. Adjoint (linearized) Stokes equations, which are characterized by a 4th order anisotropic viscosity tensor, facilitates efficient computation of the gradient. A quasi-Newton method for the solution of this optimization problem is presented, which requires the repeated solution of both nonlinear forward Stokes and linearized adjoint Stokes equations. For the solution of the nonlinear Stokes equations, we find that Newton’s method is significantly more efficient than a Picard fixed point method. Spectral analysis of the inverse operator given by the Hessian of the optimization problem reveals that the numerical eigenvalues collapse rapidly to zero, suggesting a high degree of ill-posedness of the inverse problem. To overcome this ill-posedness, we employ Tikhonov regularization (favoring smooth parameter fields) or total variation (TV) regularization (favoring piecewise-smooth parameter fields). Solution of two- and three-dimensional finite element-based model inverse problems show that a constant parameter in the constitutive law can be recovered well from surface velocity observations. Inverting for a spatially-varying parameter field leads to its reasonable recovery, in particular close to the surface. When inferring two spatially varying parameter fields, only an effective viscosity field and the total viscous dissipation are recoverable. Finally, a model of a subducting plate shows that a localized weak zone at the plate boundary can be partially recovered, especially with TV regularization.

A method for reducing the largest relative errors in Monte Carlo iterated-fission-source calculations

DOE Office of Scientific and Technical Information (OSTI.GOV)

Hunter, J. L.; Sutton, T. M.

2013-07-01

In Monte Carlo iterated-fission-source calculations relative uncertainties on local tallies tend to be larger in lower-power regions and smaller in higher-power regions. Reducing the largest uncertainties to an acceptable level simply by running a larger number of neutron histories is often prohibitively expensive. The uniform fission site method has been developed to yield a more spatially-uniform distribution of relative uncertainties. This is accomplished by biasing the density of fission neutron source sites while not biasing the solution. The method is integrated into the source iteration process, and does not require any auxiliary forward or adjoint calculations. For a given amountmore » of computational effort, the use of the method results in a reduction of the largest uncertainties relative to the standard algorithm. Two variants of the method have been implemented and tested. Both have been shown to be effective. (authors)« less

On iterative algorithms for quantitative photoacoustic tomography in the radiative transport regime

NASA Astrophysics Data System (ADS)

Wang, Chao; Zhou, Tie

2017-11-01

In this paper, we present a numerical reconstruction method for quantitative photoacoustic tomography (QPAT), based on the radiative transfer equation (RTE), which models light propagation more accurately than diffusion approximation (DA). We investigate the reconstruction of absorption coefficient and scattering coefficient of biological tissues. An improved fixed-point iterative method to retrieve the absorption coefficient, given the scattering coefficient, is proposed for its cheap computational cost; the convergence of this method is also proved. The Barzilai-Borwein (BB) method is applied to retrieve two coefficients simultaneously. Since the reconstruction of optical coefficients involves the solutions of original and adjoint RTEs in the framework of optimization, an efficient solver with high accuracy is developed from Gao and Zhao (2009 Transp. Theory Stat. Phys. 38 149-92). Simulation experiments illustrate that the improved fixed-point iterative method and the BB method are competitive methods for QPAT in the relevant cases.

Seismic tomography of the southern California crust based on spectral-element and adjoint methods

NASA Astrophysics Data System (ADS)

Tape, Carl; Liu, Qinya; Maggi, Alessia; Tromp, Jeroen

2010-01-01

We iteratively improve a 3-D tomographic model of the southern California crust using numerical simulations of seismic wave propagation based on a spectral-element method (SEM) in combination with an adjoint method. The initial 3-D model is provided by the Southern California Earthquake Center. The data set comprises three-component seismic waveforms (i.e. both body and surface waves), filtered over the period range 2-30 s, from 143 local earthquakes recorded by a network of 203 stations. Time windows for measurements are automatically selected by the FLEXWIN algorithm. The misfit function in the tomographic inversion is based on frequency-dependent multitaper traveltime differences. The gradient of the misfit function and related finite-frequency sensitivity kernels for each earthquake are computed using an adjoint technique. The kernels are combined using a source subspace projection method to compute a model update at each iteration of a gradient-based minimization algorithm. The inversion involved 16 iterations, which required 6800 wavefield simulations. The new crustal model, m16, is described in terms of independent shear (VS) and bulk-sound (VB) wave speed variations. It exhibits strong heterogeneity, including local changes of +/-30 per cent with respect to the initial 3-D model. The model reveals several features that relate to geological observations, such as sedimentary basins, exhumed batholiths, and contrasting lithologies across faults. The quality of the new model is validated by quantifying waveform misfits of full-length seismograms from 91 earthquakes that were not used in the tomographic inversion. The new model provides more accurate synthetic seismograms that will benefit seismic hazard assessment.

Adjoint Tomography of the Southern California Crust (Invited) (Invited)

NASA Astrophysics Data System (ADS)

Tape, C.; Liu, Q.; Maggi, A.; Tromp, J.

2009-12-01

We iteratively improve a three-dimensional tomographic model of the southern California crust using numerical simulations of seismic wave propagation based on a spectral-element method (SEM) in combination with an adjoint method. The initial 3D model is provided by the Southern California Earthquake Center. The dataset comprises three-component seismic waveforms (i.e. both body and surface waves), filtered over the period range 2-30 s, from 143 local earthquakes recorded by a network of 203 stations. Time windows for measurements are automatically selected by the FLEXWIN algorithm. The misfit function in the tomographic inversion is based on frequency-dependent multitaper traveltime differences. The gradient of the misfit function and related finite-frequency sensitivity kernels for each earthquake are computed using an adjoint technique. The kernels are combined using a source subspace projection method to compute a model update at each iteration of a gradient-based minimization algorithm. The inversion involved 16 iterations, which required 6800 wavefield simulations and a total of 0.8 million CPU hours. The new crustal model, m16, is described in terms of independent shear (Vs) and bulk-sound (Vb) wavespeed variations. It exhibits strong heterogeneity, including local changes of ±30% with respect to the initial 3D model. The model reveals several features that relate to geologic observations, such as sedimentary basins, exhumed batholiths, and contrasting lithologies across faults. The quality of the new model is validated by quantifying waveform misfits of full-length seismograms from 91 earthquakes that were not used in the tomographic inversion. The new model provides more accurate synthetic seismograms that will benefit seismic hazard assessment.

The efficiency of geophysical adjoint codes generated by automatic differentiation tools

NASA Astrophysics Data System (ADS)

Vlasenko, A. V.; Köhl, A.; Stammer, D.

2016-02-01

The accuracy of numerical models that describe complex physical or chemical processes depends on the choice of model parameters. Estimating an optimal set of parameters by optimization algorithms requires knowledge of the sensitivity of the process of interest to model parameters. Typically the sensitivity computation involves differentiation of the model, which can be performed by applying algorithmic differentiation (AD) tools to the underlying numerical code. However, existing AD tools differ substantially in design, legibility and computational efficiency. In this study we show that, for geophysical data assimilation problems of varying complexity, the performance of adjoint codes generated by the existing AD tools (i) Open_AD, (ii) Tapenade, (iii) NAGWare and (iv) Transformation of Algorithms in Fortran (TAF) can be vastly different. Based on simple test problems, we evaluate the efficiency of each AD tool with respect to computational speed, accuracy of the adjoint, the efficiency of memory usage, and the capability of each AD tool to handle modern FORTRAN 90-95 elements such as structures and pointers, which are new elements that either combine groups of variables or provide aliases to memory addresses, respectively. We show that, while operator overloading tools are the only ones suitable for modern codes written in object-oriented programming languages, their computational efficiency lags behind source transformation by orders of magnitude, rendering the application of these modern tools to practical assimilation problems prohibitive. In contrast, the application of source transformation tools appears to be the most efficient choice, allowing handling even large geophysical data assimilation problems. However, they can only be applied to numerical models written in earlier generations of programming languages. Our study indicates that applying existing AD tools to realistic geophysical problems faces limitations that urgently need to be solved to allow the continuous use of AD tools for solving geophysical problems on modern computer architectures.

Sensitivity kernels for viscoelastic loading based on adjoint methods

NASA Astrophysics Data System (ADS)

Al-Attar, David; Tromp, Jeroen

2014-01-01

Observations of glacial isostatic adjustment (GIA) allow for inferences to be made about mantle viscosity, ice sheet history and other related parameters. Typically, this inverse problem can be formulated as minimizing the misfit between the given observations and a corresponding set of synthetic data. When the number of parameters is large, solution of such optimization problems can be computationally challenging. A practical, albeit non-ideal, solution is to use gradient-based optimization. Although the gradient of the misfit required in such methods could be calculated approximately using finite differences, the necessary computation time grows linearly with the number of model parameters, and so this is often infeasible. A far better approach is to apply the `adjoint method', which allows the exact gradient to be calculated from a single solution of the forward problem, along with one solution of the associated adjoint problem. As a first step towards applying the adjoint method to the GIA inverse problem, we consider its application to a simpler viscoelastic loading problem in which gravitationally self-consistent ocean loading is neglected. The earth model considered is non-rotating, self-gravitating, compressible, hydrostatically pre-stressed, laterally heterogeneous and possesses a Maxwell solid rheology. We determine adjoint equations and Fréchet kernels for this problem based on a Lagrange multiplier method. Given an objective functional J defined in terms of the surface deformation fields, we show that its first-order perturbation can be written δ J = int _{MS}K_{η }δ ln η dV +int _{t0}^{t1}int _{partial M}K_{dot{σ }} δ dot{σ } dS dt, where δ ln η = δη/η denotes relative viscosity variations in solid regions MS, dV is the volume element, δ dot{σ } is the perturbation to the time derivative of the surface load which is defined on the earth model's surface ∂M and for times [t0, t1] and dS is the surface element on ∂M. The `viscosity kernel' Kη determines the linearized sensitivity of J to viscosity perturbations defined with respect to a laterally heterogeneous reference earth model, while the `rate-of-loading kernel' K_{dot{σ }} determines the sensitivity to variations in the time derivative of the surface load. By restricting attention to spherically symmetric viscosity perturbations, we also obtain a `radial viscosity kernel' overline{K}_{η } such that the associated contribution to δJ can be written int _{IS}overline{K}_{η }δ ln η dr, where IS denotes the subset of radii lying in solid regions. In order to illustrate this theory, we describe its numerical implementation in the case of a spherically symmetric earth model using a 1-D spectral element method, and calculate sensitivity kernels for a range of realistic observables.

Integrated Tsunami Database: simulation and identification of seismic tsunami sources, 3D visualization and post-disaster assessment on the shore

NASA Astrophysics Data System (ADS)

Krivorot'ko, Olga; Kabanikhin, Sergey; Marinin, Igor; Karas, Adel; Khidasheli, David

2013-04-01

One of the most important problems of tsunami investigation is the problem of seismic tsunami source reconstruction. Non-profit organization WAPMERR (http://wapmerr.org) has provided a historical database of alleged tsunami sources around the world that obtained with the help of information about seaquakes. WAPMERR also has a database of observations of the tsunami waves in coastal areas. The main idea of presentation consists of determining of the tsunami source parameters using seismic data and observations of the tsunami waves on the shore, and the expansion and refinement of the database of presupposed tsunami sources for operative and accurate prediction of hazards and assessment of risks and consequences. Also we present 3D visualization of real-time tsunami wave propagation and loss assessment, characterizing the nature of the building stock in cities at risk, and monitoring by satellite images using modern GIS technology ITRIS (Integrated Tsunami Research and Information System) developed by WAPMERR and Informap Ltd. The special scientific plug-in components are embedded in a specially developed GIS-type graphic shell for easy data retrieval, visualization and processing. The most suitable physical models related to simulation of tsunamis are based on shallow water equations. We consider the initial-boundary value problem in Ω := {(x,y) ?R2 : x ?(0,Lx ), y ?(0,Ly ), Lx,Ly > 0} for the well-known linear shallow water equations in the Cartesian coordinate system in terms of the liquid flow components in dimensional form Here ?(x,y,t) defines the free water surface vertical displacement, i.e. amplitude of a tsunami wave, q(x,y) is the initial amplitude of a tsunami wave. The lateral boundary is assumed to be a non-reflecting boundary of the domain, that is, it allows the free passage of the propagating waves. Assume that the free surface oscillation data at points (xm, ym) are given as a measured output data from tsunami records: fm(t) := ? (xm, ym,t), (xm,ym ) ?Ω, t ?(Tm1, Tm2), m = 1,2,...,M, M ?N (2) The problem of tsunami source reconstruction (inverse tsunami problem) consists of determining the unknown initial perturbation q(x,y) of the free surface defied in (1) from knowledge of the free surface oscillation data fm(t) given by (2). We present a numerical method to determine the tsunami source using measurements of the height of a passing tsunami wave. Proposed approach based on the weak solution theory for hyperbolic PDEs and adjoint problem method for minimization of the corresponding cost functional 2 J(q) = ?Aq - F? , F = (f1,...,fM ). (3) The adjoint problem is defined to obtain an explicit gradient formula for the cost functional (3). Different numerical algorithms (finite-difference approach and finite volume method) are proposed for the direct as well as adjoint problem. Conjugate gradient algorithm based on explicit gradient formula is used for numerical solution of the inverse problem (1)-(2). This work was partially supported by the Russian Foundation for Basic Research (project No. 12-01-00773) and by SB RAS interdisciplinary project 14 "Inverse Problems and Applications: Theory, Algorithms, Software".

Application of Parallel Adjoint-Based Error Estimation and Anisotropic Grid Adaptation for Three-Dimensional Aerospace Configurations

NASA Technical Reports Server (NTRS)

Lee-Rausch, E. M.; Park, M. A.; Jones, W. T.; Hammond, D. P.; Nielsen, E. J.

2005-01-01

This paper demonstrates the extension of error estimation and adaptation methods to parallel computations enabling larger, more realistic aerospace applications and the quantification of discretization errors for complex 3-D solutions. Results were shown for an inviscid sonic-boom prediction about a double-cone configuration and a wing/body segmented leading edge (SLE) configuration where the output function of the adjoint was pressure integrated over a part of the cylinder in the near field. After multiple cycles of error estimation and surface/field adaptation, a significant improvement in the inviscid solution for the sonic boom signature of the double cone was observed. Although the double-cone adaptation was initiated from a very coarse mesh, the near-field pressure signature from the final adapted mesh compared very well with the wind-tunnel data which illustrates that the adjoint-based error estimation and adaptation process requires no a priori refinement of the mesh. Similarly, the near-field pressure signature for the SLE wing/body sonic boom configuration showed a significant improvement from the initial coarse mesh to the final adapted mesh in comparison with the wind tunnel results. Error estimation and field adaptation results were also presented for the viscous transonic drag prediction of the DLR-F6 wing/body configuration, and results were compared to a series of globally refined meshes. Two of these globally refined meshes were used as a starting point for the error estimation and field-adaptation process where the output function for the adjoint was the total drag. The field-adapted results showed an improvement in the prediction of the drag in comparison with the finest globally refined mesh and a reduction in the estimate of the remaining drag error. The adjoint-based adaptation parameter showed a need for increased resolution in the surface of the wing/body as well as a need for wake resolution downstream of the fuselage and wing trailing edge in order to achieve the requested drag tolerance. Although further adaptation was required to meet the requested tolerance, no further cycles were computed in order to avoid large discrepancies between the surface mesh spacing and the refined field spacing.

Equilibrium sensitivities of the Greenland ice sheet inferred from the adjoint of the three- dimensional thermo-mechanical model SICOPOLIS

NASA Astrophysics Data System (ADS)

Heimbach, P.; Bugnion, V.

2008-12-01

We present a new and original approach to understanding the sensitivity of the Greenland ice sheet to key model parameters and environmental conditions. At the heart of this approach is the use of an adjoint ice sheet model. MacAyeal (1992) introduced adjoints in the context of applying control theory to estimate basal sliding parameters (basal shear stress, basal friction) of an ice stream model which minimize a least-squares model vs. observation misfit. Since then, this method has become widespread to fit ice stream models to the increasing number and diversity of satellite observations, and to estimate uncertain model parameters. However, no attempt has been made to extend this method to comprehensive ice sheet models. Here, we present a first step toward moving beyond limiting the use of control theory to ice stream models. We have generated an adjoint of the three-dimensional thermo-mechanical ice sheet model SICOPOLIS of Greve (1997). The adjoint was generated using the automatic differentiation (AD) tool TAF. TAF generates exact source code representing the tangent linear and adjoint model of the parent model provided. Model sensitivities are given by the partial derivatives of a scalar-valued model diagnostic or "cost function" with respect to the controls, and can be efficiently calculated via the adjoint. An effort to generate an efficient adjoint with the newly developed open-source AD tool OpenAD is also under way. To gain insight into the adjoint solutions, we explore various cost functions, such as local and domain-integrated ice temperature, total ice volume or the velocity of ice at the margins of the ice sheet. Elements of our control space include initial cold ice temperatures, surface mass balance, as well as parameters such as appear in Glen's flow law, or in the surface degree-day or basal sliding parameterizations. Sensitivity maps provide a comprehensive view, and allow a quantification of where and to which variables the ice sheet model is most sensitive to. The model used in the present study includes simplifications in the model physics, parameterizations which rely on uncertain empirical constants, and is unable to capture fast ice streams. Nevertheless, as a proof-of-concept, this method can readily be extended to incorporate higher-order physics or parameterizations (or be applied to other models). It also opens the door to ice sheet state estimation: using the model's physics jointly with field and satellite observations to estimate a best estimate of the state of the ice sheets.

Full seismic waveform tomography for upper-mantle structure in the Australasian region using adjoint methods

NASA Astrophysics Data System (ADS)

Fichtner, Andreas; Kennett, Brian L. N.; Igel, Heiner; Bunge, Hans-Peter

2009-12-01

We present a full seismic waveform tomography for upper-mantle structure in the Australasian region. Our method is based on spectral-element simulations of seismic wave propagation in 3-D heterogeneous earth models. The accurate solution of the forward problem ensures that waveform misfits are solely due to as yet undiscovered Earth structure and imprecise source descriptions, thus leading to more realistic tomographic images and source parameter estimates. To reduce the computational costs, we implement a long-wavelength equivalent crustal model. We quantify differences between the observed and the synthetic waveforms using time-frequency (TF) misfits. Their principal advantages are the separation of phase and amplitude misfits, the exploitation of complete waveform information and a quasi-linear relation to 3-D Earth structure. Fréchet kernels for the TF misfits are computed via the adjoint method. We propose a simple data compression scheme and an accuracy-adaptive time integration of the wavefields that allows us to reduce the storage requirements of the adjoint method by almost two orders of magnitude. To minimize the waveform phase misfit, we implement a pre-conditioned conjugate gradient algorithm. Amplitude information is incorporated indirectly by a restricted line search. This ensures that the cumulative envelope misfit does not increase during the inversion. An efficient pre-conditioner is found empirically through numerical experiments. It prevents the concentration of structural heterogeneity near the sources and receivers. We apply our waveform tomographic method to ~1000 high-quality vertical-component seismograms, recorded in the Australasian region between 1993 and 2008. The waveforms comprise fundamental- and higher-mode surface and long-period S body waves in the period range from 50 to 200 s. To improve the convergence of the algorithm, we implement a 3-D initial model that contains the long-wavelength features of the Australasian region. Resolution tests indicate that our algorithm converges after around 10 iterations and that both long- and short-wavelength features in the uppermost mantle are well resolved. There is evidence for effects related to the non-linearity in the inversion procedure. After 11 iterations we fit the data waveforms acceptably well; with no significant further improvements to be expected. During the inversion the total fitted seismogram length increases by 46 per cent, providing a clear indication of the efficiency and consistency of the iterative optimization algorithm. The resulting SV-wave velocity model reveals structural features of the Australasian upper mantle with great detail. We confirm the existence of a pronounced low-velocity band along the eastern margin of the continent that can be clearly distinguished against Precambrian Australia and the microcontinental Lord Howe Rise. The transition from Precambrian to Phanerozoic Australia (the Tasman Line) appears to be sharp down to at least 200 km depth. It mostly occurs further east of where it is inferred from gravity and magnetic anomalies. Also clearly visible are the Archean and Proterozoic cratons, the northward continuation of the continent and anomalously low S-wave velocities in the upper mantle in central Australia. This is, to the best of our knowledge, the first application of non-linear full seismic waveform tomography to a continental-scale problem.

Application of the adjoint optimisation of shock control bump for ONERA-M6 wing

NASA Astrophysics Data System (ADS)

Nejati, A.; Mazaheri, K.

2017-11-01

This article is devoted to the numerical investigation of the shock wave/boundary layer interaction (SWBLI) as the main factor influencing the aerodynamic performance of transonic bumped airfoils and wings. The numerical analysis is conducted for the ONERA-M6 wing through a shock control bump (SCB) shape optimisation process using the adjoint optimisation method. SWBLI is analyzed for both clean and bumped airfoils and wings, and it is shown how the modified wave structure originating from upstream of the SCB reduces the wave drag, by improving the boundary layer velocity profile downstream of the shock wave. The numerical simulation of the turbulent viscous flow and a gradient-based adjoint algorithm are used to find the optimum location and shape of the SCB for the ONERA-M6 airfoil and wing. Two different geometrical models are introduced for the 3D SCB, one with linear variations, and another with periodic variations. Both configurations result in drag reduction and improvement in the aerodynamic efficiency, but the periodic model is more effective. Although the three-dimensional flow structure involves much more complexities, the overall results are shown to be similar to the two-dimensional case.

Asynchronous Two-Level Checkpointing Scheme for Large-Scale Adjoints in the Spectral-Element Solver Nek5000

DOE Office of Scientific and Technical Information (OSTI.GOV)

Schanen, Michel; Marin, Oana; Zhang, Hong

Adjoints are an important computational tool for large-scale sensitivity evaluation, uncertainty quantification, and derivative-based optimization. An essential component of their performance is the storage/recomputation balance in which efficient checkpointing methods play a key role. We introduce a novel asynchronous two-level adjoint checkpointing scheme for multistep numerical time discretizations targeted at large-scale numerical simulations. The checkpointing scheme combines bandwidth-limited disk checkpointing and binomial memory checkpointing. Based on assumptions about the target petascale systems, which we later demonstrate to be realistic on the IBM Blue Gene/Q system Mira, we create a model of the expected performance of our checkpointing approach and validatemore » it using the highly scalable Navier-Stokes spectralelement solver Nek5000 on small to moderate subsystems of the Mira supercomputer. In turn, this allows us to predict optimal algorithmic choices when using all of Mira. We also demonstrate that two-level checkpointing is significantly superior to single-level checkpointing when adjoining a large number of time integration steps. To our knowledge, this is the first time two-level checkpointing had been designed, implemented, tuned, and demonstrated on fluid dynamics codes at large scale of 50k+ cores.« less

A result on differential inequalities and its application to higher order trajectory derivatives

NASA Technical Reports Server (NTRS)

Gunderson, R. W.

1973-01-01

A result on differential inequalities is obtained by considering the adjoint differential equation of the variational equation of the right side of the inequality. The main theorem is proved using basic results on differentiability of solutions with respect to initial conditions. The result is then applied to the problem of determining solution behavior using comparison techniques.

Solution of the advection-dispersion equation by a finite-volume eulerian-lagrangian local adjoint method

USGS Publications Warehouse

Healy, R.W.; Russell, T.F.

1992-01-01

A finite-volume Eulerian-Lagrangian local adjoint method for solution of the advection-dispersion equation is developed and discussed. The method is mass conservative and can solve advection-dominated ground-water solute-transport problems accurately and efficiently. An integrated finite-difference approach is used in the method. A key component of the method is that the integral representing the mass-storage term is evaluated numerically at the current time level. Integration points, and the mass associated with these points, are then forward tracked up to the next time level. The number of integration points required to reach a specified level of accuracy is problem dependent and increases as the sharpness of the simulated solute front increases. Integration points are generally equally spaced within each grid cell. For problems involving variable coefficients it has been found to be advantageous to include additional integration points at strategic locations in each well. These locations are determined by backtracking. Forward tracking of boundary fluxes by the method alleviates problems that are encountered in the backtracking approaches of most characteristic methods. A test problem is used to illustrate that the new method offers substantial advantages over other numerical methods for a wide range of problems.

Transmutation approximations for the application of hybrid Monte Carlo/deterministic neutron transport to shutdown dose rate analysis

DOE Office of Scientific and Technical Information (OSTI.GOV)

Biondo, Elliott D.; Wilson, Paul P. H.

In fusion energy systems (FES) neutrons born from burning plasma activate system components. The photon dose rate after shutdown from resulting radionuclides must be quantified. This shutdown dose rate (SDR) is calculated by coupling neutron transport, activation analysis, and photon transport. The size, complexity, and attenuating configuration of FES motivate the use of hybrid Monte Carlo (MC)/deterministic neutron transport. The Multi-Step Consistent Adjoint Driven Importance Sampling (MS-CADIS) method can be used to optimize MC neutron transport for coupled multiphysics problems, including SDR analysis, using deterministic estimates of adjoint flux distributions. When used for SDR analysis, MS-CADIS requires the formulation ofmore » an adjoint neutron source that approximates the transmutation process. In this work, transmutation approximations are used to derive a solution for this adjoint neutron source. It is shown that these approximations are reasonably met for typical FES neutron spectra and materials over a range of irradiation scenarios. When these approximations are met, the Groupwise Transmutation (GT)-CADIS method, proposed here, can be used effectively. GT-CADIS is an implementation of the MS-CADIS method for SDR analysis that uses a series of single-energy-group irradiations to calculate the adjoint neutron source. For a simple SDR problem, GT-CADIS provides speedups of 200 100 relative to global variance reduction with the Forward-Weighted (FW)-CADIS method and 9 ± 5 • 104 relative to analog. As a result, this work shows that GT-CADIS is broadly applicable to FES problems and will significantly reduce the computational resources necessary for SDR analysis.« less

Transmutation approximations for the application of hybrid Monte Carlo/deterministic neutron transport to shutdown dose rate analysis

DOE PAGES

Biondo, Elliott D.; Wilson, Paul P. H.

2017-05-08

In fusion energy systems (FES) neutrons born from burning plasma activate system components. The photon dose rate after shutdown from resulting radionuclides must be quantified. This shutdown dose rate (SDR) is calculated by coupling neutron transport, activation analysis, and photon transport. The size, complexity, and attenuating configuration of FES motivate the use of hybrid Monte Carlo (MC)/deterministic neutron transport. The Multi-Step Consistent Adjoint Driven Importance Sampling (MS-CADIS) method can be used to optimize MC neutron transport for coupled multiphysics problems, including SDR analysis, using deterministic estimates of adjoint flux distributions. When used for SDR analysis, MS-CADIS requires the formulation ofmore » an adjoint neutron source that approximates the transmutation process. In this work, transmutation approximations are used to derive a solution for this adjoint neutron source. It is shown that these approximations are reasonably met for typical FES neutron spectra and materials over a range of irradiation scenarios. When these approximations are met, the Groupwise Transmutation (GT)-CADIS method, proposed here, can be used effectively. GT-CADIS is an implementation of the MS-CADIS method for SDR analysis that uses a series of single-energy-group irradiations to calculate the adjoint neutron source. For a simple SDR problem, GT-CADIS provides speedups of 200 100 relative to global variance reduction with the Forward-Weighted (FW)-CADIS method and 9 ± 5 • 104 relative to analog. As a result, this work shows that GT-CADIS is broadly applicable to FES problems and will significantly reduce the computational resources necessary for SDR analysis.« less

Optimization of Aerospace Structure Subject to Damage Tolerance Criteria

NASA Technical Reports Server (NTRS)

Akgun, Mehmet A.

1999-01-01

The objective of this cooperative agreement was to seek computationally efficient ways to optimize aerospace structures subject to damage tolerance criteria. Optimization was to involve sizing as well as topology optimization. The work was done in collaboration with Steve Scotti, Chauncey Wu and Joanne Walsh at the NASA Langley Research Center. Computation of constraint sensitivity is normally the most time-consuming step of an optimization procedure. The cooperative work first focused on this issue and implemented the adjoint method of sensitivity computation in an optimization code (runstream) written in Engineering Analysis Language (EAL). The method was implemented both for bar and plate elements including buckling sensitivity for the latter. Lumping of constraints was investigated as a means to reduce the computational cost. Adjoint sensitivity computation was developed and implemented for lumped stress and buckling constraints. Cost of the direct method and the adjoint method was compared for various structures with and without lumping. The results were reported in two papers. It is desirable to optimize topology of an aerospace structure subject to a large number of damage scenarios so that a damage tolerant structure is obtained. Including damage scenarios in the design procedure is critical in order to avoid large mass penalties at later stages. A common method for topology optimization is that of compliance minimization which has not been used for damage tolerant design. In the present work, topology optimization is treated as a conventional problem aiming to minimize the weight subject to stress constraints. Multiple damage configurations (scenarios) are considered. Each configuration has its own structural stiffness matrix and, normally, requires factoring of the matrix and solution of the system of equations. Damage that is expected to be tolerated is local and represents a small change in the stiffness matrix compared to the baseline (undamaged) structure. The exact solution to a slightly modified set of equations can be obtained from the baseline solution economically without actually solving the modified system. Sherrnan-Morrison-Woodbury (SMW) formulas are matrix update formulas that allow this. SMW formulas were therefore used here to compute adjoint displacements for sensitivity computation and structural displacements in damaged configurations.

Solution of the advection-dispersion equation in two dimensions by a finite-volume Eulerian-Lagrangian localized adjoint method

USGS Publications Warehouse

Healy, R.W.; Russell, T.F.

1998-01-01

We extend the finite-volume Eulerian-Lagrangian localized adjoint method (FVELLAM) for solution of the advection-dispersion equation to two dimensions. The method can conserve mass globally and is not limited by restrictions on the size of the grid Peclet or Courant number. Therefore, it is well suited for solution of advection-dominated ground-water solute transport problems. In test problem comparisons with standard finite differences, FVELLAM is able to attain accurate solutions on much coarser space and time grids. On fine grids, the accuracy of the two methods is comparable. A critical aspect of FVELLAM (and all other ELLAMs) is evaluation of the mass storage integral from the preceding time level. In FVELLAM this may be accomplished with either a forward or backtracking approach. The forward tracking approach conserves mass globally and is the preferred approach. The backtracking approach is less computationally intensive, but not globally mass conservative. Boundary terms are systematically represented as integrals in space and time which are evaluated by a common integration scheme in conjunction with forward tracking through time. Unlike the one-dimensional case, local mass conservation cannot be guaranteed, so slight oscillations in concentration can develop, particularly in the vicinity of inflow or outflow boundaries. Published by Elsevier Science Ltd.

GASPACHO: a generic automatic solver using proximal algorithms for convex huge optimization problems

NASA Astrophysics Data System (ADS)

Goossens, Bart; Luong, Hiêp; Philips, Wilfried

2017-08-01

Many inverse problems (e.g., demosaicking, deblurring, denoising, image fusion, HDR synthesis) share various similarities: degradation operators are often modeled by a specific data fitting function while image prior knowledge (e.g., sparsity) is incorporated by additional regularization terms. In this paper, we investigate automatic algorithmic techniques for evaluating proximal operators. These algorithmic techniques also enable efficient calculation of adjoints from linear operators in a general matrix-free setting. In particular, we study the simultaneous-direction method of multipliers (SDMM) and the parallel proximal algorithm (PPXA) solvers and show that the automatically derived implementations are well suited for both single-GPU and multi-GPU processing. We demonstrate this approach for an Electron Microscopy (EM) deconvolution problem.

GPU-accelerated adjoint algorithmic differentiation

NASA Astrophysics Data System (ADS)

Gremse, Felix; Höfter, Andreas; Razik, Lukas; Kiessling, Fabian; Naumann, Uwe

2016-03-01

Many scientific problems such as classifier training or medical image reconstruction can be expressed as minimization of differentiable real-valued cost functions and solved with iterative gradient-based methods. Adjoint algorithmic differentiation (AAD) enables automated computation of gradients of such cost functions implemented as computer programs. To backpropagate adjoint derivatives, excessive memory is potentially required to store the intermediate partial derivatives on a dedicated data structure, referred to as the ;tape;. Parallelization is difficult because threads need to synchronize their accesses during taping and backpropagation. This situation is aggravated for many-core architectures, such as Graphics Processing Units (GPUs), because of the large number of light-weight threads and the limited memory size in general as well as per thread. We show how these limitations can be mediated if the cost function is expressed using GPU-accelerated vector and matrix operations which are recognized as intrinsic functions by our AAD software. We compare this approach with naive and vectorized implementations for CPUs. We use four increasingly complex cost functions to evaluate the performance with respect to memory consumption and gradient computation times. Using vectorization, CPU and GPU memory consumption could be substantially reduced compared to the naive reference implementation, in some cases even by an order of complexity. The vectorization allowed usage of optimized parallel libraries during forward and reverse passes which resulted in high speedups for the vectorized CPU version compared to the naive reference implementation. The GPU version achieved an additional speedup of 7.5 ± 4.4, showing that the processing power of GPUs can be utilized for AAD using this concept. Furthermore, we show how this software can be systematically extended for more complex problems such as nonlinear absorption reconstruction for fluorescence-mediated tomography.

GPU-Accelerated Adjoint Algorithmic Differentiation.

PubMed

Gremse, Felix; Höfter, Andreas; Razik, Lukas; Kiessling, Fabian; Naumann, Uwe

2016-03-01

Many scientific problems such as classifier training or medical image reconstruction can be expressed as minimization of differentiable real-valued cost functions and solved with iterative gradient-based methods. Adjoint algorithmic differentiation (AAD) enables automated computation of gradients of such cost functions implemented as computer programs. To backpropagate adjoint derivatives, excessive memory is potentially required to store the intermediate partial derivatives on a dedicated data structure, referred to as the "tape". Parallelization is difficult because threads need to synchronize their accesses during taping and backpropagation. This situation is aggravated for many-core architectures, such as Graphics Processing Units (GPUs), because of the large number of light-weight threads and the limited memory size in general as well as per thread. We show how these limitations can be mediated if the cost function is expressed using GPU-accelerated vector and matrix operations which are recognized as intrinsic functions by our AAD software. We compare this approach with naive and vectorized implementations for CPUs. We use four increasingly complex cost functions to evaluate the performance with respect to memory consumption and gradient computation times. Using vectorization, CPU and GPU memory consumption could be substantially reduced compared to the naive reference implementation, in some cases even by an order of complexity. The vectorization allowed usage of optimized parallel libraries during forward and reverse passes which resulted in high speedups for the vectorized CPU version compared to the naive reference implementation. The GPU version achieved an additional speedup of 7.5 ± 4.4, showing that the processing power of GPUs can be utilized for AAD using this concept. Furthermore, we show how this software can be systematically extended for more complex problems such as nonlinear absorption reconstruction for fluorescence-mediated tomography.

GPU-Accelerated Adjoint Algorithmic Differentiation

PubMed Central

Gremse, Felix; Höfter, Andreas; Razik, Lukas; Kiessling, Fabian; Naumann, Uwe

2015-01-01

Many scientific problems such as classifier training or medical image reconstruction can be expressed as minimization of differentiable real-valued cost functions and solved with iterative gradient-based methods. Adjoint algorithmic differentiation (AAD) enables automated computation of gradients of such cost functions implemented as computer programs. To backpropagate adjoint derivatives, excessive memory is potentially required to store the intermediate partial derivatives on a dedicated data structure, referred to as the “tape”. Parallelization is difficult because threads need to synchronize their accesses during taping and backpropagation. This situation is aggravated for many-core architectures, such as Graphics Processing Units (GPUs), because of the large number of light-weight threads and the limited memory size in general as well as per thread. We show how these limitations can be mediated if the cost function is expressed using GPU-accelerated vector and matrix operations which are recognized as intrinsic functions by our AAD software. We compare this approach with naive and vectorized implementations for CPUs. We use four increasingly complex cost functions to evaluate the performance with respect to memory consumption and gradient computation times. Using vectorization, CPU and GPU memory consumption could be substantially reduced compared to the naive reference implementation, in some cases even by an order of complexity. The vectorization allowed usage of optimized parallel libraries during forward and reverse passes which resulted in high speedups for the vectorized CPU version compared to the naive reference implementation. The GPU version achieved an additional speedup of 7.5 ± 4.4, showing that the processing power of GPUs can be utilized for AAD using this concept. Furthermore, we show how this software can be systematically extended for more complex problems such as nonlinear absorption reconstruction for fluorescence-mediated tomography. PMID:26941443

Enhancing adaptive sparse grid approximations and improving refinement strategies using adjoint-based a posteriori error estimates

DOE PAGES

Jakeman, J. D.; Wildey, T.

2015-01-01

In this paper we present an algorithm for adaptive sparse grid approximations of quantities of interest computed from discretized partial differential equations. We use adjoint-based a posteriori error estimates of the interpolation error in the sparse grid to enhance the sparse grid approximation and to drive adaptivity. We show that utilizing these error estimates provides significantly more accurate functional values for random samples of the sparse grid approximation. We also demonstrate that alternative refinement strategies based upon a posteriori error estimates can lead to further increases in accuracy in the approximation over traditional hierarchical surplus based strategies. Throughout this papermore » we also provide and test a framework for balancing the physical discretization error with the stochastic interpolation error of the enhanced sparse grid approximation.« less

Three-dimensional full waveform inversion of short-period teleseismic wavefields based upon the SEM-DSM hybrid method

NASA Astrophysics Data System (ADS)

Monteiller, Vadim; Chevrot, Sébastien; Komatitsch, Dimitri; Wang, Yi

2015-08-01

We present a method for high-resolution imaging of lithospheric structures based on full waveform inversion of teleseismic waveforms. We model the propagation of seismic waves using our recently developed direct solution method/spectral-element method hybrid technique, which allows us to simulate the propagation of short-period teleseismic waves through a regional 3-D model. We implement an iterative quasi-Newton method based upon the L-BFGS algorithm, where the gradient of the misfit function is computed using the adjoint-state method. Compared to gradient or conjugate-gradient methods, the L-BFGS algorithm has a much faster convergence rate. We illustrate the potential of this method on a synthetic test case that consists of a crustal model with a crustal discontinuity at 25 km depth and a sharp Moho jump. This model contains short- and long-wavelength heterogeneities along the lateral and vertical directions. The iterative inversion starts from a smooth 1-D model derived from the IASP91 reference Earth model. We invert both radial and vertical component waveforms, starting from long-period signals filtered at 10 s and gradually decreasing the cut-off period down to 1.25 s. This multiscale algorithm quickly converges towards a model that is very close to the true model, in contrast to inversions involving short-period waveforms only, which always get trapped into a local minimum of the cost function.

First- and second-order sensitivity analysis of linear and nonlinear structures

NASA Technical Reports Server (NTRS)

Haftka, R. T.; Mroz, Z.

1986-01-01

This paper employs the principle of virtual work to derive sensitivity derivatives of structural response with respect to stiffness parameters using both direct and adjoint approaches. The computations required are based on additional load conditions characterized by imposed initial strains, body forces, or surface tractions. As such, they are equally applicable to numerical or analytical solution techniques. The relative efficiency of various approaches for calculating first and second derivatives is assessed. It is shown that for the evaluation of second derivatives the most efficient approach is one that makes use of both the first-order sensitivities and adjoint vectors. Two example problems are used for demonstrating the various approaches.

The existence of semiregular solutions to elliptic spectral problems with discontinuous nonlinearities

DOE Office of Scientific and Technical Information (OSTI.GOV)

Pavlenko, V N; Potapov, D K

2015-09-30

This paper is concerned with the existence of semiregular solutions to the Dirichlet problem for an equation of elliptic type with discontinuous nonlinearity and when the differential operator is not assumed to be formally self-adjoint. Theorems on the existence of semiregular (positive and negative) solutions for the problem under consideration are given, and a principle of upper and lower solutions giving the existence of semiregular solutions is established. For positive values of the spectral parameter, elliptic spectral problems with discontinuous nonlinearities are shown to have nontrivial semiregular (positive and negative) solutions. Bibliography: 32 titles.

Variational data assimilation for tropospheric chemistry modeling

NASA Astrophysics Data System (ADS)

Elbern, Hendrik; Schmidt, Hauke; Ebel, Adolf

1997-07-01

The method of variational adjoint data assimilation has been applied to assimilate chemistry observations into a comprehensive tropospheric gas phase model. The rationale of this method is to find the correct initial values for a subsequent atmospheric chemistry model run when observations scattered in time are available. The variational adjoint technique is esteemed to be a promising tool for future advanced meteorological forecasting. The stimulating experience gained with the application of four-dimensional variational data assimilation in this research area has motivated the attempt to apply the technique to air quality modeling and analysis of the chemical state of the atmosphere. The present study describes the development and application of the adjoint of the second-generation regional acid deposition model gas phase mechanism, which is used in the European air pollution dispersion model system. Performance results of the assimilation scheme using both model-generated data and real observations are presented for tropospheric conditions. In the former case it is demonstrated that time series of only few or even one measured key species convey sufficient information to improve considerably the analysis of unobserved species which are directly coupled with the observed species. In the latter case a Lagrangian approach is adopted where trajectory calculations between two comprehensively furnished measurement sites are carried out. The method allows us to analyze initial data for air pollution modeling even when only sparse observations are available. Besides remarkable improvements of the model performance by properly analyzed initial concentrations, it is shown that the adjoint algorithm offers the feasibility to estimate the sensitivity of ozone concentrations relative to its precursors.

An optimal control method for fluid structure interaction systems via adjoint boundary pressure

NASA Astrophysics Data System (ADS)

Chirco, L.; Da Vià, R.; Manservisi, S.

2017-11-01

In recent year, in spite of the computational complexity, Fluid-structure interaction (FSI) problems have been widely studied due to their applicability in science and engineering. Fluid-structure interaction systems consist of one or more solid structures that deform by interacting with a surrounding fluid flow. FSI simulations evaluate the tensional state of the mechanical component and take into account the effects of the solid deformations on the motion of the interior fluids. The inverse FSI problem can be described as the achievement of a certain objective by changing some design parameters such as forces, boundary conditions and geometrical domain shapes. In this paper we would like to study the inverse FSI problem by using an optimal control approach. In particular we propose a pressure boundary optimal control method based on Lagrangian multipliers and adjoint variables. The objective is the minimization of a solid domain displacement matching functional obtained by finding the optimal pressure on the inlet boundary. The optimality system is derived from the first order necessary conditions by taking the Fréchet derivatives of the Lagrangian with respect to all the variables involved. The optimal solution is then obtained through a standard steepest descent algorithm applied to the optimality system. The approach presented in this work is general and could be used to assess other objective functionals and controls. In order to support the proposed approach we perform a few numerical tests where the fluid pressure on the domain inlet controls the displacement that occurs in a well defined region of the solid domain.

Generalized Birkhoffian representation of nonholonomic systems and its discrete variational algorithm

NASA Astrophysics Data System (ADS)

Liu, Shixing; Liu, Chang; Hua, Wei; Guo, Yongxin

2016-11-01

By using the discrete variational method, we study the numerical method of the general nonholonomic system in the generalized Birkhoffian framework, and construct a numerical method of generalized Birkhoffian equations called a self-adjoint-preserving algorithm. Numerical results show that it is reasonable to study the nonholonomic system by the structure-preserving algorithm in the generalized Birkhoffian framework. Project supported by the National Natural Science Foundation of China (Grant Nos. 11472124, 11572145, 11202090, and 11301350), the Doctor Research Start-up Fund of Liaoning Province, China (Grant No. 20141050), the China Postdoctoral Science Foundation (Grant No. 2014M560203), and the General Science and Technology Research Plans of Liaoning Educational Bureau, China (Grant No. L2013005).

Application of the sequential quadratic programming algorithm for reconstructing the distribution of optical parameters based on the time-domain radiative transfer equation.

PubMed

Qi, Hong; Qiao, Yao-Bin; Ren, Ya-Tao; Shi, Jing-Wen; Zhang, Ze-Yu; Ruan, Li-Ming

2016-10-17

Sequential quadratic programming (SQP) is used as an optimization algorithm to reconstruct the optical parameters based on the time-domain radiative transfer equation (TD-RTE). Numerous time-resolved measurement signals are obtained using the TD-RTE as forward model. For a high computational efficiency, the gradient of objective function is calculated using an adjoint equation technique. SQP algorithm is employed to solve the inverse problem and the regularization term based on the generalized Gaussian Markov random field (GGMRF) model is used to overcome the ill-posed problem. Simulated results show that the proposed reconstruction scheme performs efficiently and accurately.

Adjoint-based Sensitivity of Jet Noise to Near-nozzle Forcing

NASA Astrophysics Data System (ADS)

Chung, Seung Whan; Vishnampet, Ramanathan; Bodony, Daniel; Freund, Jonathan

2017-11-01

Past efforts have used optimal control theory, based on the numerical solution of the adjoint flow equations, to perturb turbulent jets in order to reduce their radiated sound. These efforts have been successful in that sound is reduced, with concomitant changes to the large-scale turbulence structures in the flow. However, they have also been inconclusive, in that the ultimate level of reduction seemed to depend upon the accuracy of the adjoint-based gradient rather than a physical limitation of the flow. The chaotic dynamics of the turbulence can degrade the smoothness of cost functional in the control-parameter space, which is necessary for gradient-based optimization. We introduce a route to overcoming this challenge, in part by leveraging the regularity and accuracy with a dual-consistent, discrete-exact adjoint formulation. We confirm its properties and use it to study the sensitivity and controllability of the acoustic radiation from a simulation of a M = 1.3 turbulent jet, whose statistics matches data. The smoothness of the cost functional over time is quantified by a minimum optimization step size beyond which the gradient cannot have a certain degree of accuracy. Based on this, we achieve a moderate level of sound reduction in the first few optimization steps. This material is based [in part] upon work supported by the Department of Energy, National Nuclear Security Administration, under Award Number DE-NA0002374.

Variational data assimilation for limited-area models: solution of the open boundary control problem and its application for the Gulf of Finland

NASA Astrophysics Data System (ADS)

Sheloput, Tatiana; Agoshkov, Valery

2017-04-01

The problem of modeling water areas with `liquid' (open) lateral boundaries is discussed. There are different known methods dealing with open boundaries in limited-area models, and one of the most efficient is data assimilation. Although this method is popular, there are not so many articles concerning its implementation for recovering boundary functions. However, the problem of specifying boundary conditions at the open boundary of a limited area is still actual and important. The mathematical model of the Baltic Sea circulation, developed in INM RAS, is considered. It is based on the system of thermo-hydrodynamic equations in the Boussinesq and hydrostatic approximations. The splitting method that is used for time approximation in the model allows to consider the data assimilation problem as a sequence of linear problems. One of such `simple' temperature (salinity) assimilation problem is investigated in the study. Using well known techniques of study and solution of inverse problems and optimal control problems [1], we propose an iterative solution algorithm and we obtain conditions for existence of the solution, for unique and dense solvability of the problem and for convergence of the iterative algorithm. The investigation shows that if observations satisfy certain conditions, the proposed algorithm converges to the solution of the boundary control problem. Particularly, it converges when observational data are given on the `liquid' boundary [2]. Theoretical results are confirmed by the results of numerical experiments. The numerical algorithm was implemented to water area of the Baltic Sea. Two numerical experiments were carried out in the Gulf of Finland: one with the application of the assimilation procedure and the other without. The analyses have shown that the surface temperature field in the first experiment is close to the observed one, while the result of the second experiment misfits. Number of iterations depends on the regularisation parameter, but generally the algorithm converges after 10 iterations. The results of the numerical experiments show that the usage of the proposed method makes sense. The work was supported by the Russian Science Foundation (project 14-11-00609, the formulation of the iterative process and numerical experiments) and by the Russian Foundation for Basic Research (project 16-01-00548, the formulation of the problem and its study). [1] Agoshkov V. I. Methods of Optimal Control and Adjoint Equations in Problems of Mathematical Physics. INM RAS, Moscow, 2003 (in Russian). [2] Agoshkov V.I., Sheloput T.O. The study and numerical solution of the problem of heat and salinity transfer assuming 'liquid' boundaries // Russ. J. Numer. Anal. Math. Modelling. 2016. Vol. 31, No. 2. P. 71-80.

Numerical Computation of Sensitivities and the Adjoint Approach

NASA Technical Reports Server (NTRS)

Lewis, Robert Michael

1997-01-01

We discuss the numerical computation of sensitivities via the adjoint approach in optimization problems governed by differential equations. We focus on the adjoint problem in its weak form. We show how one can avoid some of the problems with the adjoint approach, such as deriving suitable boundary conditions for the adjoint equation. We discuss the convergence of numerical approximations of the costate computed via the weak form of the adjoint problem and show the significance for the discrete adjoint problem.

Topology optimization of natural convection: Flow in a differentially heated cavity

NASA Astrophysics Data System (ADS)

Saglietti, Clio; Schlatter, Philipp; Berggren, Martin; Henningson, Dan

2017-11-01

The goal of the present work is to develop methods for optimization of the design of natural convection cooled heat sinks, using resolved simulation of both fluid flow and heat transfer. We rely on mathematical programming techniques combined with direct numerical simulations in order to iteratively update the topology of a solid structure towards optimality, i.e. until the design yielding the best performance is found, while satisfying a specific set of constraints. The investigated test case is a two-dimensional differentially heated cavity, in which the two vertical walls are held at different temperatures. The buoyancy force induces a swirling convective flow around a solid structure, whose topology is optimized to maximize the heat flux through the cavity. We rely on the spectral-element code Nek5000 to compute a high-order accurate solution of the natural convection flow arising from the conjugate heat transfer in the cavity. The laminar, steady-state solution of the problem is evaluated with a time-marching scheme that has an increased convergence rate; the actual iterative optimization is obtained using a steepest-decent algorithm, and the gradients are conveniently computed using the continuous adjoint equations for convective heat transfer.

Toward regional-scale adjoint tomography in the deep earth

NASA Astrophysics Data System (ADS)

Masson, Y.; Romanowicz, B. A.

2013-12-01

Thanks to the development of efficient numerical computation methods, such as the Spectral Element Method (SEM) and to the increasing power of computer clusters, it is now possible to obtain regional-scale images of the Earth's interior using adjoint-tomography (e.g. Tape, C., et al., 2009). As for now, these tomographic models are limited to the upper layers of the earth, i.e., they provide us with high-resolution images of the crust and the upper part of the mantle. Given the gigantic amount of calculation it represents, obtaing similar models at the global scale (i.e. images of the entire Earth) seems out of reach at the moment. Furthermore, it's likely that the first generation of such global adjoint tomographic models will have a resolution significantly smaller than the current regional models. In order to image regions of interests in the deep Earth, such as plumes, slabs or large low shear velocity provinces (LLSVPs), while keeping the computation tractable, we are developing new tools that will allow us to perform regional-scale adjoint-tomography at arbitrary depths. In a recent study (Masson et al., 2013), we showed that a numerical equivalent of the time reversal mirrors used in experimental acoustics permits to confine the wave propagation computations (i.e. using SEM simulations) inside the region to be imaged. With this ability to limit wave propagation modeling inside a region of interest, obtaining the adjoint sensitivity kernels needed for tomographic imaging is only two steps further. First, the local wavefield modeling needs to be coupled with field extrapolation techniques in order to obtain synthetic seismograms at the surface of the earth. These seismograms will account for the 3D structure inside the region of interest in a quasi-exact manner. We will present preliminary results where the field-extrapolation is performed using Green's function computed in a 1D Earth model thanks to the Direct Solution Method (DSM). Once synthetic seismograms can be obtained, it is possible to evaluate the misfit between observed and computed seismograms. The second step will then be to extrapolate the misfit function back into the SEM region in order to compute local adjoint sensitivity kernels. When available, these kernels will allow us to perform regional-scale adjoint tomography at arbitrary locations inside the earth. Masson Y., Cupillard P., Capdeville Y., & Romanowicz B., 2013. On the numerical implementation of time-reversal mirrors for tomographic imaging, Journal of Geophysical Research (under review). Tape, C., et al. (2009). "Adjoint tomography of the southern California crust." Science 325(5943): 988-992.

Iterative spectral methods and spectral solutions to compressible flows

NASA Technical Reports Server (NTRS)

Hussaini, M. Y.; Zang, T. A.

1982-01-01

A spectral multigrid scheme is described which can solve pseudospectral discretizations of self-adjoint elliptic problems in O(N log N) operations. An iterative technique for efficiently implementing semi-implicit time-stepping for pseudospectral discretizations of Navier-Stokes equations is discussed. This approach can handle variable coefficient terms in an effective manner. Pseudospectral solutions of compressible flow problems are presented. These include one dimensional problems and two dimensional Euler solutions. Results are given both for shock-capturing approaches and for shock-fitting ones.

Introduction to Adjoint Models

NASA Technical Reports Server (NTRS)

Errico, Ronald M.

2015-01-01

In this lecture, some fundamentals of adjoint models will be described. This includes a basic derivation of tangent linear and corresponding adjoint models from a parent nonlinear model, the interpretation of adjoint-derived sensitivity fields, a description of methods of automatic differentiation, and the use of adjoint models to solve various optimization problems, including singular vectors. Concluding remarks will attempt to correct common misconceptions about adjoint models and their utilization.

Fully automatic time-window selection using machine learning for global adjoint tomography

NASA Astrophysics Data System (ADS)

Chen, Y.; Hill, J.; Lei, W.; Lefebvre, M. P.; Bozdag, E.; Komatitsch, D.; Tromp, J.

2017-12-01

Selecting time windows from seismograms such that the synthetic measurements (from simulations) and measured observations are sufficiently close is indispensable in a global adjoint tomography framework. The increasing amount of seismic data collected everyday around the world demands "intelligent" algorithms for seismic window selection. While the traditional FLEXWIN algorithm can be "automatic" to some extent, it still requires both human input and human knowledge or experience, and thus is not deemed to be fully automatic. The goal of intelligent window selection is to automatically select windows based on a learnt engine that is built upon a huge number of existing windows generated through the adjoint tomography project. We have formulated the automatic window selection problem as a classification problem. All possible misfit calculation windows are classified as either usable or unusable. Given a large number of windows with a known selection mode (select or not select), we train a neural network to predict the selection mode of an arbitrary input window. Currently, the five features we extract from the windows are its cross-correlation value, cross-correlation time lag, amplitude ratio between observed and synthetic data, window length, and minimum STA/LTA value. More features can be included in the future. We use these features to characterize each window for training a multilayer perceptron neural network (MPNN). Training the MPNN is equivalent to solve a non-linear optimization problem. We use backward propagation to derive the gradient of the loss function with respect to the weighting matrices and bias vectors and use the mini-batch stochastic gradient method to iteratively optimize the MPNN. Numerical tests show that with a careful selection of the training data and a sufficient amount of training data, we are able to train a robust neural network that is capable of detecting the waveforms in an arbitrary earthquake data with negligible detection error compared to existing selection methods (e.g. FLEXWIN). We will introduce in detail the mathematical formulation of the window-selection-oriented MPNN and show very encouraging results when applying the new algorithm to real earthquake data.

Control surface hinge moment prediction using computational fluid dynamics

NASA Astrophysics Data System (ADS)

Simpson, Christopher David

The following research determines the feasibility of predicting control surface hinge moments using various computational methods. A detailed analysis is conducted using a 2D GA(W)-1 airfoil with a 20% plain flap. Simple hinge moment prediction methods are tested, including empirical Datcom relations and XFOIL. Steady-state and time-accurate turbulent, viscous, Navier-Stokes solutions are computed using Fun3D. Hinge moment coefficients are computed. Mesh construction techniques are discussed. An adjoint-based mesh adaptation case is also evaluated. An NACA 0012 45-degree swept horizontal stabilizer with a 25% elevator is also evaluated using Fun3D. Results are compared with experimental wind-tunnel data obtained from references. Finally, the costs of various solution methods are estimated. Results indicate that while a steady-state Navier-Stokes solution can accurately predict control surface hinge moments for small angles of attack and deflection angles, a time-accurate solution is necessary to accurately predict hinge moments in the presence of flow separation. The ability to capture the unsteady vortex shedding behavior present in moderate to large control surface deflections is found to be critical to hinge moment prediction accuracy. Adjoint-based mesh adaptation is shown to give hinge moment predictions similar to a globally-refined mesh for a steady-state 2D simulation.

Enhancing adaptive sparse grid approximations and improving refinement strategies using adjoint-based a posteriori error estimates

DOE Office of Scientific and Technical Information (OSTI.GOV)

Jakeman, J.D., E-mail: jdjakem@sandia.gov; Wildey, T.

2015-01-01

In this paper we present an algorithm for adaptive sparse grid approximations of quantities of interest computed from discretized partial differential equations. We use adjoint-based a posteriori error estimates of the physical discretization error and the interpolation error in the sparse grid to enhance the sparse grid approximation and to drive adaptivity of the sparse grid. Utilizing these error estimates provides significantly more accurate functional values for random samples of the sparse grid approximation. We also demonstrate that alternative refinement strategies based upon a posteriori error estimates can lead to further increases in accuracy in the approximation over traditional hierarchicalmore » surplus based strategies. Throughout this paper we also provide and test a framework for balancing the physical discretization error with the stochastic interpolation error of the enhanced sparse grid approximation.« less

Utilizing Adjoint-Based Error Estimates for Surrogate Models to Accurately Predict Probabilities of Events

DOE PAGES

Butler, Troy; Wildey, Timothy

2018-01-01

In thist study, we develop a procedure to utilize error estimates for samples of a surrogate model to compute robust upper and lower bounds on estimates of probabilities of events. We show that these error estimates can also be used in an adaptive algorithm to simultaneously reduce the computational cost and increase the accuracy in estimating probabilities of events using computationally expensive high-fidelity models. Specifically, we introduce the notion of reliability of a sample of a surrogate model, and we prove that utilizing the surrogate model for the reliable samples and the high-fidelity model for the unreliable samples gives preciselymore » the same estimate of the probability of the output event as would be obtained by evaluation of the original model for each sample. The adaptive algorithm uses the additional evaluations of the high-fidelity model for the unreliable samples to locally improve the surrogate model near the limit state, which significantly reduces the number of high-fidelity model evaluations as the limit state is resolved. Numerical results based on a recently developed adjoint-based approach for estimating the error in samples of a surrogate are provided to demonstrate (1) the robustness of the bounds on the probability of an event, and (2) that the adaptive enhancement algorithm provides a more accurate estimate of the probability of the QoI event than standard response surface approximation methods at a lower computational cost.« less

Utilizing Adjoint-Based Error Estimates for Surrogate Models to Accurately Predict Probabilities of Events

DOE Office of Scientific and Technical Information (OSTI.GOV)

Butler, Troy; Wildey, Timothy

In thist study, we develop a procedure to utilize error estimates for samples of a surrogate model to compute robust upper and lower bounds on estimates of probabilities of events. We show that these error estimates can also be used in an adaptive algorithm to simultaneously reduce the computational cost and increase the accuracy in estimating probabilities of events using computationally expensive high-fidelity models. Specifically, we introduce the notion of reliability of a sample of a surrogate model, and we prove that utilizing the surrogate model for the reliable samples and the high-fidelity model for the unreliable samples gives preciselymore » the same estimate of the probability of the output event as would be obtained by evaluation of the original model for each sample. The adaptive algorithm uses the additional evaluations of the high-fidelity model for the unreliable samples to locally improve the surrogate model near the limit state, which significantly reduces the number of high-fidelity model evaluations as the limit state is resolved. Numerical results based on a recently developed adjoint-based approach for estimating the error in samples of a surrogate are provided to demonstrate (1) the robustness of the bounds on the probability of an event, and (2) that the adaptive enhancement algorithm provides a more accurate estimate of the probability of the QoI event than standard response surface approximation methods at a lower computational cost.« less

Development of the WRF-CO2 4D-Var assimilation system v1.0

NASA Astrophysics Data System (ADS)

Zheng, Tao; French, Nancy H. F.; Baxter, Martin

2018-05-01

Regional atmospheric CO2 inversions commonly use Lagrangian particle trajectory model simulations to calculate the required influence function, which quantifies the sensitivity of a receptor to flux sources. In this paper, an adjoint-based four-dimensional variational (4D-Var) assimilation system, WRF-CO2 4D-Var, is developed to provide an alternative approach. This system is developed based on the Weather Research and Forecasting (WRF) modeling system, including the system coupled to chemistry (WRF-Chem), with tangent linear and adjoint codes (WRFPLUS), and with data assimilation (WRFDA), all in version 3.6. In WRF-CO2 4D-Var, CO2 is modeled as a tracer and its feedback to meteorology is ignored. This configuration allows most WRF physical parameterizations to be used in the assimilation system without incurring a large amount of code development. WRF-CO2 4D-Var solves for the optimized CO2 flux scaling factors in a Bayesian framework. Two variational optimization schemes are implemented for the system: the first uses the limited memory Broyden-Fletcher-Goldfarb-Shanno (BFGS) minimization algorithm (L-BFGS-B) and the second uses the Lanczos conjugate gradient (CG) in an incremental approach. WRFPLUS forward, tangent linear, and adjoint models are modified to include the physical and dynamical processes involved in the atmospheric transport of CO2. The system is tested by simulations over a domain covering the continental United States at 48 km × 48 km grid spacing. The accuracy of the tangent linear and adjoint models is assessed by comparing against finite difference sensitivity. The system's effectiveness for CO2 inverse modeling is tested using pseudo-observation data. The results of the sensitivity and inverse modeling tests demonstrate the potential usefulness of WRF-CO2 4D-Var for regional CO2 inversions.

Adjoint Methods for Adjusting Three-Dimensional Atmosphere and Surface Properties to Fit Multi-Angle Multi-Pixel Polarimetric Measurements

NASA Technical Reports Server (NTRS)

Martin, William G.; Cairns, Brian; Bal, Guillaume

2014-01-01

This paper derives an efficient procedure for using the three-dimensional (3D) vector radiative transfer equation (VRTE) to adjust atmosphere and surface properties and improve their fit with multi-angle/multi-pixel radiometric and polarimetric measurements of scattered sunlight. The proposed adjoint method uses the 3D VRTE to compute the measurement misfit function and the adjoint 3D VRTE to compute its gradient with respect to all unknown parameters. In the remote sensing problems of interest, the scalar-valued misfit function quantifies agreement with data as a function of atmosphere and surface properties, and its gradient guides the search through this parameter space. Remote sensing of the atmosphere and surface in a three-dimensional region may require thousands of unknown parameters and millions of data points. Many approaches would require calls to the 3D VRTE solver in proportion to the number of unknown parameters or measurements. To avoid this issue of scale, we focus on computing the gradient of the misfit function as an alternative to the Jacobian of the measurement operator. The resulting adjoint method provides a way to adjust 3D atmosphere and surface properties with only two calls to the 3D VRTE solver for each spectral channel, regardless of the number of retrieval parameters, measurement view angles or pixels. This gives a procedure for adjusting atmosphere and surface parameters that will scale to the large problems of 3D remote sensing. For certain types of multi-angle/multi-pixel polarimetric measurements, this encourages the development of a new class of three-dimensional retrieval algorithms with more flexible parametrizations of spatial heterogeneity, less reliance on data screening procedures, and improved coverage in terms of the resolved physical processes in the Earth?s atmosphere.

Application of variational principles and adjoint integrating factors for constructing numerical GFD models

NASA Astrophysics Data System (ADS)

Penenko, Vladimir; Tsvetova, Elena; Penenko, Alexey

2015-04-01

The proposed method is considered on an example of hydrothermodynamics and atmospheric chemistry models [1,2]. In the development of the existing methods for constructing numerical schemes possessing the properties of total approximation for operators of multiscale process models, we have developed a new variational technique, which uses the concept of adjoint integrating factors. The technique is as follows. First, a basic functional of the variational principle (the integral identity that unites the model equations, initial and boundary conditions) is transformed using Lagrange's identity and the second Green's formula. As a result, the action of the operators of main problem in the space of state functions is transferred to the adjoint operators defined in the space of sufficiently smooth adjoint functions. By the choice of adjoint functions the order of the derivatives becomes lower by one than those in the original equations. We obtain a set of new balance relationships that take into account the sources and boundary conditions. Next, we introduce the decomposition of the model domain into a set of finite volumes. For multi-dimensional non-stationary problems, this technique is applied in the framework of the variational principle and schemes of decomposition and splitting on the set of physical processes for each coordinate directions successively at each time step. For each direction within the finite volume, the analytical solutions of one-dimensional homogeneous adjoint equations are constructed. In this case, the solutions of adjoint equations serve as integrating factors. The results are the hybrid discrete-analytical schemes. They have the properties of stability, approximation and unconditional monotony for convection-diffusion operators. These schemes are discrete in time and analytic in the spatial variables. They are exact in case of piecewise-constant coefficients within the finite volume and along the coordinate lines of the grid area in each direction on a time step. In each direction, they have tridiagonal structure. They are solved by the sweep method. An important advantage of the discrete-analytical schemes is that the values of derivatives at the boundaries of finite volume are calculated together with the values of the unknown functions. This technique is particularly attractive for problems with dominant convection, as it does not require artificial monotonization and limiters. The same idea of integrating factors is applied in temporal dimension to the stiff systems of equations describing chemical transformation models [2]. The proposed method is applicable for the problems involving convection-diffusion-reaction operators. The work has been partially supported by the Presidium of RAS under Program 43, and by the RFBR grants 14-01-00125 and 14-01-31482. References: 1. V.V. Penenko, E.A. Tsvetova, A.V. Penenko. Variational approach and Euler's integrating factors for environmental studies// Computers and Mathematics with Applications, (2014) V.67, Issue 12, P. 2240-2256. 2. V.V.Penenko, E.A.Tsvetova. Variational methods of constructing monotone approximations for atmospheric chemistry models // Numerical analysis and applications, 2013, V. 6, Issue 3, pp 210-220.

Adjoint assimilation of altimetric, surface drifter, and hydrographic data in a quasi-geostrophic model of the Azores Current

NASA Astrophysics Data System (ADS)

Morrow, Rosemary; de Mey, Pierre

1995-12-01

The flow characteristics in the region of the Azores Current are investigated by assimilating TOPEX/POSEIDON and ERS 1 altimeter data into the multilevel Harvard quasigeostrophic (QG) model with open boundaries (Miller et al., 1983) using an adjoint variational scheme (Moore, 1991). The study site lies in the path of the Azores Current, where a branch retroflects to the south in the vicinity of the Madeira Rise. The region was the site of an intensive field program in 1993, SEMAPHORE. We had two main aims in this adjoint assimilation project. The first was to see whether the adjoint method could be applied locally to optimize an initial guess field, derived from the continous assimilation of altimetry data using optimal interpolation (OI). The second aim was to assimilate a variety of different data sets and evaluate their importance in constraining our QG model. The adjoint assimilation of surface data was effective in optimizing the initial conditions from OI. After 20 iterations the cost function was generally reduced by 50-80%, depending on the chosen data constraints. The primary adjustment process was via the barotropic mode. Altimetry proved to be a good constraint on the variable flow field, in particular, for constraining the barotropic field. The excellent data quality of the TOPEX/POSEIDON (T/P) altimeter data provided smooth and reliable forcing; but for our mesoscale study in a region of long decorrelation times O(30 days), the spatial coverage from the combined T/P and ERS 1 data sets was more important for constraining the solution and providing stable flow at all levels. Surface drifters provided an excellent constraint on both the barotropic and baroclinic model fields. More importantly, the drifters provided a reliable measure of the mean field. Hydrographic data were also applied as a constraint; in general, hydrography provided a weak but effective constraint on the vertical Rossby modes in the model. Finally, forecasts run over a 2-month period indicate that the initial conditions optimized by the 20-day adjoint assimilation provide more stable, longer-term forecasts.

A Community Terrain-Following Ocean Modeling System (ROMS)

DTIC Science & Technology

2015-09-30

funded NOPP project titled: Toward the Development of a Coupled COAMPS-ROMS Ensemble Kalman filter and adjoint with a focus on the Indian Ocean and the...surface temperature and surface salinity daily averages for 31-Jan-2014. Similarly, Figure 3 shows the sea surface height averaged solution for 31-Jan... temperature (upper panel; Celsius) and surface salinity (lower panel) for 31-Jan-2014. The refined solution for the Hudson Canyon grid is overlaid on

Sensitivity Analysis of Multidisciplinary Rotorcraft Simulations

NASA Technical Reports Server (NTRS)

Wang, Li; Diskin, Boris; Biedron, Robert T.; Nielsen, Eric J.; Bauchau, Olivier A.

2017-01-01

A multidisciplinary sensitivity analysis of rotorcraft simulations involving tightly coupled high-fidelity computational fluid dynamics and comprehensive analysis solvers is presented and evaluated. An unstructured sensitivity-enabled Navier-Stokes solver, FUN3D, and a nonlinear flexible multibody dynamics solver, DYMORE, are coupled to predict the aerodynamic loads and structural responses of helicopter rotor blades. A discretely-consistent adjoint-based sensitivity analysis available in FUN3D provides sensitivities arising from unsteady turbulent flows and unstructured dynamic overset meshes, while a complex-variable approach is used to compute DYMORE structural sensitivities with respect to aerodynamic loads. The multidisciplinary sensitivity analysis is conducted through integrating the sensitivity components from each discipline of the coupled system. Numerical results verify accuracy of the FUN3D/DYMORE system by conducting simulations for a benchmark rotorcraft test model and comparing solutions with established analyses and experimental data. Complex-variable implementation of sensitivity analysis of DYMORE and the coupled FUN3D/DYMORE system is verified by comparing with real-valued analysis and sensitivities. Correctness of adjoint formulations for FUN3D/DYMORE interfaces is verified by comparing adjoint-based and complex-variable sensitivities. Finally, sensitivities of the lift and drag functions obtained by complex-variable FUN3D/DYMORE simulations are compared with sensitivities computed by the multidisciplinary sensitivity analysis, which couples adjoint-based flow and grid sensitivities of FUN3D and FUN3D/DYMORE interfaces with complex-variable sensitivities of DYMORE structural responses.

Overdetermined shooting methods for computing standing water waves with spectral accuracy

NASA Astrophysics Data System (ADS)

Wilkening, Jon; Yu, Jia

2012-01-01

A high-performance shooting algorithm is developed to compute time-periodic solutions of the free-surface Euler equations with spectral accuracy in double and quadruple precision. The method is used to study resonance and its effect on standing water waves. We identify new nucleation mechanisms in which isolated large-amplitude solutions, and closed loops of such solutions, suddenly exist for depths below a critical threshold. We also study degenerate and secondary bifurcations related to Wilton's ripples in the traveling case, and explore the breakdown of self-similarity at the crests of extreme standing waves. In shallow water, we find that standing waves take the form of counter-propagating solitary waves that repeatedly collide quasi-elastically. In deep water with surface tension, we find that standing waves resemble counter-propagating depression waves. We also discuss the existence and non-uniqueness of solutions, and smooth versus erratic dependence of Fourier modes on wave amplitude and fluid depth. In the numerical method, robustness is achieved by posing the problem as an overdetermined nonlinear system and using either adjoint-based minimization techniques or a quadratically convergent trust-region method to minimize the objective function. Efficiency is achieved in the trust-region approach by parallelizing the Jacobian computation, so the setup cost of computing the Dirichlet-to-Neumann operator in the variational equation is not repeated for each column. Updates of the Jacobian are also delayed until the previous Jacobian ceases to be useful. Accuracy is maintained using spectral collocation with optional mesh refinement in space, a high-order Runge-Kutta or spectral deferred correction method in time and quadruple precision for improved navigation of delicate regions of parameter space as well as validation of double-precision results. Implementation issues for transferring much of the computation to a graphic processing units are briefly discussed, and the performance of the algorithm is tested for a number of hardware configurations.

Adjoint affine fusion and tadpoles

DOE Office of Scientific and Technical Information (OSTI.GOV)

Urichuk, Andrew, E-mail: andrew.urichuk@uleth.ca; Walton, Mark A., E-mail: walton@uleth.ca; International School for Advanced Studies

2016-06-15

We study affine fusion with the adjoint representation. For simple Lie algebras, elementary and universal formulas determine the decomposition of a tensor product of an integrable highest-weight representation with the adjoint representation. Using the (refined) affine depth rule, we prove that equally striking results apply to adjoint affine fusion. For diagonal fusion, a coefficient equals the number of nonzero Dynkin labels of the relevant affine highest weight, minus 1. A nice lattice-polytope interpretation follows and allows the straightforward calculation of the genus-1 1-point adjoint Verlinde dimension, the adjoint affine fusion tadpole. Explicit formulas, (piecewise) polynomial in the level, are writtenmore » for the adjoint tadpoles of all classical Lie algebras. We show that off-diagonal adjoint affine fusion is obtained from the corresponding tensor product by simply dropping non-dominant representations.« less

Solving the transport equation with quadratic finite elements: Theory and applications

DOE Office of Scientific and Technical Information (OSTI.GOV)

Ferguson, J.M.

1997-12-31

At the 4th Joint Conference on Computational Mathematics, the author presented a paper introducing a new quadratic finite element scheme (QFEM) for solving the transport equation. In the ensuing year the author has obtained considerable experience in the application of this method, including solution of eigenvalue problems, transmission problems, and solution of the adjoint form of the equation as well as the usual forward solution. He will present detailed results, and will also discuss other refinements of his transport codes, particularly for 3-dimensional problems on rectilinear and non-rectilinear grids.

Extension of the ADjoint Approach to a Laminar Navier-Stokes Solver

NASA Astrophysics Data System (ADS)

Paige, Cody

The use of adjoint methods is common in computational fluid dynamics to reduce the cost of the sensitivity analysis in an optimization cycle. The forward mode ADjoint is a combination of an adjoint sensitivity analysis method with a forward mode automatic differentiation (AD) and is a modification of the reverse mode ADjoint method proposed by Mader et al.[1]. A colouring acceleration technique is presented to reduce the computational cost increase associated with forward mode AD. The forward mode AD facilitates the implementation of the laminar Navier-Stokes (NS) equations. The forward mode ADjoint method is applied to a three-dimensional computational fluid dynamics solver. The resulting Euler and viscous ADjoint sensitivities are compared to the reverse mode Euler ADjoint derivatives and a complex-step method to demonstrate the reduced computational cost and accuracy. Both comparisons demonstrate the benefits of the colouring method and the practicality of using a forward mode AD. [1] Mader, C.A., Martins, J.R.R.A., Alonso, J.J., and van der Weide, E. (2008) ADjoint: An approach for the rapid development of discrete adjoint solvers. AIAA Journal, 46(4):863-873. doi:10.2514/1.29123.

Adjoint tomography of Empirical Green's functions from ambient noise in Southern California

NASA Astrophysics Data System (ADS)

Wang, K.; Liu, Q.; Yang, Y.; Basini, P.; Tape, C.

2017-12-01

We construct a new shear-wave velocity (Vsv) model in Southern California by adjoint tomography of Rayleigh-wave Empirical Green's functions at 5-50 s period from Z-Z component ambient noise cross-correlations. The initial model of our adjoint tomography is the isotropic Vs model M16 from Tape et al. [2010], which is generated by three-component body and surface waves at 2-30 s period from local earthquake data. Synthetic Green's functions (SGFs) from M16 show a good agreement with the Empirical Green's functions (EGFs) from ambient noise at 5-50 s and 10-50 s period bands, but have an average 1.75 s advance in time at 20-50 s. By minimizing the traveltime differences between the EGFs and SGFs using gradient-based algorithm, the initial model is refined and improved and the total misfits is reduced from the initial 1.75s to its convergent point of 0.33 s after five iterations. The final Vsv model fits EGF waveforms better than the initial model at all the three frequency bands with smaller misfit distributions. Our new Vsv model reveals some new features in the mid- and lower-crust, mainly including: (1) the mean speed of lower crust is slowed down by about 5%; (2) In the Los Angeles Basin and its Northern area, the speed is higher than the initial model throughout the crust; (3) beneath the westernmost Peninsular Range Batholith (PRB) and Sierra Nevada Batholith (SNB), we observe high shear velocities in the lower crust; (4) a shallow high-velocity zone in the mid-crust are observed beneath Salton Trough Basin. Our model also shows refined lateral velocity gradient across PRB, SNB, San Andreas Fault (SAF), which helps to understand the west-east compositional boundary in PRB, SNB, and the dip angle and the depth extent of SAF. Our study demonstrates the feasibility of adjoint tomography of ambient noise data in southern California, which is an important complement to earthquake data. The numerical solver used in adjoint tomography can provide more accurate structure sensitivity kernels than analytical methods used in traditional ambient noise tomography.

Modal expansions for infrasound propagation and their implications for ground-to-ground propagation.

PubMed

Waxler, Roger; Assink, Jelle; Velea, Doru

2017-02-01

The use of expansions in vertical eigenmodes for long range infrasound propagation modeling in the effective sound speed approximation is investigated. The question of convergence of such expansions is related to the maximum elevation angles that are required. Including atmospheric attenuation leads to a non-self-adjoint vertical eigenvalue problem. The use of leading order perturbation theory for the modal attenuation is compared to the results of numerical solutions to the non-self-adjoint eigenvalue problem and conditions under which the perturbative result is expected to be valid are obtained. Modal expansions are obtained in the frequency domain; broadband signals must be modeled through Fourier reconstruction. Such broadband signal reconstruction is investigated and the relation between bandwidth, wavetrain duration, and frequency sampling is discussed.

Frozen Gaussian approximation for 3D seismic tomography

NASA Astrophysics Data System (ADS)

Chai, Lihui; Tong, Ping; Yang, Xu

2018-05-01

Three-dimensional (3D) wave-equation-based seismic tomography is computationally challenging in large scales and high-frequency regime. In this paper, we apply the frozen Gaussian approximation (FGA) method to compute 3D sensitivity kernels and seismic tomography of high-frequency. Rather than standard ray theory used in seismic inversion (e.g. Kirchhoff migration and Gaussian beam migration), FGA is used to compute the 3D high-frequency sensitivity kernels for travel-time or full waveform inversions. Specifically, we reformulate the equations of the forward and adjoint wavefields for the purpose of convenience to apply FGA, and with this reformulation, one can efficiently compute the Green’s functions whose convolutions with source time function produce wavefields needed for the construction of 3D kernels. Moreover, a fast summation method is proposed based on local fast Fourier transform which greatly improves the speed of reconstruction as the last step of FGA algorithm. We apply FGA to both the travel-time adjoint tomography and full waveform inversion (FWI) on synthetic crosswell seismic data with dominant frequencies as high as those of real crosswell data, and confirm again that FWI requires a more sophisticated initial velocity model for the convergence than travel-time adjoint tomography. We also numerically test the accuracy of applying FGA to local earthquake tomography. This study paves the way to directly apply wave-equation-based seismic tomography methods into real data around their dominant frequencies.

Stabilized FE simulation of prototype thermal-hydraulics problems with integrated adjoint-based capabilities

NASA Astrophysics Data System (ADS)

Shadid, J. N.; Smith, T. M.; Cyr, E. C.; Wildey, T. M.; Pawlowski, R. P.

2016-09-01

A critical aspect of applying modern computational solution methods to complex multiphysics systems of relevance to nuclear reactor modeling, is the assessment of the predictive capability of specific proposed mathematical models. In this respect the understanding of numerical error, the sensitivity of the solution to parameters associated with input data, boundary condition uncertainty, and mathematical models is critical. Additionally, the ability to evaluate and or approximate the model efficiently, to allow development of a reasonable level of statistical diagnostics of the mathematical model and the physical system, is of central importance. In this study we report on initial efforts to apply integrated adjoint-based computational analysis and automatic differentiation tools to begin to address these issues. The study is carried out in the context of a Reynolds averaged Navier-Stokes approximation to turbulent fluid flow and heat transfer using a particular spatial discretization based on implicit fully-coupled stabilized FE methods. Initial results are presented that show the promise of these computational techniques in the context of nuclear reactor relevant prototype thermal-hydraulics problems.

Stabilized FE simulation of prototype thermal-hydraulics problems with integrated adjoint-based capabilities

DOE Office of Scientific and Technical Information (OSTI.GOV)

Shadid, J.N., E-mail: jnshadi@sandia.gov; Department of Mathematics and Statistics, University of New Mexico; Smith, T.M.

A critical aspect of applying modern computational solution methods to complex multiphysics systems of relevance to nuclear reactor modeling, is the assessment of the predictive capability of specific proposed mathematical models. In this respect the understanding of numerical error, the sensitivity of the solution to parameters associated with input data, boundary condition uncertainty, and mathematical models is critical. Additionally, the ability to evaluate and or approximate the model efficiently, to allow development of a reasonable level of statistical diagnostics of the mathematical model and the physical system, is of central importance. In this study we report on initial efforts tomore » apply integrated adjoint-based computational analysis and automatic differentiation tools to begin to address these issues. The study is carried out in the context of a Reynolds averaged Navier–Stokes approximation to turbulent fluid flow and heat transfer using a particular spatial discretization based on implicit fully-coupled stabilized FE methods. Initial results are presented that show the promise of these computational techniques in the context of nuclear reactor relevant prototype thermal-hydraulics problems.« less

Stabilized FE simulation of prototype thermal-hydraulics problems with integrated adjoint-based capabilities

DOE Office of Scientific and Technical Information (OSTI.GOV)

Shadid, J. N.; Smith, T. M.; Cyr, E. C.

A critical aspect of applying modern computational solution methods to complex multiphysics systems of relevance to nuclear reactor modeling, is the assessment of the predictive capability of specific proposed mathematical models. The understanding of numerical error, the sensitivity of the solution to parameters associated with input data, boundary condition uncertainty, and mathematical models is critical. Additionally, the ability to evaluate and or approximate the model efficiently, to allow development of a reasonable level of statistical diagnostics of the mathematical model and the physical system, is of central importance. In our study we report on initial efforts to apply integrated adjoint-basedmore » computational analysis and automatic differentiation tools to begin to address these issues. The study is carried out in the context of a Reynolds averaged Navier–Stokes approximation to turbulent fluid flow and heat transfer using a particular spatial discretization based on implicit fully-coupled stabilized FE methods. We present the initial results that show the promise of these computational techniques in the context of nuclear reactor relevant prototype thermal-hydraulics problems.« less

Stabilized FE simulation of prototype thermal-hydraulics problems with integrated adjoint-based capabilities

DOE PAGES

Shadid, J. N.; Smith, T. M.; Cyr, E. C.; ...

2016-05-20

A critical aspect of applying modern computational solution methods to complex multiphysics systems of relevance to nuclear reactor modeling, is the assessment of the predictive capability of specific proposed mathematical models. The understanding of numerical error, the sensitivity of the solution to parameters associated with input data, boundary condition uncertainty, and mathematical models is critical. Additionally, the ability to evaluate and or approximate the model efficiently, to allow development of a reasonable level of statistical diagnostics of the mathematical model and the physical system, is of central importance. In our study we report on initial efforts to apply integrated adjoint-basedmore » computational analysis and automatic differentiation tools to begin to address these issues. The study is carried out in the context of a Reynolds averaged Navier–Stokes approximation to turbulent fluid flow and heat transfer using a particular spatial discretization based on implicit fully-coupled stabilized FE methods. We present the initial results that show the promise of these computational techniques in the context of nuclear reactor relevant prototype thermal-hydraulics problems.« less

Assimilation of Wave and Current Data for Prediction of Inlet and River Mouth Dynamics

DTIC Science & Technology

2013-07-01

onto the Delft3D computational grid and the specification of Riemann -type boundary conditions for the boundary-normal velocity and surface elevation...conditions from time- history data from in situ tide gages. The corrections are applied to the surface-elevation contribution to the Riemann boundary...The algorithms described above are all of the strong-constraint variational variety, and make use of adjoint solvers corresponding to the various

Gradient calculations for dynamic recurrent neural networks: a survey.

PubMed

Pearlmutter, B A

1995-01-01

Surveys learning algorithms for recurrent neural networks with hidden units and puts the various techniques into a common framework. The authors discuss fixed point learning algorithms, namely recurrent backpropagation and deterministic Boltzmann machines, and nonfixed point algorithms, namely backpropagation through time, Elman's history cutoff, and Jordan's output feedback architecture. Forward propagation, an on-line technique that uses adjoint equations, and variations thereof, are also discussed. In many cases, the unified presentation leads to generalizations of various sorts. The author discusses advantages and disadvantages of temporally continuous neural networks in contrast to clocked ones continues with some "tricks of the trade" for training, using, and simulating continuous time and recurrent neural networks. The author presents some simulations, and at the end, addresses issues of computational complexity and learning speed.

FAST TRACK COMMUNICATION Quasi self-adjoint nonlinear wave equations

NASA Astrophysics Data System (ADS)

Ibragimov, N. H.; Torrisi, M.; Tracinà, R.

2010-11-01

In this paper we generalize the classification of self-adjoint second-order linear partial differential equation to a family of nonlinear wave equations with two independent variables. We find a class of quasi self-adjoint nonlinear equations which includes the self-adjoint linear equations as a particular case. The property of a differential equation to be quasi self-adjoint is important, e.g. for constructing conservation laws associated with symmetries of the differential equation.

2D Seismic Imaging of Elastic Parameters by Frequency Domain Full Waveform Inversion

NASA Astrophysics Data System (ADS)

Brossier, R.; Virieux, J.; Operto, S.

2008-12-01

Thanks to recent advances in parallel computing, full waveform inversion is today a tractable seismic imaging method to reconstruct physical parameters of the earth interior at different scales ranging from the near- surface to the deep crust. We present a massively parallel 2D frequency-domain full-waveform algorithm for imaging visco-elastic media from multi-component seismic data. The forward problem (i.e. the resolution of the frequency-domain 2D PSV elastodynamics equations) is based on low-order Discontinuous Galerkin (DG) method (P0 and/or P1 interpolations). Thanks to triangular unstructured meshes, the DG method allows accurate modeling of both body waves and surface waves in case of complex topography for a discretization of 10 to 15 cells per shear wavelength. The frequency-domain DG system is solved efficiently for multiple sources with the parallel direct solver MUMPS. The local inversion procedure (i.e. minimization of residuals between observed and computed data) is based on the adjoint-state method which allows to efficiently compute the gradient of the objective function. Applying the inversion hierarchically from the low frequencies to the higher ones defines a multiresolution imaging strategy which helps convergence towards the global minimum. In place of expensive Newton algorithm, the combined use of the diagonal terms of the approximate Hessian matrix and optimization algorithms based on quasi-Newton methods (Conjugate Gradient, LBFGS, ...) allows to improve the convergence of the iterative inversion. The distribution of forward problem solutions over processors driven by a mesh partitioning performed by METIS allows to apply most of the inversion in parallel. We shall present the main features of the parallel modeling/inversion algorithm, assess its scalability and illustrate its performances with realistic synthetic case studies.

Autofluorescence and diffuse reflectance patterns in cervical spectroscopy

NASA Astrophysics Data System (ADS)

Marin, Nena Maribel

Fluorescence and diffuse reflectance spectroscopy are two new optical technologies, which have shown promise to aid in the real time, non-invasive identification of cancers and precancers. Spectral patterns carry a fingerprint of scattering, absorption and fluorescence properties in tissue. Scattering, absorption and fluorescence in tissue are directly affected by biological features that are diagnostically significant, such as nuclear size, micro-vessel density, volume fraction of collagen fibers, tissue oxygenation and cell metabolism. Thus, analysis of spectral patterns can unlock a wealth of information directly related with the onset and progression of disease. Data from a Phase II clinical trial to assess the technical efficacy of fluorescence and diffuse reflectance spectroscopy acquired from 850 women at three clinical locations with two research grade optical devices is calibrated and analyzed. Tools to process and standardize spectra so that data from multiple spectrometers can be combined and analyzed are presented. Methodologies for calibration and quality assurance of optical systems are established to simplify design issues and ensure validity of data for future clinical trials. Empirically based algorithms, using multivariate statistical approaches are applied to spectra and evaluated as a clinical diagnostic tool. Physically based algorithms, using mathematical models of light propagation in tissue are presented. The presented mathematical model combines a diffusion theory in P3 approximation reflectance model and a 2-layer fluorescence model using exponential attenuation and diffusion theory. The resulting adjoint fluorescence and reflectance model extracts twelve optical properties characterizing fluorescence efficiency of cervical epithelium and stroma fluorophores, stromal hemoglobin and collagen absorption, oxygen saturation, and stromal scattering strength and shape. Validations with Monte Carlo simulations show that adjoint model extracted optical properties of the epithelium and the stroma can be estimated accurately. Adjoint model is applied to 926 clinical measurements from 503 patients. Mean values of extracted optical properties have demonstrated to characterize the biological changes associated with dysplastic progression. Finally, penalized logistic regression algorithms are applied to discriminate dysplastic stages in tissue based on extracted optical features. This work provides understandable and interpretable information regarding predictive and generalization ability of optical spectroscopy in neoplastic changes using a minimum subset of optical measurements. Ultimately these methodologies would facilitate the transfer of these optical technologies into clinical practice.

Efficient Algorithms for Estimating the Absorption Spectrum within Linear Response TDDFT

DOE Office of Scientific and Technical Information (OSTI.GOV)

Brabec, Jiri; Lin, Lin; Shao, Meiyue

We present a special symmetric Lanczos algorithm and a kernel polynomial method (KPM) for approximating the absorption spectrum of molecules within the linear response time-dependent density functional theory (TDDFT) framework in the product form. In contrast to existing algorithms, the new algorithms are based on reformulating the original non-Hermitian eigenvalue problem as a product eigenvalue problem and the observation that the product eigenvalue problem is self-adjoint with respect to an appropriately chosen inner product. This allows a simple symmetric Lanczos algorithm to be used to compute the desired absorption spectrum. The use of a symmetric Lanczos algorithm only requires halfmore » of the memory compared with the nonsymmetric variant of the Lanczos algorithm. The symmetric Lanczos algorithm is also numerically more stable than the nonsymmetric version. The KPM algorithm is also presented as a low-memory alternative to the Lanczos approach, but the algorithm may require more matrix-vector multiplications in practice. We discuss the pros and cons of these methods in terms of their accuracy as well as their computational and storage cost. Applications to a set of small and medium-sized molecules are also presented.« less

Efficient Algorithms for Estimating the Absorption Spectrum within Linear Response TDDFT

DOE PAGES

Brabec, Jiri; Lin, Lin; Shao, Meiyue; ...

2015-10-06

We present a special symmetric Lanczos algorithm and a kernel polynomial method (KPM) for approximating the absorption spectrum of molecules within the linear response time-dependent density functional theory (TDDFT) framework in the product form. In contrast to existing algorithms, the new algorithms are based on reformulating the original non-Hermitian eigenvalue problem as a product eigenvalue problem and the observation that the product eigenvalue problem is self-adjoint with respect to an appropriately chosen inner product. This allows a simple symmetric Lanczos algorithm to be used to compute the desired absorption spectrum. The use of a symmetric Lanczos algorithm only requires halfmore » of the memory compared with the nonsymmetric variant of the Lanczos algorithm. The symmetric Lanczos algorithm is also numerically more stable than the nonsymmetric version. The KPM algorithm is also presented as a low-memory alternative to the Lanczos approach, but the algorithm may require more matrix-vector multiplications in practice. We discuss the pros and cons of these methods in terms of their accuracy as well as their computational and storage cost. Applications to a set of small and medium-sized molecules are also presented.« less

Optimal boundary conditions for ORCA-2 model

NASA Astrophysics Data System (ADS)

Kazantsev, Eugene

2013-08-01

A 4D-Var data assimilation technique is applied to ORCA-2 configuration of the NEMO in order to identify the optimal parametrization of boundary conditions on the lateral boundaries as well as on the bottom and on the surface of the ocean. The influence of boundary conditions on the solution is analyzed both within and beyond the assimilation window. It is shown that the optimal bottom and surface boundary conditions allow us to better represent the jet streams, such as Gulf Stream and Kuroshio. Analyzing the reasons of the jets reinforcement, we notice that data assimilation has a major impact on parametrization of the bottom boundary conditions for u and v. Automatic generation of the tangent and adjoint codes is also discussed. Tapenade software is shown to be able to produce the adjoint code that can be used after a memory usage optimization.

A predictive control framework for optimal energy extraction of wind farms

NASA Astrophysics Data System (ADS)

Vali, M.; van Wingerden, J. W.; Boersma, S.; Petrović, V.; Kühn, M.

2016-09-01

This paper proposes an adjoint-based model predictive control for optimal energy extraction of wind farms. It employs the axial induction factor of wind turbines to influence their aerodynamic interactions through the wake. The performance index is defined here as the total power production of the wind farm over a finite prediction horizon. A medium-fidelity wind farm model is utilized to predict the inflow propagation in advance. The adjoint method is employed to solve the formulated optimization problem in a cost effective way and the first part of the optimal solution is implemented over the control horizon. This procedure is repeated at the next controller sample time providing the feedback into the optimization. The effectiveness and some key features of the proposed approach are studied for a two turbine test case through simulations.

Aerothermodynamic Design Sensitivities for a Reacting Gas Flow Solver on an Unstructured Mesh Using a Discrete Adjoint Formulation

NASA Astrophysics Data System (ADS)

Thompson, Kyle Bonner

An algorithm is described to efficiently compute aerothermodynamic design sensitivities using a decoupled variable set. In a conventional approach to computing design sensitivities for reacting flows, the species continuity equations are fully coupled to the conservation laws for momentum and energy. In this algorithm, the species continuity equations are solved separately from the mixture continuity, momentum, and total energy equations. This decoupling simplifies the implicit system, so that the flow solver can be made significantly more efficient, with very little penalty on overall scheme robustness. Most importantly, the computational cost of the point implicit relaxation is shown to scale linearly with the number of species for the decoupled system, whereas the fully coupled approach scales quadratically. Also, the decoupled method significantly reduces the cost in wall time and memory in comparison to the fully coupled approach. This decoupled approach for computing design sensitivities with the adjoint system is demonstrated for inviscid flow in chemical non-equilibrium around a re-entry vehicle with a retro-firing annular nozzle. The sensitivities of the surface temperature and mass flow rate through the nozzle plenum are computed with respect to plenum conditions and verified against sensitivities computed using a complex-variable finite-difference approach. The decoupled scheme significantly reduces the computational time and memory required to complete the optimization, making this an attractive method for high-fidelity design of hypersonic vehicles.

Aerodynamic Shape Optimization of Complex Aircraft Configurations via an Adjoint Formulation

NASA Technical Reports Server (NTRS)

Reuther, James; Jameson, Antony; Farmer, James; Martinelli, Luigi; Saunders, David

1996-01-01

This work describes the implementation of optimization techniques based on control theory for complex aircraft configurations. Here control theory is employed to derive the adjoint differential equations, the solution of which allows for a drastic reduction in computational costs over previous design methods (13, 12, 43, 38). In our earlier studies (19, 20, 22, 23, 39, 25, 40, 41, 42) it was shown that this method could be used to devise effective optimization procedures for airfoils, wings and wing-bodies subject to either analytic or arbitrary meshes. Design formulations for both potential flows and flows governed by the Euler equations have been demonstrated, showing that such methods can be devised for various governing equations (39, 25). In our most recent works (40, 42) the method was extended to treat wing-body configurations with a large number of mesh points, verifying that significant computational savings can be gained for practical design problems. In this paper the method is extended for the Euler equations to treat complete aircraft configurations via a new multiblock implementation. New elements include a multiblock-multigrid flow solver, a multiblock-multigrid adjoint solver, and a multiblock mesh perturbation scheme. Two design examples are presented in which the new method is used for the wing redesign of a transonic business jet.

A Least-Squares Transport Equation Compatible with Voids

DOE Office of Scientific and Technical Information (OSTI.GOV)

Hansen, Jon; Peterson, Jacob; Morel, Jim

Standard second-order self-adjoint forms of the transport equation, such as the even-parity, odd-parity, and self-adjoint angular flux equation, cannot be used in voids. Perhaps more important, they experience numerical convergence difficulties in near-voids. Here we present a new form of a second-order self-adjoint transport equation that has an advantage relative to standard forms in that it can be used in voids or near-voids. Our equation is closely related to the standard least-squares form of the transport equation with both equations being applicable in a void and having a nonconservative analytic form. However, unlike the standard least-squares form of the transportmore » equation, our least-squares equation is compatible with source iteration. It has been found that the standard least-squares form of the transport equation with a linear-continuous finite-element spatial discretization has difficulty in the thick diffusion limit. Here we extensively test the 1D slab-geometry version of our scheme with respect to void solutions, spatial convergence rate, and the intermediate and thick diffusion limits. We also define an effective diffusion synthetic acceleration scheme for our discretization. Our conclusion is that our least-squares S n formulation represents an excellent alternative to existing second-order S n transport formulations« less

Solid oxide fuel cell simulation and design optimization with numerical adjoint techniques

NASA Astrophysics Data System (ADS)

Elliott, Louie C.

This dissertation reports on the application of numerical optimization techniques as applied to fuel cell simulation and design. Due to the "multi-physics" inherent in a fuel cell, which results in a highly coupled and non-linear behavior, an experimental program to analyze and improve the performance of fuel cells is extremely difficult. This program applies new optimization techniques with computational methods from the field of aerospace engineering to the fuel cell design problem. After an overview of fuel cell history, importance, and classification, a mathematical model of solid oxide fuel cells (SOFC) is presented. The governing equations are discretized and solved with computational fluid dynamics (CFD) techniques including unstructured meshes, non-linear solution methods, numerical derivatives with complex variables, and sensitivity analysis with adjoint methods. Following the validation of the fuel cell model in 2-D and 3-D, the results of the sensitivity analysis are presented. The sensitivity derivative for a cost function with respect to a design variable is found with three increasingly sophisticated techniques: finite difference, direct differentiation, and adjoint. A design cycle is performed using a simple optimization method to improve the value of the implemented cost function. The results from this program could improve fuel cell performance and lessen the world's dependence on fossil fuels.

Seismic Imaging of VTI, HTI and TTI based on Adjoint Methods

NASA Astrophysics Data System (ADS)

Rusmanugroho, H.; Tromp, J.

2014-12-01

Recent studies show that isotropic seismic imaging based on adjoint method reduces low-frequency artifact caused by diving waves, which commonly occur in two-wave wave-equation migration, such as Reverse Time Migration (RTM). Here, we derive new expressions of sensitivity kernels for Vertical Transverse Isotropy (VTI) using the Thomsen parameters (ɛ, δ, γ) plus the P-, and S-wave speeds (α, β) as well as via the Chen & Tromp (GJI 2005) parameters (A, C, N, L, F). For Horizontal Transverse Isotropy (HTI), these parameters depend on an azimuthal angle φ, where the tilt angle θ is equivalent to 90°, and for Tilted Transverse Isotropy (TTI), these parameters depend on both the azimuth and tilt angles. We calculate sensitivity kernels for each of these two approaches. Individual kernels ("images") are numerically constructed based on the interaction between the regular and adjoint wavefields in smoothed models which are in practice estimated through Full-Waveform Inversion (FWI). The final image is obtained as a result of summing all shots, which are well distributed to sample the target model properly. The impedance kernel, which is a sum of sensitivity kernels of density and the Thomsen or Chen & Tromp parameters, looks crisp and promising for seismic imaging. The other kernels suffer from low-frequency artifacts, similar to traditional seismic imaging conditions. However, all sensitivity kernels are important for estimating the gradient of the misfit function, which, in combination with a standard gradient-based inversion algorithm, is used to minimize the objective function in FWI.

Adjoint-state inversion of electric resistivity tomography data of seawater intrusion at the Argentona coastal aquifer (Spain)

NASA Astrophysics Data System (ADS)

Fernández-López, Sheila; Carrera, Jesús; Ledo, Juanjo; Queralt, Pilar; Luquot, Linda; Martínez, Laura; Bellmunt, Fabián

2016-04-01

Seawater intrusion in aquifers is a complex phenomenon that can be characterized with the help of electric resistivity tomography (ERT) because of the low resistivity of seawater, which underlies the freshwater floating on top. The problem is complex because of the need for joint inversion of electrical and hydraulic (density dependent flow) data. Here we present an adjoint-state algorithm to treat electrical data. This method is a common technique to obtain derivatives of an objective function, depending on potentials with respect to model parameters. The main advantages of it are its simplicity in stationary problems and the reduction of computational cost respect others methodologies. The relationship between the concentration of chlorides and the resistivity values of the field is well known. Also, these resistivities are related to the values of potentials measured using ERT. Taking this into account, it will be possible to define the different resistivities zones from the field data of potential distribution using the basis of inverse problem. In this case, the studied zone is situated in Argentona (Baix Maresme, Catalonia), where the values of chlorides obtained in some wells of the zone are too high. The adjoint-state method will be used to invert the measured data using a new finite element code in C ++ language developed in an open-source framework called Kratos. Finally, the information obtained numerically with our code will be checked with the information obtained with other codes.

Adjoint-based optimization of PDEs in moving domains

NASA Astrophysics Data System (ADS)

Protas, Bartosz; Liao, Wenyuan

2008-02-01

In this investigation we address the problem of adjoint-based optimization of PDE systems in moving domains. As an example we consider the one-dimensional heat equation with prescribed boundary temperatures and heat fluxes. We discuss two methods of deriving an adjoint system necessary to obtain a gradient of a cost functional. In the first approach we derive the adjoint system after mapping the problem to a fixed domain, whereas in the second approach we derive the adjoint directly in the moving domain by employing methods of the noncylindrical calculus. We show that the operations of transforming the system from a variable to a fixed domain and deriving the adjoint do not commute and that, while the gradient information contained in both systems is the same, the second approach results in an adjoint problem with a simpler structure which is therefore easier to implement numerically. This approach is then used to solve a moving boundary optimization problem for our model system.

Photoacoustic image reconstruction via deep learning

NASA Astrophysics Data System (ADS)

Antholzer, Stephan; Haltmeier, Markus; Nuster, Robert; Schwab, Johannes

2018-02-01

Applying standard algorithms to sparse data problems in photoacoustic tomography (PAT) yields low-quality images containing severe under-sampling artifacts. To some extent, these artifacts can be reduced by iterative image reconstruction algorithms which allow to include prior knowledge such as smoothness, total variation (TV) or sparsity constraints. These algorithms tend to be time consuming as the forward and adjoint problems have to be solved repeatedly. Further, iterative algorithms have additional drawbacks. For example, the reconstruction quality strongly depends on a-priori model assumptions about the objects to be recovered, which are often not strictly satisfied in practical applications. To overcome these issues, in this paper, we develop direct and efficient reconstruction algorithms based on deep learning. As opposed to iterative algorithms, we apply a convolutional neural network, whose parameters are trained before the reconstruction process based on a set of training data. For actual image reconstruction, a single evaluation of the trained network yields the desired result. Our presented numerical results (using two different network architectures) demonstrate that the proposed deep learning approach reconstructs images with a quality comparable to state of the art iterative reconstruction methods.

Traveling wave solutions and conservation laws for nonlinear evolution equation

NASA Astrophysics Data System (ADS)

Baleanu, Dumitru; Inc, Mustafa; Yusuf, Abdullahi; Aliyu, Aliyu Isa

2018-02-01

In this work, the Riccati-Bernoulli sub-ordinary differential equation and modified tanh-coth methods are used to reach soliton solutions of the nonlinear evolution equation. We acquire new types of traveling wave solutions for the governing equation. We show that the equation is nonlinear self-adjoint by obtaining suitable substitution. Therefore, we construct conservation laws for the equation using new conservation theorem. The obtained solutions in this work may be used to explain and understand the physical nature of the wave spreads in the most dispersive medium. The constraint condition for the existence of solitons is stated. Some three dimensional figures for some of the acquired results are illustrated.

Source Parameter Estimation using the Second-order Closure Integrated Puff Model

DTIC Science & Technology

The sensor measurements are categorized as triggered and non-triggered based on the recorded concentration measurements and a threshold...concentration value. Using each measured value, sources of adjoint material are created from the triggered and non-triggered sensors, and the adjoint transport...equations are solved to predict the adjoint concentration fields. The adjoint source strength is inversely proportional to the concentration measurement

Multimodel methods for optimal control of aeroacoustics.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Chen, Guoquan; Collis, Samuel Scott

2005-01-01

A new multidomain/multiphysics computational framework for optimal control of aeroacoustic noise has been developed based on a near-field compressible Navier-Stokes solver coupled with a far-field linearized Euler solver both based on a discontinuous Galerkin formulation. In this approach, the coupling of near- and far-field domains is achieved by weakly enforcing continuity of normal fluxes across a coupling surface that encloses all nonlinearities and noise sources. For optimal control, gradient information is obtained by the solution of an appropriate adjoint problem that involves the propagation of adjoint information from the far-field to the near-field. This computational framework has been successfully appliedmore » to study optimal boundary-control of blade-vortex interaction, which is a significant noise source for helicopters on approach to landing. In the model-problem presented here, the noise propagated toward the ground is reduced by 12dB.« less

Periodic differential equations with self-adjoint monodromy operator

NASA Astrophysics Data System (ADS)

Yudovich, V. I.

2001-04-01

A linear differential equation \\dot u=A(t)u with p-periodic (generally speaking, unbounded) operator coefficient in a Euclidean or a Hilbert space \\mathbb H is considered. It is proved under natural constraints that the monodromy operator U_p is self-adjoint and strictly positive if A^*(-t)=A(t) for all t\\in\\mathbb R.It is shown that Hamiltonian systems in the class under consideration are usually unstable and, if they are stable, then the operator U_p reduces to the identity and all solutions are p-periodic.For higher frequencies averaged equations are derived. Remarkably, high-frequency modulation may double the number of critical values.General results are applied to rotational flows with cylindrical components of the velocity a_r=a_z=0, a_\\theta=\\lambda c(t)r^\\beta, \\beta<-1, c(t) is an even p-periodic function, and also to several problems of free gravitational convection of fluids in periodic fields.

Least-Squares PN Formulation of the Transport Equation Using Self-Adjoint-Angular-Flux Consistent Boundary Conditions.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Vincent M. Laboure; Yaqi Wang; Mark D. DeHart

In this paper, we study the Least-Squares (LS) PN form of the transport equation compatible with voids in the context of Continuous Finite Element Methods (CFEM).We first deriveweakly imposed boundary conditions which make the LS weak formulation equivalent to the Self-Adjoint Angular Flux (SAAF) variational formulation with a void treatment, in the particular case of constant cross-sections and a uniform mesh. We then implement this method in Rattlesnake with the Multiphysics Object Oriented Simulation Environment (MOOSE) framework using a spherical harmonics (PN) expansion to discretize in angle. We test our implementation using the Method of Manufactured Solutions (MMS) and findmore » the expected convergence behavior both in angle and space. Lastly, we investigate the impact of the global non-conservation of LS by comparing the method with SAAF on a heterogeneous test problem.« less

Toward Automatic Verification of Goal-Oriented Flow Simulations

NASA Technical Reports Server (NTRS)

Nemec, Marian; Aftosmis, Michael J.

2014-01-01

We demonstrate the power of adaptive mesh refinement with adjoint-based error estimates in verification of simulations governed by the steady Euler equations. The flow equations are discretized using a finite volume scheme on a Cartesian mesh with cut cells at the wall boundaries. The discretization error in selected simulation outputs is estimated using the method of adjoint-weighted residuals. Practical aspects of the implementation are emphasized, particularly in the formulation of the refinement criterion and the mesh adaptation strategy. Following a thorough code verification example, we demonstrate simulation verification of two- and three-dimensional problems. These involve an airfoil performance database, a pressure signature of a body in supersonic flow and a launch abort with strong jet interactions. The results show reliable estimates and automatic control of discretization error in all simulations at an affordable computational cost. Moreover, the approach remains effective even when theoretical assumptions, e.g., steady-state and solution smoothness, are relaxed.

Numerical solution of inverse scattering for near-field optics.

PubMed

Bao, Gang; Li, Peijun

2007-06-01

A novel regularized recursive linearization method is developed for a two-dimensional inverse medium scattering problem that arises in near-field optics, which reconstructs the scatterer of an inhomogeneous medium located on a substrate from data accessible through photon scanning tunneling microscopy experiments. Based on multiple frequency scattering data, the method starts from the Born approximation corresponding to weak scattering at a low frequency, and each update is obtained by continuation on the wavenumber from solutions of one forward problem and one adjoint problem of the Helmholtz equation.

Analysis and development of adjoint-based h-adaptive direct discontinuous Galerkin method for the compressible Navier-Stokes equations

NASA Astrophysics Data System (ADS)

Cheng, Jian; Yue, Huiqiang; Yu, Shengjiao; Liu, Tiegang

2018-06-01

In this paper, an adjoint-based high-order h-adaptive direct discontinuous Galerkin method is developed and analyzed for the two dimensional steady state compressible Navier-Stokes equations. Particular emphasis is devoted to the analysis of the adjoint consistency for three different direct discontinuous Galerkin discretizations: including the original direct discontinuous Galerkin method (DDG), the direct discontinuous Galerkin method with interface correction (DDG(IC)) and the symmetric direct discontinuous Galerkin method (SDDG). Theoretical analysis shows the extra interface correction term adopted in the DDG(IC) method and the SDDG method plays a key role in preserving the adjoint consistency. To be specific, for the model problem considered in this work, we prove that the original DDG method is not adjoint consistent, while the DDG(IC) method and the SDDG method can be adjoint consistent with appropriate treatment of boundary conditions and correct modifications towards the underlying output functionals. The performance of those three DDG methods is carefully investigated and evaluated through typical test cases. Based on the theoretical analysis, an adjoint-based h-adaptive DDG(IC) method is further developed and evaluated, numerical experiment shows its potential in the applications of adjoint-based adaptation for simulating compressible flows.

Universal Racah matrices and adjoint knot polynomials: Arborescent knots

NASA Astrophysics Data System (ADS)

Mironov, A.; Morozov, A.

2016-04-01

By now it is well established that the quantum dimensions of descendants of the adjoint representation can be described in a universal form, independent of a particular family of simple Lie algebras. The Rosso-Jones formula then implies a universal description of the adjoint knot polynomials for torus knots, which in particular unifies the HOMFLY (SUN) and Kauffman (SON) polynomials. For E8 the adjoint representation is also fundamental. We suggest to extend the universality from the dimensions to the Racah matrices and this immediately produces a unified description of the adjoint knot polynomials for all arborescent (double-fat) knots, including twist, 2-bridge and pretzel. Technically we develop together the universality and the "eigenvalue conjecture", which expresses the Racah and mixing matrices through the eigenvalues of the quantum R-matrix, and for dealing with the adjoint polynomials one has to extend it to the previously unknown 6 × 6 case. The adjoint polynomials do not distinguish between mutants and therefore are not very efficient in knot theory, however, universal polynomials in higher representations can probably be better in this respect.

Least Squares Shadowing sensitivity analysis of chaotic limit cycle oscillations

DOE Office of Scientific and Technical Information (OSTI.GOV)

Wang, Qiqi, E-mail: qiqi@mit.edu; Hu, Rui, E-mail: hurui@mit.edu; Blonigan, Patrick, E-mail: blonigan@mit.edu

2014-06-15

The adjoint method, among other sensitivity analysis methods, can fail in chaotic dynamical systems. The result from these methods can be too large, often by orders of magnitude, when the result is the derivative of a long time averaged quantity. This failure is known to be caused by ill-conditioned initial value problems. This paper overcomes this failure by replacing the initial value problem with the well-conditioned “least squares shadowing (LSS) problem”. The LSS problem is then linearized in our sensitivity analysis algorithm, which computes a derivative that converges to the derivative of the infinitely long time average. We demonstrate ourmore » algorithm in several dynamical systems exhibiting both periodic and chaotic oscillations.« less

The DANTE Boltzmann transport solver: An unstructured mesh, 3-D, spherical harmonics algorithm compatible with parallel computer architectures

DOE Office of Scientific and Technical Information (OSTI.GOV)

McGhee, J.M.; Roberts, R.M.; Morel, J.E.

1997-06-01

A spherical harmonics research code (DANTE) has been developed which is compatible with parallel computer architectures. DANTE provides 3-D, multi-material, deterministic, transport capabilities using an arbitrary finite element mesh. The linearized Boltzmann transport equation is solved in a second order self-adjoint form utilizing a Galerkin finite element spatial differencing scheme. The core solver utilizes a preconditioned conjugate gradient algorithm. Other distinguishing features of the code include options for discrete-ordinates and simplified spherical harmonics angular differencing, an exact Marshak boundary treatment for arbitrarily oriented boundary faces, in-line matrix construction techniques to minimize memory consumption, and an effective diffusion based preconditioner formore » scattering dominated problems. Algorithm efficiency is demonstrated for a massively parallel SIMD architecture (CM-5), and compatibility with MPP multiprocessor platforms or workstation clusters is anticipated.« less

Feedback control for unsteady flow and its application to the stochastic Burgers equation

NASA Technical Reports Server (NTRS)

Choi, Haecheon; Temam, Roger; Moin, Parviz; Kim, John

1993-01-01

The study applies mathematical methods of control theory to the problem of control of fluid flow with the long-range objective of developing effective methods for the control of turbulent flows. Model problems are employed through the formalism and language of control theory to present the procedure of how to cast the problem of controlling turbulence into a problem in optimal control theory. Methods of calculus of variations through the adjoint state and gradient algorithms are used to present a suboptimal control and feedback procedure for stationary and time-dependent problems. Two types of controls are investigated: distributed and boundary controls. Several cases of both controls are numerically simulated to investigate the performances of the control algorithm. Most cases considered show significant reductions of the costs to be minimized. The dependence of the control algorithm on the time-descretization method is discussed.

Self-adjointness of deformed unbounded operators

DOE Office of Scientific and Technical Information (OSTI.GOV)

Much, Albert

2015-09-15

We consider deformations of unbounded operators by using the novel construction tool of warped convolutions. By using the Kato-Rellich theorem, we show that unbounded self-adjoint deformed operators are self-adjoint if they satisfy a certain condition. This condition proves itself to be necessary for the oscillatory integral to be well-defined. Moreover, different proofs are given for self-adjointness of deformed unbounded operators in the context of quantum mechanics and quantum field theory.

Higher representations on the lattice: Numerical simulations, SU(2) with adjoint fermions

DOE Office of Scientific and Technical Information (OSTI.GOV)

Del Debbio, Luigi; Patella, Agostino; Pica, Claudio

2010-05-01

We discuss the lattice formulation of gauge theories with fermions in arbitrary representations of the color group and present in detail the implementation of the hybrid Monte Carlo (HMC)/rational HMC algorithm for simulating dynamical fermions. We discuss the validation of the implementation through an extensive set of tests and the stability of simulations by monitoring the distribution of the lowest eigenvalue of the Wilson-Dirac operator. Working with two flavors of Wilson fermions in the adjoint representation, benchmark results for realistic lattice simulations are presented. Runs are performed on different lattice sizes ranging from 4{sup 3}x8 to 24{sup 3}x64 sites. Formore » the two smallest lattices we also report the measured values of benchmark mesonic observables. These results can be used as a baseline for rapid cross-checks of simulations in higher representations. The results presented here are the first steps toward more extensive investigations with controlled systematic errors, aiming at a detailed understanding of the phase structure of these theories, and of their viability as candidates for strong dynamics beyond the standard model.« less

Sensitivity of Lumped Constraints Using the Adjoint Method

NASA Technical Reports Server (NTRS)

Akgun, Mehmet A.; Haftka, Raphael T.; Wu, K. Chauncey; Walsh, Joanne L.

1999-01-01

Adjoint sensitivity calculation of stress, buckling and displacement constraints may be much less expensive than direct sensitivity calculation when the number of load cases is large. Adjoint stress and displacement sensitivities are available in the literature. Expressions for local buckling sensitivity of isotropic plate elements are derived in this study. Computational efficiency of the adjoint method is sensitive to the number of constraints and, therefore, the method benefits from constraint lumping. A continuum version of the Kreisselmeier-Steinhauser (KS) function is chosen to lump constraints. The adjoint and direct methods are compared for three examples: a truss structure, a simple HSCT wing model, and a large HSCT model. These sensitivity derivatives are then used in optimization.

Almost commuting self-adjoint matrices: The real and self-dual cases

NASA Astrophysics Data System (ADS)

Loring, Terry A.; Sørensen, Adam P. W.

2016-08-01

We show that a pair of almost commuting self-adjoint, symmetric matrices is close to a pair of commuting self-adjoint, symmetric matrices (in a uniform way). Moreover, we prove that the same holds with self-dual in place of symmetric and also for paths of self-adjoint matrices. Since a symmetric, self-adjoint matrix is real, we get a real version of Huaxin Lin’s famous theorem on almost commuting matrices. Similarly, the self-dual case gives a version for matrices over the quaternions. To prove these results, we develop a theory of semiprojectivity for real C*-algebras and also examine various definitions of low-rank for real C*-algebras.

A numerical scheme to solve unstable boundary value problems

NASA Technical Reports Server (NTRS)

Kalnay-Rivas, E.

1977-01-01

The considered scheme makes it possible to determine an unstable steady state solution in cases in which, because of lack of symmetry, such a solution cannot be obtained analytically, and other time integration or relaxation schemes, because of instability, fail to converge. The iterative solution of a single complex equation is discussed and a nonlinear system of equations is considered. Described applications of the scheme are related to a steady state solution with shear instability, an unstable nonlinear Ekman boundary layer, and the steady state solution of a baroclinic atmosphere with asymmetric forcing. The scheme makes use of forward and backward time integrations of the original spatial differential operators and of an approximation of the adjoint operators. Only two computations of the time derivative per iteration are required.

Assessing the Tangent Linear Behaviour of Common Tracer Transport Schemes and Their Use in a Linearised Atmospheric General Circulation Model

NASA Technical Reports Server (NTRS)

Holdaway, Daniel; Kent, James

2015-01-01

The linearity of a selection of common advection schemes is tested and examined with a view to their use in the tangent linear and adjoint versions of an atmospheric general circulation model. The schemes are tested within a simple offline one-dimensional periodic domain as well as using a simplified and complete configuration of the linearised version of NASA's Goddard Earth Observing System version 5 (GEOS-5). All schemes which prevent the development of negative values and preserve the shape of the solution are confirmed to have nonlinear behaviour. The piecewise parabolic method (PPM) with certain flux limiters, including that used by default in GEOS-5, is found to support linear growth near the shocks. This property can cause the rapid development of unrealistically large perturbations within the tangent linear and adjoint models. It is shown that these schemes with flux limiters should not be used within the linearised version of a transport scheme. The results from tests using GEOS-5 show that the current default scheme (a version of PPM) is not suitable for the tangent linear and adjoint model, and that using a linear third-order scheme for the linearised model produces better behaviour. Using the third-order scheme for the linearised model improves the correlations between the linear and non-linear perturbation trajectories for cloud liquid water and cloud liquid ice in GEOS-5.

Towards adjoint-based inversion of time-dependent mantle convection with nonlinear viscosity

NASA Astrophysics Data System (ADS)

Li, Dunzhu; Gurnis, Michael; Stadler, Georg

2017-04-01

We develop and study an adjoint-based inversion method for the simultaneous recovery of initial temperature conditions and viscosity parameters in time-dependent mantle convection from the current mantle temperature and historic plate motion. Based on a realistic rheological model with temperature-dependent and strain-rate-dependent viscosity, we formulate the inversion as a PDE-constrained optimization problem. The objective functional includes the misfit of surface velocity (plate motion) history, the misfit of the current mantle temperature, and a regularization for the uncertain initial condition. The gradient of this functional with respect to the initial temperature and the uncertain viscosity parameters is computed by solving the adjoint of the mantle convection equations. This gradient is used in a pre-conditioned quasi-Newton minimization algorithm. We study the prospects and limitations of the inversion, as well as the computational performance of the method using two synthetic problems, a sinking cylinder and a realistic subduction model. The subduction model is characterized by the migration of a ridge toward a trench whereby both plate motions and subduction evolve. The results demonstrate: (1) for known viscosity parameters, the initial temperature can be well recovered, as in previous initial condition-only inversions where the effective viscosity was given; (2) for known initial temperature, viscosity parameters can be recovered accurately, despite the existence of trade-offs due to ill-conditioning; (3) for the joint inversion of initial condition and viscosity parameters, initial condition and effective viscosity can be reasonably recovered, but the high dimension of the parameter space and the resulting ill-posedness may limit recovery of viscosity parameters.

A hybrid approach to generating search subspaces in dynamically constrained 4-dimensional data assimilation

NASA Astrophysics Data System (ADS)

Yaremchuk, Max; Martin, Paul; Beattie, Christopher

2017-09-01

Development and maintenance of the linearized and adjoint code for advanced circulation models is a challenging issue, requiring a significant proportion of total effort in operational data assimilation (DA). The ensemble-based DA techniques provide a derivative-free alternative, which appears to be competitive with variational methods in many practical applications. This article proposes a hybrid scheme for generating the search subspaces in the adjoint-free 4-dimensional DA method (a4dVar) that does not use a predefined ensemble. The method resembles 4dVar in that the optimal solution is strongly constrained by model dynamics and search directions are supplied iteratively using information from the current and previous model trajectories generated in the process of optimization. In contrast to 4dVar, which produces a single search direction from exact gradient information, a4dVar employs an ensemble of directions to form a subspace in order to proceed. In the earlier versions of a4dVar, search subspaces were built using the leading EOFs of either the model trajectory or the projections of the model-data misfits onto the range of the background error covariance (BEC) matrix at the current iteration. In the present study, we blend both approaches and explore a hybrid scheme of ensemble generation in order to improve the performance and flexibility of the algorithm. In addition, we introduce balance constraints into the BEC structure and periodically augment the search ensemble with BEC eigenvectors to avoid repeating minimization over already explored subspaces. Performance of the proposed hybrid a4dVar (ha4dVar) method is compared with that of standard 4dVar in a realistic regional configuration assimilating real data into the Navy Coastal Ocean Model (NCOM). It is shown that the ha4dVar converges faster than a4dVar and can be potentially competitive with 4dvar both in terms of the required computational time and the forecast skill.

3D CSEM inversion based on goal-oriented adaptive finite element method

NASA Astrophysics Data System (ADS)

Zhang, Y.; Key, K.

2016-12-01

We present a parallel 3D frequency domain controlled-source electromagnetic inversion code name MARE3DEM. Non-linear inversion of observed data is performed with the Occam variant of regularized Gauss-Newton optimization. The forward operator is based on the goal-oriented finite element method that efficiently calculates the responses and sensitivity kernels in parallel using a data decomposition scheme where independent modeling tasks contain different frequencies and subsets of the transmitters and receivers. To accommodate complex 3D conductivity variation with high flexibility and precision, we adopt the dual-grid approach where the forward mesh conforms to the inversion parameter grid and is adaptively refined until the forward solution converges to the desired accuracy. This dual-grid approach is memory efficient, since the inverse parameter grid remains independent from fine meshing generated around the transmitter and receivers by the adaptive finite element method. Besides, the unstructured inverse mesh efficiently handles multiple scale structures and allows for fine-scale model parameters within the region of interest. Our mesh generation engine keeps track of the refinement hierarchy so that the map of conductivity and sensitivity kernel between the forward and inverse mesh is retained. We employ the adjoint-reciprocity method to calculate the sensitivity kernels which establish a linear relationship between changes in the conductivity model and changes in the modeled responses. Our code uses a direcy solver for the linear systems, so the adjoint problem is efficiently computed by re-using the factorization from the primary problem. Further computational efficiency and scalability is obtained in the regularized Gauss-Newton portion of the inversion using parallel dense matrix-matrix multiplication and matrix factorization routines implemented with the ScaLAPACK library. We show the scalability, reliability and the potential of the algorithm to deal with complex geological scenarios by applying it to the inversion of synthetic marine controlled source EM data generated for a complex 3D offshore model with significant seafloor topography.

Development and Applications of the FV3 GEOS-5 Adjoint Modeling System

NASA Technical Reports Server (NTRS)

Holdaway, Daniel; Kim, Jong G.; Lin, Shian-Jiann; Errico, Ron; Gelaro, Ron; Kent, James; Coy, Larry; Doyle, Jim; Goldstein, Alex

2017-01-01

GMAO has developed a highly sophisticated adjoint modeling system based on the most recent version of the finite volume cubed sphere (FV3) dynamical core. This provides a mechanism for investigating sensitivity to initial conditions and examining observation impacts. It also allows for the computation of singular vectors and for the implementation of hybrid 4DVAR. In this work we will present the scientific assessment of the new adjoint system and show results from a number of research application of the adjoint system.

Skyshine analysis using energy and angular dependent dose-contribution fluxes obtained from air-over-ground adjoint calculation.

PubMed

Uematsu, Mikio; Kurosawa, Masahiko

2005-01-01

A generalised and convenient skyshine dose analysis method has been developed based on forward-adjoint folding technique. In the method, the air penetration data were prepared by performing an adjoint DOT3.5 calculation with cylindrical air-over-ground geometry having an adjoint point source (importance of unit flux to dose rate at detection point) in the centre. The accuracy of the present method was certified by comparing with DOT3.5 forward calculation. The adjoint flux data can be used as generalised radiation skyshine data for all sorts of nuclear facilities. Moreover, the present method supplies plenty of energy-angular dependent contribution flux data, which will be useful for detailed shielding design of facilities.

Reconfigurable Model Execution in the OpenMDAO Framework

NASA Technical Reports Server (NTRS)

Hwang, John T.

2017-01-01

NASA's OpenMDAO framework facilitates constructing complex models and computing their derivatives for multidisciplinary design optimization. Decomposing a model into components that follow a prescribed interface enables OpenMDAO to assemble multidisciplinary derivatives from the component derivatives using what amounts to the adjoint method, direct method, chain rule, global sensitivity equations, or any combination thereof, using the MAUD architecture. OpenMDAO also handles the distribution of processors among the disciplines by hierarchically grouping the components, and it automates the data transfer between components that are on different processors. These features have made OpenMDAO useful for applications in aircraft design, satellite design, wind turbine design, and aircraft engine design, among others. This paper presents new algorithms for OpenMDAO that enable reconfigurable model execution. This concept refers to dynamically changing, during execution, one or more of: the variable sizes, solution algorithm, parallel load balancing, or set of variables-i.e., adding and removing components, perhaps to switch to a higher-fidelity sub-model. Any component can reconfigure at any point, even when running in parallel with other components, and the reconfiguration algorithm presented here performs the synchronized updates to all other components that are affected. A reconfigurable software framework for multidisciplinary design optimization enables new adaptive solvers, adaptive parallelization, and new applications such as gradient-based optimization with overset flow solvers and adaptive mesh refinement. Benchmarking results demonstrate the time savings for reconfiguration compared to setting up the model again from scratch, which can be significant in large-scale problems. Additionally, the new reconfigurability feature is applied to a mission profile optimization problem for commercial aircraft where both the parametrization of the mission profile and the time discretization are adaptively refined, resulting in computational savings of roughly 10% and the elimination of oscillations in the optimized altitude profile.

Simulation of a class of hazardous situations in the ICS «INM RAS - Baltic Sea»

NASA Astrophysics Data System (ADS)

Zakharova, Natalia; Agoshkov, Valery; Aseev, Nikita; Parmuzin, Eugene; Sheloput, Tateana; Shutyaev, Victor

2017-04-01

Development of Informational Computational Systems (ICS) for data assimilation procedures is one of multidisciplinary problems. To study and solve these problems one needs to apply modern results from different disciplines and recent developments in mathematical modeling, theory of adjoint equations and optimal control, inverse problems, numerical methods theory, numerical algebra, scientific computing and processing of satellite data. In this work the results on the ICS development for PC-ICS "INM RAS - Baltic Sea" are presented. We discuss practical problems studied by ICS. The System includes numerical model of the Baltic Sea thermodynamics, the new oil spill model describing the propagation of a slick at the Sea surface (Agoshkov, Aseev et al., 2014) and the optimal ship route calculating block (Agoshkov, Zayachkovsky et al., 2014). The ICS is based on the INMOM numerical model of the Baltic Sea thermodynamics (Zalesny et al., 2013). It is possible to calculate main hydrodynamic parameters (temperature, salinity, velocities, sea level) using user-friendly interface of the ICS. The System includes data assimilation procedures (Agoshkov, 2003, Parmuzin, Agoshkov, 2012) and one can use the block of variational assimilation of the sea surface temperature in order to obtain main hydrodynamic parameters. Main possibilities of the ICS and several numerical experiments are presented in the work. By the problem of risk control is meant a problem of determination of optimal resources quantity which are necessary for decreasing the risk to some acceptable value. Mass of oil slick is chosen as a function of control. For the realization of the random variable the quadratic "functional of cost" is introduced. It comprises cleaning costs and deviation of damage of oil pollution from its acceptable value. The problem of minimization of this functional is solved based on the methods of optimal control and the theory of adjoint equations. The solution of this problem is explicitly found. The study was supported by the Russian Foundation for Basic Research (project 16-31-00510) and by the Russian Science Foundation (project №14-11-00609). V. I. Agoshkov, Methods of Optimal Control and Adjoint Equations in Problems of Mathematical Physics. INM RAS, Moscow, 2003 (in Russian). V. B. Zalesny, A. V. Gusev, V. O. Ivchenko, R. Tamsalu, and R. Aps, Numerical model of the Baltic Sea circulation. Russ. J. Numer. Anal. Math. Modelling 28 (2013), No. 1, 85-100. V.I. Agoshkov, A.O. Zayachkovskiy, R. Aps, P. Kujala, and J. Rytkönen. Risk theory based solution to the problem of optimal vessel route // Russian Journal of Numerical Analysis and Mathematical Modelling. 2014. Volume 29, Issue 2, Pages 69-78. Agoshkov, V., Aseev, N., Aps, R., Kujala, P., Rytkönen, J., Zalesny, V. The problem of control of oil pollution risk in the Baltic Sea // Russian Journal of Numerical Analysis and Mathematical Modelling. 2014. Volume 29, Issue 2, Pages 93-105. E. I. Parmuzin and V. I. Agoshkov, Numerical solution of the variational assimilation problem for sea surface temperature in the model of the Black Sea dynamics. Russ. J. Numer. Anal. Math. Modelling 27 (2012), No. 1, 69-94. Olof Liungman and Johan Mattsson. Scientic Documentation of Seatrack Web; physical processes, algorithms and references, 2011.

Tangent Adjoint Methods In a Higher-Order Space-Time Discontinuous-Galerkin Solver For Turbulent Flows

NASA Technical Reports Server (NTRS)

Diosady, Laslo; Murman, Scott; Blonigan, Patrick; Garai, Anirban

2017-01-01

Presented space-time adjoint solver for turbulent compressible flows. Confirmed failure of traditional sensitivity methods for chaotic flows. Assessed rate of exponential growth of adjoint for practical 3D turbulent simulation. Demonstrated failure of short-window sensitivity approximations.

Topology optimization of thermal fluid flows with an adjoint Lattice Boltzmann Method

NASA Astrophysics Data System (ADS)

Dugast, Florian; Favennec, Yann; Josset, Christophe; Fan, Yilin; Luo, Lingai

2018-07-01

This paper presents an adjoint Lattice Boltzmann Method (LBM) coupled with the Level-Set Method (LSM) for topology optimization of thermal fluid flows. The adjoint-state formulation implies discrete velocity directions in order to take into account the LBM boundary conditions. These boundary conditions are introduced at the beginning of the adjoint-state method as the LBM residuals, so that the adjoint-state boundary conditions can appear directly during the adjoint-state equation formulation. The proposed method is tested with 3 numerical examples concerning thermal fluid flows, but with different objectives: minimization of the mean temperature in the domain, maximization of the heat evacuated by the fluid, and maximization of the heat exchange with heated solid parts. This latter example, treated in several articles, is used to validate our method. In these optimization problems, a limitation of the maximal pressure drop and of the porosity (number of fluid elements) is also applied. The obtained results demonstrate that the method is robust and effective for solving topology optimization of thermal fluid flows.

Adjoint-Based Methodology for Time-Dependent Optimization

NASA Technical Reports Server (NTRS)

Yamaleev, N. K.; Diskin, B.; Nielsen, E. J.

2008-01-01

This paper presents a discrete adjoint method for a broad class of time-dependent optimization problems. The time-dependent adjoint equations are derived in terms of the discrete residual of an arbitrary finite volume scheme which approximates unsteady conservation law equations. Although only the 2-D unsteady Euler equations are considered in the present analysis, this time-dependent adjoint method is applicable to the 3-D unsteady Reynolds-averaged Navier-Stokes equations with minor modifications. The discrete adjoint operators involving the derivatives of the discrete residual and the cost functional with respect to the flow variables are computed using a complex-variable approach, which provides discrete consistency and drastically reduces the implementation and debugging cycle. The implementation of the time-dependent adjoint method is validated by comparing the sensitivity derivative with that obtained by forward mode differentiation. Our numerical results show that O(10) optimization iterations of the steepest descent method are needed to reduce the objective functional by 3-6 orders of magnitude for test problems considered.

Airfoil Design Using a Coupled Euler and Integral Boundary Layer Method with Adjoint Based Sensitivities

NASA Technical Reports Server (NTRS)

Edwards, S.; Reuther, J.; Chattot, J. J.

1997-01-01

The objective of this paper is to present a control theory approach for the design of airfoils in the presence of viscous compressible flows. A coupled system of the integral boundary layer and the Euler equations is solved to provide rapid flow simulations. An adjunct approach consistent with the complete coupled state equations is employed to obtain the sensitivities needed to drive a numerical optimization algorithm. Design to target pressure distribution is demonstrated on an RAE 2822 airfoil at transonic speed.

A fast finite-difference algorithm for topology optimization of permanent magnets

NASA Astrophysics Data System (ADS)

Abert, Claas; Huber, Christian; Bruckner, Florian; Vogler, Christoph; Wautischer, Gregor; Suess, Dieter

2017-09-01

We present a finite-difference method for the topology optimization of permanent magnets that is based on the fast-Fourier-transform (FFT) accelerated computation of the stray-field. The presented method employs the density approach for topology optimization and uses an adjoint method for the gradient computation. Comparison to various state-of-the-art finite-element implementations shows a superior performance and accuracy. Moreover, the presented method is very flexible and easy to implement due to various preexisting FFT stray-field implementations that can be used.

Self-dual monopoles and toda molecules

NASA Astrophysics Data System (ADS)

Ganoulis, N.; Goddard, P.; Olive, D.

1982-07-01

Stable static solutions to a gauge field theory with a Higgs field in the adjoint representation and with vanishing self-coupling are self-dual in the sense of Bogomolny. Leznov and Saveliev showed that a specific form of spherical symmetry reduces these equations to a modified form of the Toda molecule equations associated with the overall gauge symmetry G. Values of the constants of integration are found in terms of the distant Higgs field, guaranteeing regularity of the solution at the origin. The expressions hold for any simple Lie group G, depending on G via its root system.

Convective heat transfer for a gaseous slip flow in micropipe and parallel-plate microchannel with uniform wall heat flux: effect of axial heat conduction

NASA Astrophysics Data System (ADS)

Haddout, Y.; Essaghir, E.; Oubarra, A.; Lahjomri, J.

2017-12-01

Thermally developing laminar slip flow through a micropipe and a parallel plate microchannel, with axial heat conduction and uniform wall heat flux, is studied analytically by using a powerful method of self-adjoint formalism. This method results from a decomposition of the elliptic energy equation into a system of two first-order partial differential equations. The advantage of this method over other methods, resides in the fact that the decomposition procedure leads to a selfadjoint problem although the initial problem is apparently not a self-adjoint one. The solution is an extension of prior studies and considers a first order slip model boundary conditions at the fluid-wall interface. The analytical expressions for the developing temperature and local Nusselt number in the thermal entrance region are obtained in the general case. Therefore, the solution obtained could be extended easily to any hydrodynamically developed flow and arbitrary heat flux distribution. The analytical results obtained are compared for select simplified cases with available numerical calculations and they both agree. The results show that the heat transfer characteristics of flow in the thermal entrance region are strongly influenced by the axial heat conduction and rarefaction effects which are respectively characterized by Péclet and Knudsen numbers.

Convective heat transfer for a gaseous slip flow in micropipe and parallel-plate microchannel with uniform wall heat flux: effect of axial heat conduction

NASA Astrophysics Data System (ADS)

Haddout, Y.; Essaghir, E.; Oubarra, A.; Lahjomri, J.

2018-06-01

Thermally developing laminar slip flow through a micropipe and a parallel plate microchannel, with axial heat conduction and uniform wall heat flux, is studied analytically by using a powerful method of self-adjoint formalism. This method results from a decomposition of the elliptic energy equation into a system of two first-order partial differential equations. The advantage of this method over other methods, resides in the fact that the decomposition procedure leads to a selfadjoint problem although the initial problem is apparently not a self-adjoint one. The solution is an extension of prior studies and considers a first order slip model boundary conditions at the fluid-wall interface. The analytical expressions for the developing temperature and local Nusselt number in the thermal entrance region are obtained in the general case. Therefore, the solution obtained could be extended easily to any hydrodynamically developed flow and arbitrary heat flux distribution. The analytical results obtained are compared for select simplified cases with available numerical calculations and they both agree. The results show that the heat transfer characteristics of flow in the thermal entrance region are strongly influenced by the axial heat conduction and rarefaction effects which are respectively characterized by Péclet and Knudsen numbers.

A Minimum Fuel Based Estimator for Maneuver and Natrual Dynamics Reconstruction

NASA Astrophysics Data System (ADS)

Lubey, D.; Scheeres, D.

2013-09-01

The vast and growing population of objects in Earth orbit (active and defunct spacecraft, orbital debris, etc.) offers many unique challenges when it comes to tracking these objects and associating the resulting observations. Complicating these challenges are the inaccurate natural dynamical models of these objects, the active maneuvers of spacecraft that deviate them from their ballistic trajectories, and the fact that spacecraft are tracked and operated by separate agencies. Maneuver detection and reconstruction algorithms can help with each of these issues by estimating mismodeled and unmodeled dynamics through indirect observation of spacecraft. It also helps to verify the associations made by an object correlation algorithm or aid in making those associations, which is essential when tracking objects in orbit. The algorithm developed in this study applies an Optimal Control Problem (OCP) Distance Metric approach to the problems of Maneuver Reconstruction and Dynamics Estimation. This was first developed by Holzinger, Scheeres, and Alfriend (2011), with a subsequent study by Singh, Horwood, and Poore (2012). This method estimates the minimum fuel control policy rather than the state as a typical Kalman Filter would. This difference ensures that the states are connected through a given dynamical model and allows for automatic covariance manipulation, which can help to prevent filter saturation. Using a string of measurements (either verified or hypothesized to correlate with one another), the algorithm outputs a corresponding string of adjoint and state estimates with associated noise. Post-processing techniques are implemented, which when applied to the adjoint estimates can remove noise and expose unmodeled maneuvers and mismodeled natural dynamics. Specifically, the estimated controls are used to determine spacecraft dependent accelerations (atmospheric drag and solar radiation pressure) using an adapted form of the Optimal Control based natural dynamics estimation scheme developed by Lubey and Scheeres (2012). In order to allow for direct comparison, the estimator developed here was modeled after a typical Kalman Filter. The estimator forces the terminal state to lie on a manifold that satisfies the least squares with a priori information cost function, thus establishing a link with a typical Kalman filter. Terms are collected into a pseudo-Kalman Gain, which creates an equivalent form in the state estimates and covariances between the two estimators. While the two estimators share common roots, the inclusion of control in the Minimum Fuel Estimator gives it special properties. For instance, the inclusion of adjoint noise can help to automatically prevent filter saturation in a manner similar to a State Noise Compensation Algorithm. This property is quite important when considering dynamics mismodeling as filter saturation will cause estimate divergence for mismodeled systems. Additional properties and alternative forms of the estimator are also explored in this study. Several implementations of this estimator are given in this paper. It is applied to LEO, GEO, and GTO orbits with drag and SRP mismodeling. The inclusion of unmodeled maneuvers is also considered. These numerical simulations verify the mathematical properties of this estimator, and demonstrate the advantages that this estimator has over typical Kalman Filters.

Improved Adjoint-Operator Learning For A Neural Network

NASA Technical Reports Server (NTRS)

Toomarian, Nikzad; Barhen, Jacob

1995-01-01

Improved method of adjoint-operator learning reduces amount of computation and associated computational memory needed to make electronic neural network learn temporally varying pattern (e.g., to recognize moving object in image) in real time. Method extension of method described in "Adjoint-Operator Learning for a Neural Network" (NPO-18352).

Adjoint method and runaway electron avalanche

DOE PAGES

Liu, Chang; Brennan, Dylan P.; Boozer, Allen H.; ...

2016-12-16

The adjoint method for the study of runaway electron dynamics in momentum space Liu et al (2016 Phys. Plasmas 23 010702) is rederived using the Green's function method, for both the runaway probability function (RPF) and the expected loss time (ELT). The RPF and ELT obtained using the adjoint method are presented, both with and without the synchrotron radiation reaction force. In conclusion, the adjoint method is then applied to study the runaway electron avalanche. Both the critical electric field and the growth rate for the avalanche are calculated using this fast and novel approach.

Sensitivities of Greenland ice sheet volume inferred from an ice sheet adjoint model

NASA Astrophysics Data System (ADS)

Heimbach, P.; Bugnion, V.

2009-04-01

We present a new and original approach to understanding the sensitivity of the Greenland ice sheet to key model parameters and environmental conditions. At the heart of this approach is the use of an adjoint ice sheet model. Since its introduction by MacAyeal (1992), the adjoint method has become widespread to fit ice stream models to the increasing number and diversity of satellite observations, and to estimate uncertain model parameters such as basal conditions. However, no attempt has been made to extend this method to comprehensive ice sheet models. As a first step toward the use of adjoints of comprehensive three-dimensional ice sheet models we have generated an adjoint of the ice sheet model SICOPOLIS of Greve (1997). The adjoint was generated by means of the automatic differentiation (AD) tool TAF. The AD tool generates exact source code representing the tangent linear and adjoint model of the nonlinear parent model provided. Model sensitivities are given by the partial derivatives of a scalar-valued model diagnostic with respect to the controls, and can be efficiently calculated via the adjoint. By way of example, we determine the sensitivity of the total Greenland ice volume to various control variables, such as spatial fields of basal flow parameters, surface and basal forcings, and initial conditions. Reliability of the adjoint was tested through finite-difference perturbation calculations for various control variables and perturbation regions. Besides confirming qualitative aspects of ice sheet sensitivities, such as expected regional variations, we detect regions where model sensitivities are seemingly unexpected or counter-intuitive, albeit ``real'' in the sense of actual model behavior. An example is inferred regions where sensitivities of ice sheet volume to basal sliding coefficient are positive, i.e. where a local increase in basal sliding parameter increases the ice sheet volume. Similarly, positive ice temperature sensitivities in certain parts of the ice sheet are found (in most regions it is negativ, i.e. an increase in temperature decreases ice sheet volume), the detection of which seems highly unlikely if only conventional perturbation experiments had been used. An effort to generate an efficient adjoint with the newly developed open-source AD tool OpenAD is also under way. Available adjoint code generation tools now open up a variety of novel model applications, notably with regard to sensitivity and uncertainty analyses and ice sheet state estimation or data assimilation.

Seismic Full Waveform Modeling & Imaging in Attenuating Media

NASA Astrophysics Data System (ADS)

Guo, Peng

Seismic attenuation strongly affects seismic waveforms by amplitude loss and velocity dispersion. Without proper inclusion of Q parameters, errors can be introduced for seismic full waveform modeling and imaging. Three different (Carcione's, Robertsson's, and the generalized Robertsson's) isotropic viscoelastic wave equations based on the generalized standard linear solid (GSLS) are evaluated. The second-order displacement equations are derived, and used to demonstrate that, with the same stress relaxation times, these viscoelastic formulations are equivalent. By introducing separate memory variables for P and S relaxation functions, Robertsson's formulation is generalized to allow different P and S wave stress relaxation times, which improves the physical consistency of the Qp and Qs modelled in the seismograms.The three formulations have comparable computational cost. 3D seismic finite-difference forward modeling is applied to anisotropic viscoelastic media. The viscoelastic T-matrix (a dynamic effective medium theory) relates frequency-dependent anisotropic attenuation and velocity to reservoir properties in fractured HTI media, based on the meso-scale fluid flow attenuation mechanism. The seismic signatures resulting from changing viscoelastic reservoir properties are easily visible. Analysis of 3D viscoelastic seismograms suggests that anisotropic attenuation is a potential tool for reservoir characterization. To compensate the Q effects during reverse-time migration (RTM) in viscoacoustic and viscoelastic media, amplitudes need to be compensated during wave propagation; the propagation velocity of the Q-compensated wavefield needs to be the same as in the attenuating wavefield, to restore the phase information. Both amplitude and phase can be compensated when the velocity dispersion and the amplitude loss are decoupled. For wave equations based on the GSLS, because Q effects are coupled in the memory variables, Q-compensated wavefield propagates faster than the attenuating wavefield, and introduce unwanted phase shift. Numerical examples show that there are phase (depth) shifts in the Q-compensated RTM images from the GSLS equation. An adjoint-based least-squares reverse-time migration is proposed for viscoelastic media (Q-LSRTM), to compensate the attenuation losses in P and S images. The viscoelastic adjoint operator, and the P and S modulus perturbation imaging conditions are derived using the adjoint-state method and an augmented Lagrangian functional. Q-LSRTM solves the viscoelastic linearized modeling operator for synthetic data, and the adjoint operator is used for back propagating the data residual. Q-LSRTM is capable of iteratively updating the P and S modulus perturbations,in the direction of minimizing data residuals, and attenuation loss is iteratively compensated. A novel Q compensation approach is developed for adjoint seismic imaging by pseudodifferential scaling. With a correct Q model included in the migration algorithm, propagation effects, including the Q effects, can be compensated with the application of the inverse Hessian to the RTM image. Pseudodifferential scaling is used to efficiently approximate the action of the inverse Hessian. Numerical examples indicate that the adjoint RTM images with pseudodifferential scaling approximate the true model perturbation, and can be used as well-conditioned gradients for least-squares imaging.

Burst wait time simulation of CALIBAN reactor at delayed super-critical state

DOE Office of Scientific and Technical Information (OSTI.GOV)

Humbert, P.; Authier, N.; Richard, B.

2012-07-01

In the past, the super prompt critical wait time probability distribution was measured on CALIBAN fast burst reactor [4]. Afterwards, these experiments were simulated with a very good agreement by solving the non-extinction probability equation [5]. Recently, the burst wait time probability distribution has been measured at CEA-Valduc on CALIBAN at different delayed super-critical states [6]. However, in the delayed super-critical case the non-extinction probability does not give access to the wait time distribution. In this case it is necessary to compute the time dependent evolution of the full neutron count number probability distribution. In this paper we present themore » point model deterministic method used to calculate the probability distribution of the wait time before a prescribed count level taking into account prompt neutrons and delayed neutron precursors. This method is based on the solution of the time dependent adjoint Kolmogorov master equations for the number of detections using the generating function methodology [8,9,10] and inverse discrete Fourier transforms. The obtained results are then compared to the measurements and Monte-Carlo calculations based on the algorithm presented in [7]. (authors)« less

Adjoint-Based, Three-Dimensional Error Prediction and Grid Adaptation

NASA Technical Reports Server (NTRS)

Park, Michael A.

2002-01-01

Engineering computational fluid dynamics (CFD) analysis and design applications focus on output functions (e.g., lift, drag). Errors in these output functions are generally unknown and conservatively accurate solutions may be computed. Computable error estimates can offer the possibility to minimize computational work for a prescribed error tolerance. Such an estimate can be computed by solving the flow equations and the linear adjoint problem for the functional of interest. The computational mesh can be modified to minimize the uncertainty of a computed error estimate. This robust mesh-adaptation procedure automatically terminates when the simulation is within a user specified error tolerance. This procedure for estimating and adapting to error in a functional is demonstrated for three-dimensional Euler problems. An adaptive mesh procedure that links to a Computer Aided Design (CAD) surface representation is demonstrated for wing, wing-body, and extruded high lift airfoil configurations. The error estimation and adaptation procedure yielded corrected functions that are as accurate as functions calculated on uniformly refined grids with ten times as many grid points.

Adjoint optimization of natural convection problems: differentially heated cavity

NASA Astrophysics Data System (ADS)

Saglietti, Clio; Schlatter, Philipp; Monokrousos, Antonios; Henningson, Dan S.

2017-12-01

Optimization of natural convection-driven flows may provide significant improvements to the performance of cooling devices, but a theoretical investigation of such flows has been rarely done. The present paper illustrates an efficient gradient-based optimization method for analyzing such systems. We consider numerically the natural convection-driven flow in a differentially heated cavity with three Prandtl numbers (Pr=0.15{-}7) at super-critical conditions. All results and implementations were done with the spectral element code Nek5000. The flow is analyzed using linear direct and adjoint computations about a nonlinear base flow, extracting in particular optimal initial conditions using power iteration and the solution of the full adjoint direct eigenproblem. The cost function for both temperature and velocity is based on the kinetic energy and the concept of entransy, which yields a quadratic functional. Results are presented as a function of Prandtl number, time horizons and weights between kinetic energy and entransy. In particular, it is shown that the maximum transient growth is achieved at time horizons on the order of 5 time units for all cases, whereas for larger time horizons the adjoint mode is recovered as optimal initial condition. For smaller time horizons, the influence of the weights leads either to a concentric temperature distribution or to an initial condition pattern that opposes the mean shear and grows according to the Orr mechanism. For specific cases, it could also been shown that the computation of optimal initial conditions leads to a degenerate problem, with a potential loss of symmetry. In these situations, it turns out that any initial condition lying in a specific span of the eigenfunctions will yield exactly the same transient amplification. As a consequence, the power iteration converges very slowly and fails to extract all possible optimal initial conditions. According to the authors' knowledge, this behavior is illustrated here for the first time.

Bose-Fermi degeneracies in large N adjoint QCD

DOE PAGES

Basar, Gokce; Cherman, Aleksey; McGady, David

2015-07-06

Here, we analyze the large N limit of adjoint QCD, an SU( N) gauge theory with N f flavors of massless adjoint Majorana fermions, compactified on S 3 × S 1. We focus on the weakly-coupled confining small- S 3 regime. If the fermions are given periodic boundary conditions on S 1, we show that there are large cancellations between bosonic and fermionic contributions to the twisted partition function. These cancellations follow a pattern previously seen in the context of misaligned supersymmetry, and lead to the absence of Hagedorn instabilities for any S 1 size L, even though the bosonicmore » and fermionic densities of states both have Hagedorn growth. Adjoint QCD stays in the confining phase for any L ~ N 0, explaining how it is able to enjoy large N volume independence for any L. The large N boson-fermion cancellations take place in a setting where adjoint QCD is manifestly non-supersymmetric at any finite N, and are consistent with the recent conjecture that adjoint QCD has emergent fermionic symmetries in the large N limit.« less

Assessing the Impact of Observations on Numerical Weather Forecasts Using the Adjoint Method

NASA Technical Reports Server (NTRS)

Gelaro, Ronald

2012-01-01

The adjoint of a data assimilation system provides a flexible and efficient tool for estimating observation impacts on short-range weather forecasts. The impacts of any or all observations can be estimated simultaneously based on a single execution of the adjoint system. The results can be easily aggregated according to data type, location, channel, etc., making this technique especially attractive for examining the impacts of new hyper-spectral satellite instruments and for conducting regular, even near-real time, monitoring of the entire observing system. This talk provides a general overview of the adjoint method, including the theoretical basis and practical implementation of the technique. Results are presented from the adjoint-based observation impact monitoring tool in NASA's GEOS-5 global atmospheric data assimilation and forecast system. When performed in conjunction with standard observing system experiments (OSEs), the adjoint results reveal both redundancies and dependencies between observing system impacts as observations are added or removed from the assimilation system. Understanding these dependencies may be important for optimizing the use of the current observational network and defining requirements for future observing systems

Adjoint tomography and centroid-moment tensor inversion of the Kanto region, Japan

NASA Astrophysics Data System (ADS)

Miyoshi, T.

2017-12-01

A three-dimensional seismic wave speed model in the Kanto region of Japan was developed using adjoint tomography based on large computing. Starting with a model based on previous travel time tomographic results, we inverted the waveforms obtained at seismic broadband stations from 140 local earthquakes in the Kanto region to obtain the P- and S-wave speeds Vp and Vs. The synthetic displacements were calculated using the spectral element method (SEM; e.g. Komatitsch and Tromp 1999; Peter et al. 2011) in which the Kanto region was parameterized using 16 million grid points. The model parameters Vp and Vs were updated iteratively by Newton's method using the misfit and Hessian kernels until the misfit between the observed and synthetic waveforms was minimized. The proposed model reveals several anomalous areas with extremely low Vs values in comparison with those of the initial model. The synthetic waveforms obtained using the newly proposed model for the selected earthquakes show better fit than the initial model to the observed waveforms in different period ranges within 5-30 s. In the present study, all centroid times of the source solutions were determined using time shifts based on cross correlation to prevent high computing resources before the structural inversion. Additionally, parameters of centroid-moment solutions were fully determined using the SEM assuming the 3D structure (e.g. Liu et al. 2004). As a preliminary result, new solutions were basically same as their initial solutions. This may indicate that the 3D structure is not effective for the source estimation. Acknowledgements: This study was supported by JSPS KAKENHI Grant Number 16K21699.

Discrete Adjoint-Based Design Optimization of Unsteady Turbulent Flows on Dynamic Unstructured Grids

NASA Technical Reports Server (NTRS)

Nielsen, Eric J.; Diskin, Boris; Yamaleev, Nail K.

2009-01-01

An adjoint-based methodology for design optimization of unsteady turbulent flows on dynamic unstructured grids is described. The implementation relies on an existing unsteady three-dimensional unstructured grid solver capable of dynamic mesh simulations and discrete adjoint capabilities previously developed for steady flows. The discrete equations for the primal and adjoint systems are presented for the backward-difference family of time-integration schemes on both static and dynamic grids. The consistency of sensitivity derivatives is established via comparisons with complex-variable computations. The current work is believed to be the first verified implementation of an adjoint-based optimization methodology for the true time-dependent formulation of the Navier-Stokes equations in a practical computational code. Large-scale shape optimizations are demonstrated for turbulent flows over a tiltrotor geometry and a simulated aeroelastic motion of a fighter jet.

Finite-frequency sensitivity kernels for global seismic wave propagation based upon adjoint methods

NASA Astrophysics Data System (ADS)

Liu, Qinya; Tromp, Jeroen

2008-07-01

We determine adjoint equations and Fréchet kernels for global seismic wave propagation based upon a Lagrange multiplier method. We start from the equations of motion for a rotating, self-gravitating earth model initially in hydrostatic equilibrium, and derive the corresponding adjoint equations that involve motions on an earth model that rotates in the opposite direction. Variations in the misfit function χ then may be expressed as , where δlnm = δm/m denotes relative model perturbations in the volume V, δlnd denotes relative topographic variations on solid-solid or fluid-solid boundaries Σ, and ∇Σδlnd denotes surface gradients in relative topographic variations on fluid-solid boundaries ΣFS. The 3-D Fréchet kernel Km determines the sensitivity to model perturbations δlnm, and the 2-D kernels Kd and Kd determine the sensitivity to topographic variations δlnd. We demonstrate also how anelasticity may be incorporated within the framework of adjoint methods. Finite-frequency sensitivity kernels are calculated by simultaneously computing the adjoint wavefield forward in time and reconstructing the regular wavefield backward in time. Both the forward and adjoint simulations are based upon a spectral-element method. We apply the adjoint technique to generate finite-frequency traveltime kernels for global seismic phases (P, Pdiff, PKP, S, SKS, depth phases, surface-reflected phases, surface waves, etc.) in both 1-D and 3-D earth models. For 1-D models these adjoint-generated kernels generally agree well with results obtained from ray-based methods. However, adjoint methods do not have the same theoretical limitations as ray-based methods, and can produce sensitivity kernels for any given phase in any 3-D earth model. The Fréchet kernels presented in this paper illustrate the sensitivity of seismic observations to structural parameters and topography on internal discontinuities. These kernels form the basis of future 3-D tomographic inversions.

Use of adjoint methods in the probabilistic finite element approach to fracture mechanics

NASA Technical Reports Server (NTRS)

Liu, Wing Kam; Besterfield, Glen; Lawrence, Mark; Belytschko, Ted

1988-01-01

The adjoint method approach to probabilistic finite element methods (PFEM) is presented. When the number of objective functions is small compared to the number of random variables, the adjoint method is far superior to the direct method in evaluating the objective function derivatives with respect to the random variables. The PFEM is extended to probabilistic fracture mechanics (PFM) using an element which has the near crack-tip singular strain field embedded. Since only two objective functions (i.e., mode I and II stress intensity factors) are needed for PFM, the adjoint method is well suited.

The Tangent Linear and Adjoint of the FV3 Dynamical Core: Development and Applications

NASA Technical Reports Server (NTRS)

Holdaway, Daniel

2018-01-01

GMAO (NASA's Global Modeling and Assimilation Office) has developed a highly sophisticated adjoint modeling system based on the most recent version of the finite volume cubed sphere (FV3) dynamical core. This provides a mechanism for investigating sensitivity to initial conditions and examining observation impacts. It also allows for the computation of singular vectors and for the implementation of hybrid 4DVAR (4-Dimensional Variational Assimilation). In this work we will present the scientific assessment of the new adjoint system and show results from a number of research application of the adjoint system.

Towards the Early Detection of Breast Cancer in Young Women

DTIC Science & Technology

2005-10-01

T. Shiina, and F. Tranquart. Progress in Freehand Elastography of the Breast . IEICE Transactions on Information and Systems, E85D (1):5–14, 2002. [3...Meaney, Naomi R. Miller, Tsuyoshi Shiina, and Francois Tranquart. Progress in freehand elastography of the breast . IEICE Transactions on Information...solution of the non-linear inverse elasticity problem 28 [26] Liew HL and Pinsky PM. Recovery of shear modulus in elastography using an adjoint method

Four-Dimensional Data Assimilation Using the Adjoint Method

NASA Astrophysics Data System (ADS)

Bao, Jian-Wen

The calculus of variations is used to confirm that variational four-dimensional data assimilation (FDDA) using the adjoint method can be implemented when the numerical model equations have a finite number of first-order discontinuous points. These points represent the on/off switches associated with physical processes, for which the Jacobian matrix of the model equation does not exist. Numerical evidence suggests that, in some situations when the adjoint method is used for FDDA, the temperature field retrieved using horizontal wind data is numerically not unique. A physical interpretation of this type of non-uniqueness of the retrieval is proposed in terms of energetics. The adjoint equations of a numerical model can also be used for model-parameter estimation. A general computational procedure is developed to determine the size and distribution of any internal model parameter. The procedure is then applied to a one-dimensional shallow -fluid model in the context of analysis-nudging FDDA: the weighting coefficients used by the Newtonian nudging technique are determined. The sensitivity of these nudging coefficients to the optimal objectives and constraints is investigated. Experiments of FDDA using the adjoint method are conducted using the dry version of the hydrostatic Penn State/NCAR mesoscale model (MM4) and its adjoint. The minimization procedure converges and the initialization experiment is successful. Temperature-retrieval experiments involving an assimilation of the horizontal wind are also carried out using the adjoint of MM4.

An Efficient Method Coupling Kernel Principal Component Analysis with Adjoint-Based Optimal Control and Its Goal-Oriented Extensions

NASA Astrophysics Data System (ADS)

Thimmisetty, C.; Talbot, C.; Tong, C. H.; Chen, X.

2016-12-01

The representativeness of available data poses a significant fundamental challenge to the quantification of uncertainty in geophysical systems. Furthermore, the successful application of machine learning methods to geophysical problems involving data assimilation is inherently constrained by the extent to which obtainable data represent the problem considered. We show how the adjoint method, coupled with optimization based on methods of machine learning, can facilitate the minimization of an objective function defined on a space of significantly reduced dimension. By considering uncertain parameters as constituting a stochastic process, the Karhunen-Loeve expansion and its nonlinear extensions furnish an optimal basis with respect to which optimization using L-BFGS can be carried out. In particular, we demonstrate that kernel PCA can be coupled with adjoint-based optimal control methods to successfully determine the distribution of material parameter values for problems in the context of channelized deformable media governed by the equations of linear elasticity. Since certain subsets of the original data are characterized by different features, the convergence rate of the method in part depends on, and may be limited by, the observations used to furnish the kernel principal component basis. By determining appropriate weights for realizations of the stochastic random field, then, one may accelerate the convergence of the method. To this end, we present a formulation of Weighted PCA combined with a gradient-based means using automatic differentiation to iteratively re-weight observations concurrent with the determination of an optimal reduced set control variables in the feature space. We demonstrate how improvements in the accuracy and computational efficiency of the weighted linear method can be achieved over existing unweighted kernel methods, and discuss nonlinear extensions of the algorithm.

Crustal wavespeed structure of North Texas and Oklahoma based on ambient noise cross-correlation functions and adjoint tomography

NASA Astrophysics Data System (ADS)

Zhu, H.

2017-12-01

Recently, seismologists observed increasing seismicity in North Texas and Oklahoma. Based on seismic observations and other geophysical measurements, some studies suggested possible links between the increasing seismicity and wastewater injection during unconventional oil and gas exploration. To better monitor seismic events and investigate their mechanisms, we need an accurate 3D crustal wavespeed model for North Texas and Oklahoma. Considering the uneven distribution of earthquakes in this region, seismic tomography with local earthquake records have difficulties to achieve good illumination. To overcome this limitation, in this study, ambient noise cross-correlation functions are used to constrain subsurface variations in wavespeeds. I use adjoint tomography to iteratively fit frequency-dependent phase differences between observed and predicted band-limited Green's functions. The spectral-element method is used to numerically calculate the band-limited Green's functions and the adjoint method is used to calculate misfit gradients with respect to wavespeeds. 25 preconditioned conjugate gradient iterations are used to update model parameters and minimize data misfits. Features in the new crustal model M25 correlates with geological units in the study region, including the Llano uplift, the Anadarko basin and the Ouachita orogenic front. In addition, these seismic anomalies correlate with gravity and magnetic observations. This new model can be used to better constrain earthquake source parameters in North Texas and Oklahoma, such as epicenter location and moment tensor solutions, which are important for investigating potential relations between seismicity and unconventional oil and gas exploration.

Preconditioned conjugate residual methods for the solution of spectral equations

NASA Technical Reports Server (NTRS)

Wong, Y. S.; Zang, T. A.; Hussaini, M. Y.

1986-01-01

Conjugate residual methods for the solution of spectral equations are described. An inexact finite-difference operator is introduced as a preconditioner in the iterative procedures. Application of these techniques is limited to problems for which the symmetric part of the coefficient matrix is positive definite. Although the spectral equation is a very ill-conditioned and full matrix problem, the computational effort of the present iterative methods for solving such a system is comparable to that for the sparse matrix equations obtained from the application of either finite-difference or finite-element methods to the same problems. Numerical experiments are shown for a self-adjoint elliptic partial differential equation with Dirichlet boundary conditions, and comparison with other solution procedures for spectral equations is presented.

Comparing mass balance and adjoint methods for inverse modeling of nitrogen dioxide columns for global nitrogen oxide emissions

NASA Astrophysics Data System (ADS)

Cooper, Matthew; Martin, Randall V.; Padmanabhan, Akhila; Henze, Daven K.

2017-04-01

Satellite observations offer information applicable to top-down constraints on emission inventories through inverse modeling. Here we compare two methods of inverse modeling for emissions of nitrogen oxides (NOx) from nitrogen dioxide (NO2) columns using the GEOS-Chem chemical transport model and its adjoint. We treat the adjoint-based 4D-Var modeling approach for estimating top-down emissions as a benchmark against which to evaluate variations on the mass balance method. We use synthetic NO2 columns generated from known NOx emissions to serve as "truth." We find that error in mass balance inversions can be reduced by up to a factor of 2 with an iterative process that uses finite difference calculations of the local sensitivity of NO2 columns to a change in emissions. In a simplified experiment to recover local emission perturbations, horizontal smearing effects due to NOx transport are better resolved by the adjoint approach than by mass balance. For more complex emission changes, or at finer resolution, the iterative finite difference mass balance and adjoint methods produce similar global top-down inventories when inverting hourly synthetic observations, both reducing the a priori error by factors of 3-4. Inversions of simulated satellite observations from low Earth and geostationary orbits also indicate that both the mass balance and adjoint inversions produce similar results, reducing a priori error by a factor of 3. As the iterative finite difference mass balance method provides similar accuracy as the adjoint method, it offers the prospect of accurately estimating top-down NOx emissions using models that do not have an adjoint.

Robustness of reduced-order observer-based controllers in transitional 2D Blasius boundary layers

NASA Astrophysics Data System (ADS)

Belson, Brandt; Semeraro, Onofrio; Rowley, Clarence; Pralits, Jan; Henningson, Dan

2011-11-01

In this work, we seek to delay transition in the Blasius boundary layer. We trip the flow with an upstream disturbance and dampen the growth of the resulting structures downstream. The observer-based controllers use a single sensor and a single localized body force near the wall. To formulate the controllers, we first find a reduced-order model of the system via the Eigensystem Realization Algorithm (ERA), then find the H2 optimal controller for this reduced-order system. We find the resulting controllers are effective only when the sensor is upstream of the actuator (in a feedforward configuration), but as is expected, are sensitive to model uncertainty. When the sensor is downstream of the actuator (in a feedback configuration), the reduced-order observer-based controllers are not robust and ineffective on the full system. In order to investigate the robustness properties of the system, an iterative technique called the adjoint of the direct adjoint (ADA) is employed to find a full-dimensional H2 optimal controller. This avoids the reduced-order modelling step and serves as a reference point. ADA is promising for investigating the lack of robustness previously mentioned.

Adjoint-based optimization of mechanical performance in polycrystalline materials and structures through texture control

DOE Office of Scientific and Technical Information (OSTI.GOV)

Gu, Grace; Brown, Judith Alice; Bishop, Joseph E.

The texture of a polycrystalline material refers to the preferred orientation of the grains within the material. In metallic materials, texture can significantly affect the mechanical properties such as elastic moduli, yield stress, strain hardening, and fracture toughness. Recent advances in additive manufacturing of metallic materials offer the possibility in the not too distant future of controlling the spatial variation of texture. In this work, we investigate the advantages, in terms of mechanical performance, of allowing the texture to vary spatially. We use an adjoint-based gradient optimization algorithm within a finite element solver (COMSOL) to optimize several engineering quantities ofmore » interest in a simple structure (hole in a plate) and loading (uniaxial tension) condition. As a first step to general texture optimization, we consider the idealized case of a pure fiber texture in which the homogenized properties are transversely isotropic. In this special case, the only spatially varying design variables are the three Euler angles that prescribe the orientation of the homogenized material at each point within the structure. This work paves a new way to design metallic materials for tunable mechanical properties at the microstructure level.« less

A Hybrid Forward-Adjoint Data Assimilation Method for Reconstructing the Temporal Evolution of Mantle Dynamics

NASA Astrophysics Data System (ADS)

Zhou, Q.; Liu, L.

2017-12-01

Quantifying past mantle dynamic processes represents a major challenge in understanding the temporal evolution of the solid earth. Mantle convection modeling with data assimilation is one of the most powerful tools to investigate the dynamics of plate subduction and mantle convection. Although various data assimilation methods, both forward and inverse, have been created, these methods all have limitations in their capabilities to represent the real earth. Pure forward models tend to miss important mantle structures due to the incorrect initial condition and thus may lead to incorrect mantle evolution. In contrast, pure tomography-based models cannot effectively resolve the fine slab structure and would fail to predict important subduction-zone dynamic processes. Here we propose a hybrid data assimilation method that combines the unique power of the sequential and adjoint algorithms, which can properly capture the detailed evolution of the downgoing slab and the tomographically constrained mantle structures, respectively. We apply this new method to reconstructing mantle dynamics below the western U.S. while considering large lateral viscosity variations. By comparing this result with those from several existing data assimilation methods, we demonstrate that the hybrid modeling approach recovers the realistic 4-D mantle dynamics to the best.

On the role of self-adjointness in the continuum formulation of topological quantum phases

NASA Astrophysics Data System (ADS)

Tanhayi Ahari, Mostafa; Ortiz, Gerardo; Seradjeh, Babak

2016-11-01

Topological quantum phases of matter are characterized by an intimate relationship between the Hamiltonian dynamics away from the edges and the appearance of bound states localized at the edges of the system. Elucidating this correspondence in the continuum formulation of topological phases, even in the simplest case of a one-dimensional system, touches upon fundamental concepts and methods in quantum mechanics that are not commonly discussed in textbooks, in particular the self-adjoint extensions of a Hermitian operator. We show how such topological bound states can be derived in a prototypical one-dimensional system. Along the way, we provide a pedagogical exposition of the self-adjoint extension method as well as the role of symmetries in correctly formulating the continuum, field-theory description of topological matter with boundaries. Moreover, we show that self-adjoint extensions can be characterized generally in terms of a conserved local current associated with the self-adjoint operator.

Two- and four-dimensional representations of the PT - and CPT -symmetric fermionic algebras

NASA Astrophysics Data System (ADS)

Beygi, Alireza; Klevansky, S. P.; Bender, Carl M.

2018-03-01

Fermionic systems differ from their bosonic counterparts, the main difference with regard to symmetry considerations being that T2=-1 for fermionic systems. In PT -symmetric quantum mechanics an operator has both PT and CPT adjoints. Fermionic operators η , which are quadratically nilpotent (η2=0 ), and algebras with PT and CPT adjoints can be constructed. These algebras obey different anticommutation relations: η ηPT+ηPTη =-1 , where ηPT is the PT adjoint of η , and η ηCPT+ηCPTη =1 , where ηCPT is the CPT adjoint of η . This paper presents matrix representations for the operator η and its PT and CPT adjoints in two and four dimensions. A PT -symmetric second-quantized Hamiltonian modeled on quantum electrodynamics that describes a system of interacting fermions and bosons is constructed within this framework and is solved exactly.

Design of Rail Instrumentation for Wind Tunnel Sonic Boom Measurements and Computational-Experimental Comparisons

NASA Technical Reports Server (NTRS)

Cliff, Susan E.; Elmiligui, A.; Aftosmis, M.; Morgenstern, J.; Durston, D.; Thomas, S.

2012-01-01

An innovative pressure rail concept for wind tunnel sonic boom testing of modern aircraft configurations with very low overpressures was designed with an adjoint-based solution-adapted Cartesian grid method. The computational method requires accurate free-air calculations of a test article as well as solutions modeling the influence of rail and tunnel walls. Specialized grids for accurate Euler and Navier-Stokes sonic boom computations were used on several test articles including complete aircraft models with flow-through nacelles. The computed pressure signatures are compared with recent results from the NASA 9- x 7-foot Supersonic Wind Tunnel using the advanced rail design.

Admitting the Inadmissible: Adjoint Formulation for Incomplete Cost Functionals in Aerodynamic Optimization

NASA Technical Reports Server (NTRS)

Arian, Eyal; Salas, Manuel D.

1997-01-01

We derive the adjoint equations for problems in aerodynamic optimization which are improperly considered as "inadmissible." For example, a cost functional which depends on the density, rather than on the pressure, is considered "inadmissible" for an optimization problem governed by the Euler equations. We show that for such problems additional terms should be included in the Lagrangian functional when deriving the adjoint equations. These terms are obtained from the restriction of the interior PDE to the control surface. Demonstrations of the explicit derivation of the adjoint equations for "inadmissible" cost functionals are given for the potential, Euler, and Navier-Stokes equations.

Adjoint sensitivity analysis of plasmonic structures using the FDTD method.

PubMed

Zhang, Yu; Ahmed, Osman S; Bakr, Mohamed H

2014-05-15

We present an adjoint variable method for estimating the sensitivities of arbitrary responses with respect to the parameters of dispersive discontinuities in nanoplasmonic devices. Our theory is formulated in terms of the electric field components at the vicinity of perturbed discontinuities. The adjoint sensitivities are computed using at most one extra finite-difference time-domain (FDTD) simulation regardless of the number of parameters. Our approach is illustrated through the sensitivity analysis of an add-drop coupler consisting of a square ring resonator between two parallel waveguides. The computed adjoint sensitivities of the scattering parameters are compared with those obtained using the accurate but computationally expensive central finite difference approach.

Operator pencil passing through a given operator

DOE Office of Scientific and Technical Information (OSTI.GOV)

Biggs, A., E-mail: khudian@manchester.ac.uk, E-mail: adam.biggs@student.manchester.ac.uk; Khudaverdian, H. M., E-mail: khudian@manchester.ac.uk, E-mail: adam.biggs@student.manchester.ac.uk

Let Δ be a linear differential operator acting on the space of densities of a given weight λ{sub 0} on a manifold M. One can consider a pencil of operators Π-circumflex(Δ)=(Δ{sub λ}) passing through the operator Δ such that any Δ{sub λ} is a linear differential operator acting on densities of weight λ. This pencil can be identified with a linear differential operator Δ-circumflex acting on the algebra of densities of all weights. The existence of an invariant scalar product in the algebra of densities implies a natural decomposition of operators, i.e., pencils of self-adjoint and anti-self-adjoint operators. We studymore » lifting maps that are on one hand equivariant with respect to divergenceless vector fields, and, on the other hand, with values in self-adjoint or anti-self-adjoint operators. In particular, we analyze the relation between these two concepts, and apply it to the study of diff (M)-equivariant liftings. Finally, we briefly consider the case of liftings equivariant with respect to the algebra of projective transformations and describe all regular self-adjoint and anti-self-adjoint liftings. Our constructions can be considered as a generalisation of equivariant quantisation.« less

THE SMALLEST FIELD OF DEFINITION OF A SUBGROUP OF THE GROUP \\mathrm{PSL}_2

NASA Astrophysics Data System (ADS)

Vinberg, È. B.

1995-02-01

As previously proved by the author, for each semisimple algebraic group of adjoint type that is dense in the Zariski topology there exists a smallest field of definition which is an invariant of the class of commensurable subgroups. In the present paper an algorithm is given for finding the smallest field of definition of a dense finitely generated subgroup of the group \\mathrm{PSL}_2(\\mathbb{C}). A criterion for arithmeticity of a lattice in \\mathrm{PSL}_2(\\mathbb{R}) or \\mathrm{PSL}_2(\\mathbb{C}) in terms of this field is presented.Bibliography: 7 titles.

Optimal startup control of a jacketed tubular reactor.

NASA Technical Reports Server (NTRS)

Hahn, D. R.; Fan, L. T.; Hwang, C. L.

1971-01-01

The optimal startup policy of a jacketed tubular reactor, in which a first-order, reversible, exothermic reaction takes place, is presented. A distributed maximum principle is presented for determining weak necessary conditions for optimality of a diffusional distributed parameter system. A numerical technique is developed for practical implementation of the distributed maximum principle. This involves the sequential solution of the state and adjoint equations, in conjunction with a functional gradient technique for iteratively improving the control function.

A new least-squares transport equation compatible with voids

DOE Office of Scientific and Technical Information (OSTI.GOV)

Hansen, J. B.; Morel, J. E.

2013-07-01

We define a new least-squares transport equation that is applicable in voids, can be solved using source iteration with diffusion-synthetic acceleration, and requires only the solution of an independent set of second-order self-adjoint equations for each direction during each source iteration. We derive the equation, discretize it using the S{sub n} method in conjunction with a linear-continuous finite-element method in space, and computationally demonstrate various of its properties. (authors)

Stability-Constrained Aerodynamic Shape Optimization with Applications to Flying Wings

NASA Astrophysics Data System (ADS)

Mader, Charles Alexander

A set of techniques is developed that allows the incorporation of flight dynamics metrics as an additional discipline in a high-fidelity aerodynamic optimization. Specifically, techniques for including static stability constraints and handling qualities constraints in a high-fidelity aerodynamic optimization are demonstrated. These constraints are developed from stability derivative information calculated using high-fidelity computational fluid dynamics (CFD). Two techniques are explored for computing the stability derivatives from CFD. One technique uses an automatic differentiation adjoint technique (ADjoint) to efficiently and accurately compute a full set of static and dynamic stability derivatives from a single steady solution. The other technique uses a linear regression method to compute the stability derivatives from a quasi-unsteady time-spectral CFD solution, allowing for the computation of static, dynamic and transient stability derivatives. Based on the characteristics of the two methods, the time-spectral technique is selected for further development, incorporated into an optimization framework, and used to conduct stability-constrained aerodynamic optimization. This stability-constrained optimization framework is then used to conduct an optimization study of a flying wing configuration. This study shows that stability constraints have a significant impact on the optimal design of flying wings and that, while static stability constraints can often be satisfied by modifying the airfoil profiles of the wing, dynamic stability constraints can require a significant change in the planform of the aircraft in order for the constraints to be satisfied.

Methodology for sensitivity analysis, approximate analysis, and design optimization in CFD for multidisciplinary applications

NASA Technical Reports Server (NTRS)

Taylor, Arthur C., III; Hou, Gene W.

1994-01-01

The straightforward automatic-differentiation and the hand-differentiated incremental iterative methods are interwoven to produce a hybrid scheme that captures some of the strengths of each strategy. With this compromise, discrete aerodynamic sensitivity derivatives are calculated with the efficient incremental iterative solution algorithm of the original flow code. Moreover, the principal advantage of automatic differentiation is retained (i.e., all complicated source code for the derivative calculations is constructed quickly with accuracy). The basic equations for second-order sensitivity derivatives are presented; four methods are compared. Each scheme requires that large systems are solved first for the first-order derivatives and, in all but one method, for the first-order adjoint variables. Of these latter three schemes, two require no solutions of large systems thereafter. For the other two for which additional systems are solved, the equations and solution procedures are analogous to those for the first order derivatives. From a practical viewpoint, implementation of the second-order methods is feasible only with software tools such as automatic differentiation, because of the extreme complexity and large number of terms. First- and second-order sensitivities are calculated accurately for two airfoil problems, including a turbulent flow example; both geometric-shape and flow-condition design variables are considered. Several methods are tested; results are compared on the basis of accuracy, computational time, and computer memory. For first-order derivatives, the hybrid incremental iterative scheme obtained with automatic differentiation is competitive with the best hand-differentiated method; for six independent variables, it is at least two to four times faster than central finite differences and requires only 60 percent more memory than the original code; the performance is expected to improve further in the future.

Ambient noise adjoint tomography for a linear array in North China

NASA Astrophysics Data System (ADS)

Zhang, C.; Yao, H.; Liu, Q.; Yuan, Y. O.; Zhang, P.; Feng, J.; Fang, L.

2017-12-01

Ambient noise tomography based on dispersion data and ray theory has been widely utilized for imaging crustal structures. In order to improve the inversion accuracy, ambient noise tomography based on the 3D adjoint approach or full waveform inversion has been developed recently, however, the computational cost is tremendous. In this study we present 2D ambient noise adjoint tomography for a linear array in north China with significant computational efficiency compared to 3D ambient noise adjoint tomography. During the preprocessing, we first convert the observed data in 3D media, i.e., surface-wave empirical Green's functions (EGFs) from ambient noise cross-correlation, to the reconstructed EGFs in 2D media using a 3D/2D transformation scheme. Different from the conventional steps of measuring phase dispersion, the 2D adjoint tomography refines 2D shear wave speeds along the profile directly from the reconstructed Rayleigh wave EGFs in the period band 6-35s. With the 2D initial model extracted from the 3D model from traditional ambient noise tomography, adjoint tomography updates the model by minimizing the frequency-dependent Rayleigh wave traveltime misfits between the reconstructed EGFs and synthetic Green function (SGFs) in 2D media generated by the spectral-element method (SEM), with a preconditioned conjugate gradient method. The multitaper traveltime difference measurement is applied in four period bands during the inversion: 20-35s, 15-30s, 10-20s and 6-15s. The recovered model shows more detailed crustal structures with pronounced low velocity anomaly in the mid-lower crust beneath the junction of Taihang Mountains and Yin-Yan Mountains compared with the initial model. This low velocity structure may imply the possible intense crust-mantle interactions, probably associated with the magmatic underplating during the Mesozoic to Cenozoic evolution of the region. To our knowledge, it's first time that ambient noise adjoint tomography is implemented in 2D media. Considering the intensive computational cost and storage of 3D adjoint tomography, this 2D ambient noise adjoint tomography has potential advantages to get high-resolution 2D crustal structures with limited computational resource.

Parallelized Three-Dimensional Resistivity Inversion Using Finite Elements And Adjoint State Methods

NASA Astrophysics Data System (ADS)

Schaa, Ralf; Gross, Lutz; Du Plessis, Jaco

2015-04-01

The resistivity method is one of the oldest geophysical exploration methods, which employs one pair of electrodes to inject current into the ground and one or more pairs of electrodes to measure the electrical potential difference. The potential difference is a non-linear function of the subsurface resistivity distribution described by an elliptic partial differential equation (PDE) of the Poisson type. Inversion of measured potentials solves for the subsurface resistivity represented by PDE coefficients. With increasing advances in multichannel resistivity acquisition systems (systems with more than 60 channels and full waveform recording are now emerging), inversion software require efficient storage and solver algorithms. We developed the finite element solver Escript, which provides a user-friendly programming environment in Python to solve large-scale PDE-based problems (see https://launchpad.net/escript-finley). Using finite elements, highly irregular shaped geology and topography can readily be taken into account. For the 3D resistivity problem, we have implemented the secondary potential approach, where the PDE is decomposed into a primary potential caused by the source current and the secondary potential caused by changes in subsurface resistivity. The primary potential is calculated analytically, and the boundary value problem for the secondary potential is solved using nodal finite elements. This approach removes the singularity caused by the source currents and provides more accurate 3D resistivity models. To solve the inversion problem we apply a 'first optimize then discretize' approach using the quasi-Newton scheme in form of the limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) method (see Gross & Kemp 2013). The evaluation of the cost function requires the solution of the secondary potential PDE for each source current and the solution of the corresponding adjoint-state PDE for the cost function gradients with respect to the subsurface resistivity. The Hessian of the regularization term is used as preconditioner which requires an additional PDE solution in each iteration step. As it turns out, the relevant PDEs are naturally formulated in the finite element framework. Using the domain decomposition method provided in Escript, the inversion scheme has been parallelized for distributed memory computers with multi-core shared memory nodes. We show numerical examples from simple layered models to complex 3D models and compare with the results from other methods. The inversion scheme is furthermore tested on a field data example to characterise localised freshwater discharge in a coastal environment.. References: L. Gross and C. Kemp (2013) Large Scale Joint Inversion of Geophysical Data using the Finite Element Method in escript. ASEG Extended Abstracts 2013, http://dx.doi.org/10.1071/ASEG2013ab306

Adjoint Inversion for Extended Earthquake Source Kinematics From Very Dense Strong Motion Data

NASA Astrophysics Data System (ADS)

Ampuero, J. P.; Somala, S.; Lapusta, N.

2010-12-01

Addressing key open questions about earthquake dynamics requires a radical improvement of the robustness and resolution of seismic observations of large earthquakes. Proposals for a new generation of earthquake observation systems include the deployment of “community seismic networks” of low-cost accelerometers in urban areas and the extraction of strong ground motions from high-rate optical images of the Earth's surface recorded by a large space telescope in geostationary orbit. Both systems could deliver strong motion data with a spatial density orders of magnitude higher than current seismic networks. In particular, a “space seismometer” could sample the seismic wave field at a spatio-temporal resolution of 100 m, 1 Hz over areas several 100 km wide with an amplitude resolution of few cm/s in ground velocity. The amount of data to process would be immensely larger than what current extended source inversion algorithms can handle, which hampers the quantitative assessment of the cost-benefit trade-offs that can guide the practical design of the proposed earthquake observation systems. We report here on the development of a scalable source imaging technique based on iterative adjoint inversion and its application to the proof-of-concept of a space seismometer. We generated synthetic ground motions for M7 earthquake rupture scenarios based on dynamic rupture simulations on a vertical strike-slip fault embedded in an elastic half-space. A range of scenarios include increasing levels of complexity and interesting features such as supershear rupture speed. The resulting ground shaking is then processed accordingly to what would be captured by an optical satellite. Based on the resulting data, we perform source inversion by an adjoint/time-reversal method. The gradient of a cost function quantifying the waveform misfit between data and synthetics is efficiently obtained by applying the time-reversed ground velocity residuals as surface force sources, back-propagating onto the locked fault plane through a seismic wave simulation and recording the fault shear stress, which is the adjoint field of the fault slip-rate. Restricting the procedure to a single iteration is known as imaging. The source reconstructed by imaging reproduces the original forward model quite well in the shallow part of the fault. However, the deeper part of the earthquake source is not well reproduced, due to the lack of data on the side and bottom boundaries of our computational domain. To resolve this issue, we are implementing the complete iterative procedure and we will report on the convergence aspects of the adjoint iterations. Our current work is also directed towards addressing the lack of data on other boundaries of our domain and improving the source reconstruction by including teleseismic data for those boundaries and non-negativity constraints on the dominant slip-rate component.

Running coupling from gluon and ghost propagators in the Landau gauge: Yang-Mills theories with adjoint fermions

NASA Astrophysics Data System (ADS)

Bergner, Georg; Piemonte, Stefano

2018-04-01

Non-Abelian gauge theories with fermions transforming in the adjoint representation of the gauge group (AdjQCD) are a fundamental ingredient of many models that describe the physics beyond the Standard Model. Two relevant examples are N =1 supersymmetric Yang-Mills (SYM) theory and minimal walking technicolor, which are gauge theories coupled to one adjoint Majorana and two adjoint Dirac fermions, respectively. While confinement is a property of N =1 SYM, minimal walking technicolor is expected to be infrared conformal. We study the propagators of ghost and gluon fields in the Landau gauge to compute the running coupling in the MiniMom scheme. We analyze several different ensembles of lattice Monte Carlo simulations for the SU(2) adjoint QCD with Nf=1 /2 ,1 ,3 /2 , and 2 Dirac fermions. We show how the running of the coupling changes as the number of interacting fermions is increased towards the conformal window.

The continuous adjoint approach to the k-ε turbulence model for shape optimization and optimal active control of turbulent flows

NASA Astrophysics Data System (ADS)

Papoutsis-Kiachagias, E. M.; Zymaris, A. S.; Kavvadias, I. S.; Papadimitriou, D. I.; Giannakoglou, K. C.

2015-03-01

The continuous adjoint to the incompressible Reynolds-averaged Navier-Stokes equations coupled with the low Reynolds number Launder-Sharma k-ε turbulence model is presented. Both shape and active flow control optimization problems in fluid mechanics are considered, aiming at minimum viscous losses. In contrast to the frequently used assumption of frozen turbulence, the adjoint to the turbulence model equations together with appropriate boundary conditions are derived, discretized and solved. This is the first time that the adjoint equations to the Launder-Sharma k-ε model have been derived. Compared to the formulation that neglects turbulence variations, the impact of additional terms and equations is evaluated. Sensitivities computed using direct differentiation and/or finite differences are used for comparative purposes. To demonstrate the need for formulating and solving the adjoint to the turbulence model equations, instead of merely relying upon the 'frozen turbulence assumption', the gain in the optimization turnaround time offered by the proposed method is quantified.

Technical Note: Adjoint formulation of the TOMCAT atmospheric transport scheme in the Eulerian backtracking framework (RETRO-TOM)

NASA Astrophysics Data System (ADS)

Haines, P. E.; Esler, J. G.; Carver, G. D.

2014-06-01

A new methodology for the formulation of an adjoint to the transport component of the chemistry transport model TOMCAT is described and implemented in a new model, RETRO-TOM. The Eulerian backtracking method is used, allowing the forward advection scheme (Prather's second-order moments) to be efficiently exploited in the backward adjoint calculations. Prather's scheme is shown to be time symmetric, suggesting the possibility of high accuracy. To attain this accuracy, however, it is necessary to make a careful treatment of the "density inconsistency" problem inherent to offline transport models. The results are verified using a series of test experiments. These demonstrate the high accuracy of RETRO-TOM when compared with direct forward sensitivity calculations, at least for problems in which flux limiters in the advection scheme are not required. RETRO-TOM therefore combines the flexibility and stability of a "finite difference of adjoint" formulation with the accuracy of an "adjoint of finite difference" formulation.

Technical Note: Adjoint formulation of the TOMCAT atmospheric transport scheme in the Eulerian backtracking framework (RETRO-TOM)

NASA Astrophysics Data System (ADS)

Haines, P. E.; Esler, J. G.; Carver, G. D.

2014-01-01

A new methodology for the formulation of an adjoint to the transport component of the chemistry transport model TOMCAT is described and implemented in a new model RETRO-TOM. The Eulerian backtracking method is used, allowing the forward advection scheme (Prather's second-order moments), to be efficiently exploited in the backward adjoint calculations. Prather's scheme is shown to be time-symmetric suggesting the possibility of high accuracy. To attain this accuracy, however, it is necessary to make a careful treatment of the "density inconsistency" problem inherent to offline transport models. The results are verified using a series of test experiments. These demonstrate the high accuracy of RETRO-TOM when compared with direct forward sensitivity calculations, at least for problems in which flux-limiters in the advection scheme are not required. RETRO-TOM therefore combines the flexibility and stability of a "finite difference of adjoint" formulation with the accuracy of an "adjoint of finite difference" formulation.

On the theory of self-adjoint extensions of symmetric operators and its applications to quantum physics

NASA Astrophysics Data System (ADS)

Ibort, A.; Pérez-Pardo, J. M.

2015-04-01

This is a series of five lectures around the common subject of the construction of self-adjoint extensions of symmetric operators and its applications to Quantum Physics. We will try to offer a brief account of some recent ideas in the theory of self-adjoint extensions of symmetric operators on Hilbert spaces and their applications to a few specific problems in Quantum Mechanics.

The solution of the problem of oil spill risk control in the Baltic Sea taking into account the processes of oil propagation and degradation

NASA Astrophysics Data System (ADS)

Aseev, Nikita; Agoshkov, Valery

2015-04-01

The report is devoted to the one approach to the problem of oil spill risk control of protected areas in the Baltic Sea (Aseev et al., 2014). By the problem of risk control is meant a problem of determination of optimal resources quantity which are necessary for decreasing the risk to some acceptable value. It is supposed that only moment of accident is a random variable. Mass of oil slick is chosen as a function of control. For the realization of the random variable the quadratic 'functional of cost' is introduced. It comprises cleaning costs and deviation of damage of oil pollution from its acceptable value. The problem of minimization of this functional is solved based on the methods of optimal control and the theory of adjoint equations (Agoshkov, 2003, Agoshkov et al., 2012). The solution of this problem is explicitly found. In order to solve the realistic problem of oil spill risk control in the Baltic Sea the 2d model of oil spill propagation on the sea surface based on the Seatrack Web model (Liungman, Mattson, 2011) is developed. The model takes into account such processes as oil transportation by sea currents and wind, turbulent diffusion, spreading, evaporation from sea surface, dispersion and formation of emulsion 'water-in-oil'. The model allows to calculate basic oil slick parameters: localization, mass, volume, thickness, density of oil, water content and viscosity of emulsion. The results of several numerical experiments in the Baltic Sea using the model and the methodology of oil spill risk control are presented. Along with moment of accident other parameters of oil spill and environment could be chosen as a random variables. The methodology of solution of oil spill risk control problem will remain the same but the computational complexity will increase. Conversion of the function of control to quantity of resources with a glance to methods of pollution removal should be processed. As a result, the developed 2d model of oil spill propagation combined with the methodology of solution of oil spill risk control problem could provide the basis for oil spill simulation systems, systems of evaluation and control of oil spill risk and damage in seas or decision support systems. References V.I. Agoshkov. The methods of optimal control and adjoint equations in problems of mathematical physics. // Moscow: INM RAS, 2003, 256 p. (in Russian). V.I. Agoshkov, N.A. Aseev, I.S. Novikov. The methods of investigation and solution of the problems of local sources and local or integral observations. // Moscow: INM RAS, 2012. 151 p. (in Russian). N.A. Aseev, V.I. Agoshkov, V.B. Zalesny, R. Aps, P. Kujala, and J. Rytkonen. The problem of control of oil pollution risk in the Baltic Sea // Russ. J. Numer. Analysis and Math. Modelling, 2014, V 29, No. 2, 93-105. O. Liungman, J. Mattson. Scientific documentation of Seatrack Web; physical processes, algorithms and references, 2011. // https://stw-helcom.smhi.se/

Boundary-Layer Stability Analysis of the Mean Flows Obtained Using Unstructured Grids

NASA Technical Reports Server (NTRS)

Liao, Wei; Malik, Mujeeb R.; Lee-Rausch, Elizabeth M.; Li, Fei; Nielsen, Eric J.; Buning, Pieter G.; Chang, Chau-Lyan; Choudhari, Meelan M.

2012-01-01

Boundary-layer stability analyses of mean flows extracted from unstructured-grid Navier- Stokes solutions have been performed. A procedure has been developed to extract mean flow profiles from the FUN3D unstructured-grid solutions. Extensive code-to-code validations have been performed by comparing the extracted mean ows as well as the corresponding stability characteristics to the predictions based on structured-grid solutions. Comparisons are made on a range of problems from a simple at plate to a full aircraft configuration-a modified Gulfstream-III with a natural laminar flow glove. The future aim of the project is to extend the adjoint-based design capability in FUN3D to include natural laminar flow and laminar flow control by integrating it with boundary-layer stability analysis codes, such as LASTRAC.

Large-Scale Optimization for Bayesian Inference in Complex Systems

DOE Office of Scientific and Technical Information (OSTI.GOV)

Willcox, Karen; Marzouk, Youssef

2013-11-12

The SAGUARO (Scalable Algorithms for Groundwater Uncertainty Analysis and Robust Optimization) Project focused on the development of scalable numerical algorithms for large-scale Bayesian inversion in complex systems that capitalize on advances in large-scale simulation-based optimization and inversion methods. The project was a collaborative effort among MIT, the University of Texas at Austin, Georgia Institute of Technology, and Sandia National Laboratories. The research was directed in three complementary areas: efficient approximations of the Hessian operator, reductions in complexity of forward simulations via stochastic spectral approximations and model reduction, and employing large-scale optimization concepts to accelerate sampling. The MIT--Sandia component of themore » SAGUARO Project addressed the intractability of conventional sampling methods for large-scale statistical inverse problems by devising reduced-order models that are faithful to the full-order model over a wide range of parameter values; sampling then employs the reduced model rather than the full model, resulting in very large computational savings. Results indicate little effect on the computed posterior distribution. On the other hand, in the Texas--Georgia Tech component of the project, we retain the full-order model, but exploit inverse problem structure (adjoint-based gradients and partial Hessian information of the parameter-to-observation map) to implicitly extract lower dimensional information on the posterior distribution; this greatly speeds up sampling methods, so that fewer sampling points are needed. We can think of these two approaches as ``reduce then sample'' and ``sample then reduce.'' In fact, these two approaches are complementary, and can be used in conjunction with each other. Moreover, they both exploit deterministic inverse problem structure, in the form of adjoint-based gradient and Hessian information of the underlying parameter-to-observation map, to achieve their speedups.« less

Final Report: Large-Scale Optimization for Bayesian Inference in Complex Systems

DOE Office of Scientific and Technical Information (OSTI.GOV)

Ghattas, Omar

2013-10-15

The SAGUARO (Scalable Algorithms for Groundwater Uncertainty Analysis and Robust Optimiza- tion) Project focuses on the development of scalable numerical algorithms for large-scale Bayesian inversion in complex systems that capitalize on advances in large-scale simulation-based optimiza- tion and inversion methods. Our research is directed in three complementary areas: efficient approximations of the Hessian operator, reductions in complexity of forward simulations via stochastic spectral approximations and model reduction, and employing large-scale optimization concepts to accelerate sampling. Our efforts are integrated in the context of a challenging testbed problem that considers subsurface reacting flow and transport. The MIT component of the SAGUAROmore » Project addresses the intractability of conventional sampling methods for large-scale statistical inverse problems by devising reduced-order models that are faithful to the full-order model over a wide range of parameter values; sampling then employs the reduced model rather than the full model, resulting in very large computational savings. Results indicate little effect on the computed posterior distribution. On the other hand, in the Texas-Georgia Tech component of the project, we retain the full-order model, but exploit inverse problem structure (adjoint-based gradients and partial Hessian information of the parameter-to- observation map) to implicitly extract lower dimensional information on the posterior distribution; this greatly speeds up sampling methods, so that fewer sampling points are needed. We can think of these two approaches as "reduce then sample" and "sample then reduce." In fact, these two approaches are complementary, and can be used in conjunction with each other. Moreover, they both exploit deterministic inverse problem structure, in the form of adjoint-based gradient and Hessian information of the underlying parameter-to-observation map, to achieve their speedups.« less

Crustal wavespeed structure of North Texas and Oklahoma based on ambient noise cross-correlation functions and adjoint tomography

NASA Astrophysics Data System (ADS)

Zhu, Hejun

2018-04-01

Recently, seismologists observed increasing seismicity in North Texas and Oklahoma. Based on seismic observations and other geophysical measurements, numerous studies suggested links between the increasing seismicity and wastewater injection during unconventional oil and gas exploration. To better monitor seismic events and investigate their triggering mechanisms, we need an accurate 3D crustal wavespeed model for the study region. Considering the uneven distribution of earthquakes in this area, seismic tomography with local earthquake records have difficulties achieving even illumination. To overcome this limitation, in this study, ambient noise cross-correlation functions are used to constrain subsurface variations in wavespeeds. I use adjoint tomography to iteratively fit frequency-dependent phase differences between observed and predicted band-limited Green's functions. The spectral-element method is used to numerically calculate the band-limited Green's functions and the adjoint method is used to calculate misfit gradients with respect to wavespeeds. Twenty five preconditioned conjugate gradient iterations are used to update model parameters and minimize data misfits. Features in the new crustal model TO25 correlates well with geological provinces in the study region, including the Llano uplift, the Anadarko basin and the Ouachita orogenic front, etc. In addition, there are relatively good correlations between seismic results with gravity and magnetic observations. This new crustal model can be used to better constrain earthquake source parameters in North Texas and Oklahoma, such as epicenter location as well as moment tensor solutions, which are important for investigating triggering mechanisms between these induced earthquakes and unconventional oil and gas exploration activities.

The development of three-dimensional adjoint method for flow control with blowing in convergent-divergent nozzle flows

NASA Astrophysics Data System (ADS)

Sikarwar, Nidhi

The noise produced by the low bypass ratio turbofan engines used to power fighter aircraft is a problem for communities near military bases and for personnel working in close proximity to the aircraft. For example, carrier deck personnel are subject to noise exposure that can result in Noise-Induced Hearing Loss which in-turn results in over a billion dollars of disability payments by the Veterans Administration. Several methods have been proposed to reduce the jet noise at the source. These methods include microjet injection of air or water downstream of the jet exit, chevrons, and corrugated nozzle inserts. The last method involves the insertion of corrugated seals into the diverging section of a military-style convergent-divergent jet nozzle (to replace the existing seals). This has been shown to reduce both the broadband shock-associated noise as well as the mixing noise in the peak noise radiation direction. However, the original inserts were designed to be effective for a take-off condition where the jet is over-expanded. The nozzle performance would be expected to degrade at other conditions, such as in cruise at altitude. A new method has been proposed to achieve the same effects as corrugated seals, but using fluidic inserts. This involves injection of air, at relatively low pressures and total mass flow rates, into the diverging section of the nozzle. These fluidic inserts" deflect the flow in the same way as the mechanical inserts. The fluidic inserts represent an active control method, since the injectors can be modified or turned off depending on the jet operating conditions. Noise reductions in the peak noise direction of 5 to 6 dB have been achieved and broadband shock-associated noise is effectively suppressed. There are multiple parameters to be considered in the design of the fluidic inserts. This includes the number and location of the injectors and the pressures and mass flow rates to be used. These could be optimized on an ad hoc basis with multiple experiments or numerical simulations. Alternatively an inverse design method can be used. An adjoint optimization method can be used to achieve the optimum blowing rate. It is shown that the method works for both geometry optimization and active control of the flow in order to deflect the flow in desirable ways. An adjoint optimization method is described. It is used to determine the blowing distribution in the diverging section of a convergent-divergent nozzle that gives a desired pressure distribution in the nozzle. Both the direct and adjoint problems and their associated boundary conditions are developed. The adjoint method is used to determine the blowing distribution required to minimize the shock strength in the nozzle to achieve a known target pressure and to achieve close to an ideally expanded flow pressure. A multi-block structured solver is developed to calculate the flow solution and associated adjoint variables. Two and three-dimensional calculations are performed for internal and external of the nozzle domains. A two step MacCormack scheme based on predictor- corrector technique is was used for some calculations. The four and five stage Runge-Kutta schemes are also used to artificially march in time. A modified Runge-Kutta scheme is used to accelerate the convergence to a steady state. Second order artificial dissipation has been added to stabilize the calculations. The steepest decent method has been used for the optimization of the blowing velocity after the gradients of the cost function with respect to the blowing velocity are calculated using adjoint method. Several examples are given of the optimization of blowing using the adjoint method.

Preliminary Results from the Application of Automated Adjoint Code Generation to CFL3D

NASA Technical Reports Server (NTRS)

Carle, Alan; Fagan, Mike; Green, Lawrence L.

1998-01-01

This report describes preliminary results obtained using an automated adjoint code generator for Fortran to augment a widely-used computational fluid dynamics flow solver to compute derivatives. These preliminary results with this augmented code suggest that, even in its infancy, the automated adjoint code generator can accurately and efficiently deliver derivatives for use in transonic Euler-based aerodynamic shape optimization problems with hundreds to thousands of independent design variables.

Global Modeling and Data Assimilation. Volume 11; Documentation of the Tangent Linear and Adjoint Models of the Relaxed Arakawa-Schubert Moisture Parameterization of the NASA GEOS-1 GCM; 5.2

NASA Technical Reports Server (NTRS)

Suarez, Max J. (Editor); Yang, Wei-Yu; Todling, Ricardo; Navon, I. Michael

1997-01-01

A detailed description of the development of the tangent linear model (TLM) and its adjoint model of the Relaxed Arakawa-Schubert moisture parameterization package used in the NASA GEOS-1 C-Grid GCM (Version 5.2) is presented. The notational conventions used in the TLM and its adjoint codes are described in detail.

Output-Adaptive Tetrahedral Cut-Cell Validation for Sonic Boom Prediction

NASA Technical Reports Server (NTRS)

Park, Michael A.; Darmofal, David L.

2008-01-01

A cut-cell approach to Computational Fluid Dynamics (CFD) that utilizes the median dual of a tetrahedral background grid is described. The discrete adjoint is also calculated, which permits adaptation based on improving the calculation of a specified output (off-body pressure signature) in supersonic inviscid flow. These predicted signatures are compared to wind tunnel measurements on and off the configuration centerline 10 body lengths below the model to validate the method for sonic boom prediction. Accurate mid-field sonic boom pressure signatures are calculated with the Euler equations without the use of hybrid grid or signature propagation methods. Highly-refined, shock-aligned anisotropic grids were produced by this method from coarse isotropic grids created without prior knowledge of shock locations. A heuristic reconstruction limiter provided stable flow and adjoint solution schemes while producing similar signatures to Barth-Jespersen and Venkatakrishnan limiters. The use of cut-cells with an output-based adaptive scheme completely automated this accurate prediction capability after a triangular mesh is generated for the cut surface. This automation drastically reduces the manual intervention required by existing methods.

Optimization of Regional Geodynamic Models for Mantle Dynamics

NASA Astrophysics Data System (ADS)

Knepley, M.; Isaac, T.; Jadamec, M. A.

2016-12-01

The SubductionGenerator program is used to construct high resolution, 3D regional thermal structures for mantle convection simulations using a variety of data sources, including sea floor ages and geographically referenced 3D slab locations based on seismic observations. The initial bulk temperature field is constructed using a half-space cooling model or plate cooling model, and related smoothing functions based on a diffusion length-scale analysis. In this work, we seek to improve the 3D thermal model and test different model geometries and dynamically driven flow fields using constraints from observed seismic velocities and plate motions. Through a formal adjoint analysis, we construct the primal-dual version of the multi-objective PDE-constrained optimization problem for the plate motions and seismic misfit. We have efficient, scalable preconditioners for both the forward and adjoint problems based upon a block preconditioning strategy, and a simple gradient update is used to improve the control residual. The full optimal control problem is formulated on a nested hierarchy of grids, allowing a nonlinear multigrid method to accelerate the solution.

On quantum integrability of the Landau-Lifshitz model

NASA Astrophysics Data System (ADS)

Melikyan, A.; Pinzul, A.

2009-10-01

We investigate the quantum integrability of the Landau-Lifshitz (LL) model and solve the long-standing problem of finding the local quantum Hamiltonian for the arbitrary n-particle sector. The particular difficulty of the LL model quantization, which arises due to the ill-defined operator product, is dealt with by simultaneously regularizing the operator product and constructing the self-adjoint extensions of a very particular structure. The diagonalizibility difficulties of the Hamiltonian of the LL model, due to the highly singular nature of the quantum-mechanical Hamiltonian, are also resolved in our method for the arbitrary n-particle sector. We explicitly demonstrate the consistency of our construction with the quantum inverse scattering method due to Sklyanin [Lett. Math. Phys. 15, 357 (1988)] and give a prescription to systematically construct the general solution, which explains and generalizes the puzzling results of Sklyanin for the particular two-particle sector case. Moreover, we demonstrate the S-matrix factorization and show that it is a consequence of the discontinuity conditions on the functions involved in the construction of the self-adjoint extensions.

On predicting receptivity to surface roughness in a compressible infinite swept wing boundary layer

NASA Astrophysics Data System (ADS)

Thomas, Christian; Mughal, Shahid; Ashworth, Richard

2017-03-01

The receptivity of crossflow disturbances on an infinite swept wing is investigated using solutions of the adjoint linearised Navier-Stokes equations. The adjoint based method for predicting the magnitude of stationary disturbances generated by randomly distributed surface roughness is described, with the analysis extended to include both surface curvature and compressible flow effects. Receptivity is predicted for a broad spectrum of spanwise wavenumbers, variable freestream Reynolds numbers, and subsonic Mach numbers. Curvature is found to play a significant role in the receptivity calculations, while compressible flow effects are only found to marginally affect the initial size of the crossflow instability. A Monte Carlo type analysis is undertaken to establish the mean amplitude and variance of crossflow disturbances generated by the randomly distributed surface roughness. Mean amplitudes are determined for a range of flow parameters that are maximised for roughness distributions containing a broad spectrum of roughness wavelengths, including those that are most effective in generating stationary crossflow disturbances. A control mechanism is then developed where the short scale roughness wavelengths are damped, leading to significant reductions in the receptivity amplitude.

Using Adjoint Methods to Improve 3-D Velocity Models of Southern California

NASA Astrophysics Data System (ADS)

Liu, Q.; Tape, C.; Maggi, A.; Tromp, J.

2006-12-01

We use adjoint methods popular in climate and ocean dynamics to calculate Fréchet derivatives for tomographic inversions in southern California. The Fréchet derivative of an objective function χ(m), where m denotes the Earth model, may be written in the generic form δχ=int Km(x) δln m(x) d3x, where δln m=δ m/m denotes the relative model perturbation. For illustrative purposes, we construct the 3-D finite-frequency banana-doughnut kernel Km, corresponding to the misfit of a single traveltime measurement, by simultaneously computing the 'adjoint' wave field s† forward in time and reconstructing the regular wave field s backward in time. The adjoint wave field is produced by using the time-reversed velocity at the receiver as a fictitious source, while the regular wave field is reconstructed on the fly by propagating the last frame of the wave field saved by a previous forward simulation backward in time. The approach is based upon the spectral-element method, and only two simulations are needed to produce density, shear-wave, and compressional-wave sensitivity kernels. This method is applied to the SCEC southern California velocity model. Various density, shear-wave, and compressional-wave sensitivity kernels are presented for different phases in the seismograms. We also generate 'event' kernels for Pnl, S and surface waves, which are the Fréchet kernels of misfit functions that measure the P, S or surface wave traveltime residuals at all the receivers simultaneously for one particular event. Effectively, an event kernel is a sum of weighted Fréchet kernels, with weights determined by the associated traveltime anomalies. By the nature of the 3-D simulation, every event kernel is also computed based upon just two simulations, i.e., its construction costs the same amount of computation time as an individual banana-doughnut kernel. One can think of the sum of the event kernels for all available earthquakes, called the 'misfit' kernel, as a graphical representation of the gradient of the misfit function. With the capability of computing both the value of the misfit function and its gradient, which assimilates the traveltime anomalies, we are ready to use a non-linear conjugate gradient algorithm to iteratively improve velocity models of southern California.

Adjoint-based Simultaneous Estimation Method of Fault Slip and Asthenosphere Viscosity Using Large-Scale Finite Element Simulation of Viscoelastic Deformation

NASA Astrophysics Data System (ADS)

Agata, R.; Ichimura, T.; Hori, T.; Hirahara, K.; Hashimoto, C.; Hori, M.

2016-12-01

Estimation of the coseismic/postseismic slip using postseismic deformation observation data is an important topic in the field of geodetic inversion. Estimation methods for this purpose are expected to be improved by introducing numerical simulation tools (e.g. finite element (FE) method) of viscoelastic deformation, in which the computation model is of high fidelity to the available high-resolution crustal data. The authors have proposed a large-scale simulation method using such FE high-fidelity models (HFM), assuming use of a large-scale computation environment such as the K computer in Japan (Ichimura et al. 2016). On the other hand, the values of viscosity in the heterogeneous viscoelastic structure in the high-fidelity model are not trivial. In this study, we developed an adjoint-based optimization method incorporating HFM, in which fault slip and asthenosphere viscosity are simultaneously estimated. We carried out numerical experiments using synthetic crustal deformation data. We constructed an HFM in the domain of 2048x1536x850 km, which includes the Tohoku region in northeast Japan based on Ichimura et al. (2013). We used the model geometry data set of JTOPO30 (2003), Koketsu et al. (2008) and CAMP standard model (Hashimoto et al. 2004). The geometry of crustal structures in HFM is in 1km resolution, resulting in 36 billion degrees-of-freedom. Synthetic crustal deformation data due to prescribed coseismic slip and after slips in the location of GEONET, GPS/A observation points, and S-net are used. The target inverse analysis is formulated as minimization of L2 norm of the difference between the FE simulation results and the observation data with respect to viscosity and fault slip, combining the quasi-Newton algorithm with the adjoint method. Use of this combination decreases the necessary number of forward analyses in the optimization calculation. As a result, we are now able to finish the estimation using 2560 computer nodes of the K computer for less than 17 hours. Thus, the target inverse analysis is completed in a realistic time because of the combination of the fast solver and the adjoint method. In the future, we would like to apply the method to the actual data.

Development of adaptive observation strategy using retrospective optimal interpolation

NASA Astrophysics Data System (ADS)

Noh, N.; Kim, S.; Song, H.; Lim, G.

2011-12-01

Retrospective optimal interpolation (ROI) is a method that is used to minimize cost functions with multiple minima without using adjoint models. Song and Lim (2011) perform the experiments to reduce the computational costs for implementing ROI by transforming the control variables into eigenvectors of background error covariance. We adapt the ROI algorithm to compute sensitivity estimates of severe weather events over the Korean peninsula. The eigenvectors of the ROI algorithm is modified every time the observations are assimilated. This implies that the modified eigenvectors shows the error distribution of control variables which are updated by assimilating observations. So, We can estimate the effects of the specific observations. In order to verify the adaptive observation strategy, High-impact weather over the Korean peninsula is simulated and interpreted using WRF modeling system and sensitive regions for each high-impact weather is calculated. The effects of assimilation for each observation type is discussed.

Quantum stochastic calculus associated with quadratic quantum noises

DOE Office of Scientific and Technical Information (OSTI.GOV)

Ji, Un Cig, E-mail: uncigji@chungbuk.ac.kr; Sinha, Kalyan B., E-mail: kbs-jaya@yahoo.co.in

2016-02-15

We first study a class of fundamental quantum stochastic processes induced by the generators of a six dimensional non-solvable Lie †-algebra consisting of all linear combinations of the generalized Gross Laplacian and its adjoint, annihilation operator, creation operator, conservation, and time, and then we study the quantum stochastic integrals associated with the class of fundamental quantum stochastic processes, and the quantum Itô formula is revisited. The existence and uniqueness of solution of a quantum stochastic differential equation is proved. The unitarity conditions of solutions of quantum stochastic differential equations associated with the fundamental processes are examined. The quantum stochastic calculusmore » extends the Hudson-Parthasarathy quantum stochastic calculus.« less

A-Posteriori Error Estimates for Mixed Finite Element and Finite Volume Methods for Problems Coupled Through a Boundary with Non-Matching Grids

DTIC Science & Technology

2013-08-01

both MFE and GFV, are often similar in size. As a gross measure of the effect of geometric projection and of the use of quadrature, we also report the...interest MFE ∑(e,ψ) or GFV ∑(e,ψ). Tables 1 and 2 show this using coarse and fine forward solutions. Table 1. The forward problem with solution (4.1) is run...adjoint data components ψu and ψp are constant everywhere and ψξ = 0. adj. grid MFE ∑(e,ψ) ∑MFEi ratio GFV ∑(e,ψ) ∑GFV i ratio 20x20 : 32x32 1.96E−3

Green's formula and variational principles for cosmic-ray transport with application to rotating and shearing flows

NASA Technical Reports Server (NTRS)

Webb, G. M.; Jokipii, J. R.; Morfill, G. E.

1994-01-01

Green's theorem and Green's formula for the diffusive cosmic-ray transport equation in relativistic flows are derived. Green's formula gives the solution of the transport equation in terms of the Green's function of the adjoint transport equation, and in terms of distributed sources throughout the region R of interest, plus terms involving the particle intensity and streaming on the boundary. The adjoint transport equation describes the time-reversed particle transport. An Euler-Lagrange variational principle is then obtained for both the mean scattering frame distribution function f, and its adjoint f(dagger). Variations of the variational functional with respect to f(dagger) yield the transport equation, whereas variations of f yield the adjoint transport equation. The variational principle, when combined with Noether's theorem, yields the conservation law associated with Green's theorem. An investigation of the transport equation for steady, azimuthal, rotating flows suggests the introduction of a new independent variable H to replace the comoving frame momentum variable p'. For the case of rigid rotating flows, H is conserved and is shown to be analogous to the Hamiltonian for a bead on a rigidly rotating wire. The variable H corresponds to a balance between the centrifugal force and the particle inertia in the rotating frame. The physical interpretation of H includes a discussion of nonrelativistic and special relativistic rotating flows as well as the cases of aziuthal, differentially rotating flows about Schwarzs-child and Kerr black holes. Green's formula is then applied to the problem of the acceleration of ultra-high-energy cosmic rays by galactic rotation. The model for galactic rotation assumes an angular velocity law Omega = Omega(sub 0)(omega(sub 0)/omega), where omega denotes radial distance from the axis of rotation. Green's functions for the galactic rotation problem are used to investigate the spectrum of accelerated particles arising from monoenergetic and truncated power-law sources. We conclude that it is possible to accelerate particles beyond the knee by galactic rotation, but not in sufficient number to adequately explain the observed spectrum.

Inverse Regional Modeling with Adjoint-Free Technique

NASA Astrophysics Data System (ADS)

Yaremchuk, M.; Martin, P.; Panteleev, G.; Beattie, C.

2016-02-01

The ongoing parallelization trend in computer technologies facilitates the use ensemble methods in geophysical data assimilation. Of particular interest are ensemble techniques which do not require the development of tangent linear numerical models and their adjoints for optimization. These ``adjoint-free'' methods minimize the cost function within the sequence of subspaces spanned by a carefully chosen sets perturbations of the control variables. In this presentation, an adjoint-free variational technique (a4dVar) is demonstrated in an application estimating initial conditions of two numerical models: the Navy Coastal Ocean Model (NCOM), and the surface wave model (WAM). With the NCOM, performance of both adjoint and adjoint-free 4dVar data assimilation techniques is compared in application to the hydrographic surveys and velocity observations collected in the Adriatic Sea in 2006. Numerical experiments have shown that a4dVar is capable of providing forecast skill similar to that of conventional 4dVar at comparable computational expense while being less susceptible to excitation of ageostrophic modes that are not supported by observations. Adjoint-free technique constrained by the WAM model is tested in a series of data assimilation experiments with synthetic observations in the southern Chukchi Sea. The types of considered observations are directional spectra estimated from point measurements by stationary buoys, significant wave height (SWH) observations by coastal high-frequency radars and along-track SWH observations by satellite altimeters. The a4dVar forecast skill is shown to be 30-40% better than the skill of the sequential assimilaiton method based on optimal interpolation which is currently used in operations. Prospects of further development of the a4dVar methods in regional applications are discussed.

Transient sensitivities of sea ice export through the Canadian Arctic Archipelago inferred from a coupled ocean/sea-ice adjoint model

NASA Astrophysics Data System (ADS)

Heimbach, P.; Losch, M.; Menemenlis, D.; Campin, J.; Hill, C.

2008-12-01

The sensitivity of sea-ice export through the Canadian Arctic Archipelago (CAA), measured in terms of its solid freshwater export through Lancaster Sound, to changes in various elements of the ocean and sea-ice state, and to elements of the atmospheric forcing fields through time and space is assessed by means of a coupled ocean/sea-ice adjoint model. The adjoint model furnishes full spatial sensitivity maps (also known as Lagrange multipliers) of the export metric to a variety of model variables at any chosen point in time, providing the unique capability to quantify major drivers of sea-ice export variability. The underlying model is the MIT ocean general circulation model (MITgcm), which is coupled to a Hibler-type dynamic/thermodynamic sea-ice model. The configuration is based on the Arctic face of the ECCO3 high-resolution cubed-sphere model, but coarsened to 36-km horizontal grid spacing. The adjoint of the coupled system has been derived by means of automatic differentiation using the software tool TAF. Finite perturbation simulations are performed to check the information provided by the adjoint. The sea-ice model's performance in the presence of narrow straits is assessed with different sea-ice lateral boundary conditions. The adjoint sensitivity clearly exposes the role of the model trajectory and the transient nature of the problem. The complex interplay between forcing, dynamics, and boundary condition is demonstrated in the comparison between the different calculations. The study is a step towards fully coupled adjoint-based ocean/sea-ice state estimation at basin to global scales as part of the ECCO efforts.

Global Seismic Imaging Based on Adjoint Tomography

NASA Astrophysics Data System (ADS)

Bozdag, E.; Lefebvre, M.; Lei, W.; Peter, D. B.; Smith, J. A.; Zhu, H.; Komatitsch, D.; Tromp, J.

2013-12-01

Our aim is to perform adjoint tomography at the scale of globe to image the entire planet. We have started elastic inversions with a global data set of 253 CMT earthquakes with moment magnitudes in the range 5.8 ≤ Mw ≤ 7 and used GSN stations as well as some local networks such as USArray, European stations, etc. Using an iterative pre-conditioned conjugate gradient scheme, we initially set the aim to obtain a global crustal and mantle model with confined transverse isotropy in the upper mantle. Global adjoint tomography has so far remained a challenge mainly due to computational limitations. Recent improvements in our 3D solvers (e.g., a GPU version) and access to high-performance computational centers (e.g., ORNL's Cray XK7 "Titan" system) now enable us to perform iterations with higher-resolution (T > 9 s) and longer-duration (200 min) simulations to accommodate high-frequency body waves and major-arc surface waves, respectively, which help improve data coverage. The remaining challenge is the heavy I/O traffic caused by the numerous files generated during the forward/adjoint simulations and the pre- and post-processing stages of our workflow. We improve the global adjoint tomography workflow by adopting the ADIOS file format for our seismic data as well as models, kernels, etc., to improve efficiency on high-performance clusters. Our ultimate aim is to use data from all available networks and earthquakes within the magnitude range of our interest (5.5 ≤ Mw ≤ 7) which requires a solid framework to manage big data in our global adjoint tomography workflow. We discuss the current status and future of global adjoint tomography based on our initial results as well as practical issues such as handling big data in inversions and on high-performance computing systems.

An adjoint view on flux consistency and strong wall boundary conditions to the Navier–Stokes equations

DOE Office of Scientific and Technical Information (OSTI.GOV)

Stück, Arthur, E-mail: arthur.stueck@dlr.de

2015-11-15

Inconsistent discrete expressions in the boundary treatment of Navier–Stokes solvers and in the definition of force objective functionals can lead to discrete-adjoint boundary treatments that are not a valid representation of the boundary conditions to the corresponding adjoint partial differential equations. The underlying problem is studied for an elementary 1D advection–diffusion problem first using a node-centred finite-volume discretisation. The defect of the boundary operators in the inconsistently defined discrete-adjoint problem leads to oscillations and becomes evident with the additional insight of the continuous-adjoint approach. A homogenisation of the discretisations for the primal boundary treatment and the force objective functional yieldsmore » second-order functional accuracy and eliminates the defect in the discrete-adjoint boundary treatment. Subsequently, the issue is studied for aerodynamic Reynolds-averaged Navier–Stokes problems in conjunction with a standard finite-volume discretisation on median-dual grids and a strong implementation of noslip walls, found in many unstructured general-purpose flow solvers. Going out from a base-line discretisation of force objective functionals which is independent of the boundary treatment in the flow solver, two improved flux-consistent schemes are presented; based on either body wall-defined or farfield-defined control-volumes they resolve the dual inconsistency. The behaviour of the schemes is investigated on a sequence of grids in 2D and 3D.« less

Practical global oceanic state estimation

NASA Astrophysics Data System (ADS)

Wunsch, Carl; Heimbach, Patrick

2007-06-01

The problem of oceanographic state estimation, by means of an ocean general circulation model (GCM) and a multitude of observations, is described and contrasted with the meteorological process of data assimilation. In practice, all such methods reduce, on the computer, to forms of least-squares. The global oceanographic problem is at the present time focussed primarily on smoothing, rather than forecasting, and the data types are unlike meteorological ones. As formulated in the consortium Estimating the Circulation and Climate of the Ocean (ECCO), an automatic differentiation tool is used to calculate the so-called adjoint code of the GCM, and the method of Lagrange multipliers used to render the problem one of unconstrained least-squares minimization. Major problems today lie less with the numerical algorithms (least-squares problems can be solved by many means) than with the issues of data and model error. Results of ongoing calculations covering the period of the World Ocean Circulation Experiment, and including among other data, satellite altimetry from TOPEX/POSEIDON, Jason-1, ERS- 1/2, ENVISAT, and GFO, a global array of profiling floats from the Argo program, and satellite gravity data from the GRACE mission, suggest that the solutions are now useful for scientific purposes. Both methodology and applications are developing in a number of different directions.

Numerical convergence and validation of the DIMP inverse particle transport model

DOE PAGES

Nelson, Noel; Azmy, Yousry

2017-09-01

The data integration with modeled predictions (DIMP) model is a promising inverse radiation transport method for solving the special nuclear material (SNM) holdup problem. Unlike previous methods, DIMP is a completely passive nondestructive assay technique that requires no initial assumptions regarding the source distribution or active measurement time. DIMP predicts the most probable source location and distribution through Bayesian inference and quasi-Newtonian optimization of predicted detector re-sponses (using the adjoint transport solution) with measured responses. DIMP performs well with for-ward hemispherical collimation and unshielded measurements, but several considerations are required when using narrow-view collimated detectors. DIMP converged well to themore » correct source distribution as the number of synthetic responses increased. DIMP also performed well for the first experimental validation exercise after applying a collimation factor, and sufficiently reducing the source search vol-ume's extent to prevent the optimizer from getting stuck in local minima. DIMP's simple point detector response function (DRF) is being improved to address coplanar false positive/negative responses, and an angular DRF is being considered for integration with the next version of DIMP to account for highly collimated responses. Overall, DIMP shows promise for solving the SNM holdup inverse problem, especially once an improved optimization algorithm is implemented.« less

Spectral-element simulations of wave propagation in complex exploration-industry models: Imaging and adjoint tomography

NASA Astrophysics Data System (ADS)

Luo, Y.; Nissen-Meyer, T.; Morency, C.; Tromp, J.

2008-12-01

Seismic imaging in the exploration industry is often based upon ray-theoretical migration techniques (e.g., Kirchhoff) or other ideas which neglect some fraction of the seismic wavefield (e.g., wavefield continuation for acoustic-wave first arrivals) in the inversion process. In a companion paper we discuss the possibility of solving the full physical forward problem (i.e., including visco- and poroelastic, anisotropic media) using the spectral-element method. With such a tool at hand, we can readily apply the adjoint method to tomographic inversions, i.e., iteratively improving an initial 3D background model to fit the data. In the context of this inversion process, we draw connections between kernels in adjoint tomography and basic imaging principles in migration. We show that the images obtained by migration are nothing but particular kinds of adjoint kernels (mainly density kernels). Migration is basically a first step in the iterative inversion process of adjoint tomography. We apply the approach to basic 2D problems involving layered structures, overthrusting faults, topography, salt domes, and poroelastic regions.

An adjoint method of sensitivity analysis for residual vibrations of structures subject to impacts

NASA Astrophysics Data System (ADS)

Yan, Kun; Cheng, Gengdong

2018-03-01

For structures subject to impact loads, the residual vibration reduction is more and more important as the machines become faster and lighter. An efficient sensitivity analysis of residual vibration with respect to structural or operational parameters is indispensable for using a gradient based optimization algorithm, which reduces the residual vibration in either active or passive way. In this paper, an integrated quadratic performance index is used as the measure of the residual vibration, since it globally measures the residual vibration response and its calculation can be simplified greatly with Lyapunov equation. Several sensitivity analysis approaches for performance index were developed based on the assumption that the initial excitations of residual vibration were given and independent of structural design. Since the resulting excitations by the impact load often depend on structural design, this paper aims to propose a new efficient sensitivity analysis method for residual vibration of structures subject to impacts to consider the dependence. The new method is developed by combining two existing methods and using adjoint variable approach. Three numerical examples are carried out and demonstrate the accuracy of the proposed method. The numerical results show that the dependence of initial excitations on structural design variables may strongly affects the accuracy of sensitivities.

SISYPHUS: A high performance seismic inversion factory

NASA Astrophysics Data System (ADS)

Gokhberg, Alexey; Simutė, Saulė; Boehm, Christian; Fichtner, Andreas

2016-04-01

In the recent years the massively parallel high performance computers became the standard instruments for solving the forward and inverse problems in seismology. The respective software packages dedicated to forward and inverse waveform modelling specially designed for such computers (SPECFEM3D, SES3D) became mature and widely available. These packages achieve significant computational performance and provide researchers with an opportunity to solve problems of bigger size at higher resolution within a shorter time. However, a typical seismic inversion process contains various activities that are beyond the common solver functionality. They include management of information on seismic events and stations, 3D models, observed and synthetic seismograms, pre-processing of the observed signals, computation of misfits and adjoint sources, minimization of misfits, and process workflow management. These activities are time consuming, seldom sufficiently automated, and therefore represent a bottleneck that can substantially offset performance benefits provided by even the most powerful modern supercomputers. Furthermore, a typical system architecture of modern supercomputing platforms is oriented towards the maximum computational performance and provides limited standard facilities for automation of the supporting activities. We present a prototype solution that automates all aspects of the seismic inversion process and is tuned for the modern massively parallel high performance computing systems. We address several major aspects of the solution architecture, which include (1) design of an inversion state database for tracing all relevant aspects of the entire solution process, (2) design of an extensible workflow management framework, (3) integration with wave propagation solvers, (4) integration with optimization packages, (5) computation of misfits and adjoint sources, and (6) process monitoring. The inversion state database represents a hierarchical structure with branches for the static process setup, inversion iterations, and solver runs, each branch specifying information at the event, station and channel levels. The workflow management framework is based on an embedded scripting engine that allows definition of various workflow scenarios using a high-level scripting language and provides access to all available inversion components represented as standard library functions. At present the SES3D wave propagation solver is integrated in the solution; the work is in progress for interfacing with SPECFEM3D. A separate framework is designed for interoperability with an optimization module; the workflow manager and optimization process run in parallel and cooperate by exchanging messages according to a specially designed protocol. A library of high-performance modules implementing signal pre-processing, misfit and adjoint computations according to established good practices is included. Monitoring is based on information stored in the inversion state database and at present implements a command line interface; design of a graphical user interface is in progress. The software design fits well into the common massively parallel system architecture featuring a large number of computational nodes running distributed applications under control of batch-oriented resource managers. The solution prototype has been implemented on the "Piz Daint" supercomputer provided by the Swiss Supercomputing Centre (CSCS).

A Matrix-Free Algorithm for Multidisciplinary Design Optimization

NASA Astrophysics Data System (ADS)

Lambe, Andrew Borean

Multidisciplinary design optimization (MDO) is an approach to engineering design that exploits the coupling between components or knowledge disciplines in a complex system to improve the final product. In aircraft design, MDO methods can be used to simultaneously design the outer shape of the aircraft and the internal structure, taking into account the complex interaction between the aerodynamic forces and the structural flexibility. Efficient strategies are needed to solve such design optimization problems and guarantee convergence to an optimal design. This work begins with a comprehensive review of MDO problem formulations and solution algorithms. First, a fundamental MDO problem formulation is defined from which other formulations may be obtained through simple transformations. Using these fundamental problem formulations, decomposition methods from the literature are reviewed and classified. All MDO methods are presented in a unified mathematical notation to facilitate greater understanding. In addition, a novel set of diagrams, called extended design structure matrices, are used to simultaneously visualize both data communication and process flow between the many software components of each method. For aerostructural design optimization, modern decomposition-based MDO methods cannot efficiently handle the tight coupling between the aerodynamic and structural states. This fact motivates the exploration of methods that can reduce the computational cost. A particular structure in the direct and adjoint methods for gradient computation motivates the idea of a matrix-free optimization method. A simple matrix-free optimizer is developed based on the augmented Lagrangian algorithm. This new matrix-free optimizer is tested on two structural optimization problems and one aerostructural optimization problem. The results indicate that the matrix-free optimizer is able to efficiently solve structural and multidisciplinary design problems with thousands of variables and constraints. On the aerostructural test problem formulated with thousands of constraints, the matrix-free optimizer is estimated to reduce the total computational time by up to 90% compared to conventional optimizers.

A Matrix-Free Algorithm for Multidisciplinary Design Optimization

NASA Astrophysics Data System (ADS)

Lambe, Andrew Borean

Multidisciplinary design optimization (MDO) is an approach to engineering design that exploits the coupling between components or knowledge disciplines in a complex system to improve the final product. In aircraft design, MDO methods can be used to simultaneously design the outer shape of the aircraft and the internal structure, taking into account the complex interaction between the aerodynamic forces and the structural flexibility. Efficient strategies are needed to solve such design optimization problems and guarantee convergence to an optimal design. This work begins with a comprehensive review of MDO problem formulations and solution algorithms. First, a fundamental MDO problem formulation is defined from which other formulations may be obtained through simple transformations. Using these fundamental problem formulations, decomposition methods from the literature are reviewed and classified. All MDO methods are presented in a unified mathematical notation to facilitate greater understanding. In addition, a novel set of diagrams, called extended design structure matrices, are used to simultaneously visualize both data communication and process flow between the many software components of each method. For aerostructural design optimization, modern decomposition-based MDO methods cannot efficiently handle the tight coupling between the aerodynamic and structural states. This fact motivates the exploration of methods that can reduce the computational cost. A particular structure in the direct and adjoint methods for gradient computation. motivates the idea of a matrix-free optimization method. A simple matrix-free optimizer is developed based on the augmented Lagrangian algorithm. This new matrix-free optimizer is tested on two structural optimization problems and one aerostructural optimization problem. The results indicate that the matrix-free optimizer is able to efficiently solve structural and multidisciplinary design problems with thousands of variables and constraints. On the aerostructural test problem formulated with thousands of constraints, the matrix-free optimizer is estimated to reduce the total computational time by up to 90% compared to conventional optimizers.

Small-inclusion asymptotic of misfit functionals for inverse problems in acoustics

NASA Astrophysics Data System (ADS)

Guzina, Bojan B.; Bonnet, Marc

2006-10-01

The aim of this study is an extension and employment of the concept of topological derivative as it pertains to the nucleation of infinitesimal inclusions in a reference (i.e. background) acoustic medium. The developments are motivated by the need to develop a preliminary indicator functional that would aid the solution of inverse scattering problems in terms of a rational initial 'guess' about the geometry and material characteristics of a hidden (finite) obstacle; an information that is often required by iterative minimization algorithms. To this end the customary definition of topological derivative, which quantifies the sensitivity of a given cost functional with respect to the creation of an infinitesimal hole, is adapted to permit the nucleation of a dissimilar acoustic medium. On employing the Green's function for the background domain, computation of topological sensitivity for the three-dimensional Helmholtz equation is reduced to the solution of a reference, Laplace transmission problem. Explicit formulae are given for the nucleating inclusions of spherical and ellipsoidal shapes. For generality the developments are also presented in an alternative, adjoint-field setting that permits nucleation of inclusions in an infinite, semi-infinite or finite background medium. Through numerical examples, it is shown that the featured topological sensitivity could be used, in the context of inverse scattering, as an effective obstacle indicator through an assembly of sampling points where it attains pronounced negative values. On varying a material characteristic (density) of the nucleating obstacle, it is also shown that the proposed methodology can be used as a preparatory tool for both geometric and material identification.

The Prediction of Broadband Shock-Associated Noise Including Propagation Effects

NASA Technical Reports Server (NTRS)

Miller, Steven; Morris, Philip J.

2011-01-01

An acoustic analogy is developed based on the Euler equations for broadband shock- associated noise (BBSAN) that directly incorporates the vector Green's function of the linearized Euler equations and a steady Reynolds-Averaged Navier-Stokes solution (SRANS) as the mean flow. The vector Green's function allows the BBSAN propagation through the jet shear layer to be determined. The large-scale coherent turbulence is modeled by two-point second order velocity cross-correlations. Turbulent length and time scales are related to the turbulent kinetic energy and dissipation. An adjoint vector Green's function solver is implemented to determine the vector Green's function based on a locally parallel mean flow at streamwise locations of the SRANS solution. However, the developed acoustic analogy could easily be based on any adjoint vector Green's function solver, such as one that makes no assumptions about the mean flow. The newly developed acoustic analogy can be simplified to one that uses the Green's function associated with the Helmholtz equation, which is consistent with the formulation of Morris and Miller (AIAAJ 2010). A large number of predictions are generated using three different nozzles over a wide range of fully expanded Mach numbers and jet stagnation temperatures. These predictions are compared with experimental data from multiple jet noise labs. In addition, two models for the so-called 'fine-scale' mixing noise are included in the comparisons. Improved BBSAN predictions are obtained relative to other models that do not include the propagation effects, especially in the upstream direction of the jet.

Inversion of geothermal heat flux in a thermomechanically coupled nonlinear Stokes ice sheet model

DOE Office of Scientific and Technical Information (OSTI.GOV)

Zhu, Hongyu; Petra, Noemi; Stadler, Georg

We address the inverse problem of inferring the basal geothermal heat flux from surface velocity observations using a steady-state thermomechanically coupled nonlinear Stokes ice flow model. This is a challenging inverse problem since the map from basal heat flux to surface velocity observables is indirect: the heat flux is a boundary condition for the thermal advection–diffusion equation, which couples to the nonlinear Stokes ice flow equations; together they determine the surface ice flow velocity. This multiphysics inverse problem is formulated as a nonlinear least-squares optimization problem with a cost functional that includes the data misfit between surface velocity observations andmore » model predictions. A Tikhonov regularization term is added to render the problem well posed. We derive adjoint-based gradient and Hessian expressions for the resulting partial differential equation (PDE)-constrained optimization problem and propose an inexact Newton method for its solution. As a consequence of the Petrov–Galerkin discretization of the energy equation, we show that discretization and differentiation do not commute; that is, the order in which we discretize the cost functional and differentiate it affects the correctness of the gradient. Using two- and three-dimensional model problems, we study the prospects for and limitations of the inference of the geothermal heat flux field from surface velocity observations. The results show that the reconstruction improves as the noise level in the observations decreases and that short-wavelength variations in the geothermal heat flux are difficult to recover. We analyze the ill-posedness of the inverse problem as a function of the number of observations by examining the spectrum of the Hessian of the cost functional. Motivated by the popularity of operator-split or staggered solvers for forward multiphysics problems – i.e., those that drop two-way coupling terms to yield a one-way coupled forward Jacobian – we study the effect on the inversion of a one-way coupling of the adjoint energy and Stokes equations. Here, we show that taking such a one-way coupled approach for the adjoint equations can lead to an incorrect gradient and premature termination of optimization iterations. This is due to loss of a descent direction stemming from inconsistency of the gradient with the contours of the cost functional. Nevertheless, one may still obtain a reasonable approximate inverse solution particularly if important features of the reconstructed solution emerge early in optimization iterations, before the premature termination.« less

Inversion of geothermal heat flux in a thermomechanically coupled nonlinear Stokes ice sheet model

DOE PAGES

Zhu, Hongyu; Petra, Noemi; Stadler, Georg; ...

2016-07-13

We address the inverse problem of inferring the basal geothermal heat flux from surface velocity observations using a steady-state thermomechanically coupled nonlinear Stokes ice flow model. This is a challenging inverse problem since the map from basal heat flux to surface velocity observables is indirect: the heat flux is a boundary condition for the thermal advection–diffusion equation, which couples to the nonlinear Stokes ice flow equations; together they determine the surface ice flow velocity. This multiphysics inverse problem is formulated as a nonlinear least-squares optimization problem with a cost functional that includes the data misfit between surface velocity observations andmore » model predictions. A Tikhonov regularization term is added to render the problem well posed. We derive adjoint-based gradient and Hessian expressions for the resulting partial differential equation (PDE)-constrained optimization problem and propose an inexact Newton method for its solution. As a consequence of the Petrov–Galerkin discretization of the energy equation, we show that discretization and differentiation do not commute; that is, the order in which we discretize the cost functional and differentiate it affects the correctness of the gradient. Using two- and three-dimensional model problems, we study the prospects for and limitations of the inference of the geothermal heat flux field from surface velocity observations. The results show that the reconstruction improves as the noise level in the observations decreases and that short-wavelength variations in the geothermal heat flux are difficult to recover. We analyze the ill-posedness of the inverse problem as a function of the number of observations by examining the spectrum of the Hessian of the cost functional. Motivated by the popularity of operator-split or staggered solvers for forward multiphysics problems – i.e., those that drop two-way coupling terms to yield a one-way coupled forward Jacobian – we study the effect on the inversion of a one-way coupling of the adjoint energy and Stokes equations. Here, we show that taking such a one-way coupled approach for the adjoint equations can lead to an incorrect gradient and premature termination of optimization iterations. This is due to loss of a descent direction stemming from inconsistency of the gradient with the contours of the cost functional. Nevertheless, one may still obtain a reasonable approximate inverse solution particularly if important features of the reconstructed solution emerge early in optimization iterations, before the premature termination.« less

Inversion of geothermal heat flux in a thermomechanically coupled nonlinear Stokes ice sheet model

NASA Astrophysics Data System (ADS)

Zhu, Hongyu; Petra, Noemi; Stadler, Georg; Isaac, Tobin; Hughes, Thomas J. R.; Ghattas, Omar

2016-07-01

We address the inverse problem of inferring the basal geothermal heat flux from surface velocity observations using a steady-state thermomechanically coupled nonlinear Stokes ice flow model. This is a challenging inverse problem since the map from basal heat flux to surface velocity observables is indirect: the heat flux is a boundary condition for the thermal advection-diffusion equation, which couples to the nonlinear Stokes ice flow equations; together they determine the surface ice flow velocity. This multiphysics inverse problem is formulated as a nonlinear least-squares optimization problem with a cost functional that includes the data misfit between surface velocity observations and model predictions. A Tikhonov regularization term is added to render the problem well posed. We derive adjoint-based gradient and Hessian expressions for the resulting partial differential equation (PDE)-constrained optimization problem and propose an inexact Newton method for its solution. As a consequence of the Petrov-Galerkin discretization of the energy equation, we show that discretization and differentiation do not commute; that is, the order in which we discretize the cost functional and differentiate it affects the correctness of the gradient. Using two- and three-dimensional model problems, we study the prospects for and limitations of the inference of the geothermal heat flux field from surface velocity observations. The results show that the reconstruction improves as the noise level in the observations decreases and that short-wavelength variations in the geothermal heat flux are difficult to recover. We analyze the ill-posedness of the inverse problem as a function of the number of observations by examining the spectrum of the Hessian of the cost functional. Motivated by the popularity of operator-split or staggered solvers for forward multiphysics problems - i.e., those that drop two-way coupling terms to yield a one-way coupled forward Jacobian - we study the effect on the inversion of a one-way coupling of the adjoint energy and Stokes equations. We show that taking such a one-way coupled approach for the adjoint equations can lead to an incorrect gradient and premature termination of optimization iterations. This is due to loss of a descent direction stemming from inconsistency of the gradient with the contours of the cost functional. Nevertheless, one may still obtain a reasonable approximate inverse solution particularly if important features of the reconstructed solution emerge early in optimization iterations, before the premature termination.

Sonic Boom Mitigation Through Aircraft Design and Adjoint Methodology

NASA Technical Reports Server (NTRS)

Rallabhandi, Siriam K.; Diskin, Boris; Nielsen, Eric J.

2012-01-01

This paper presents a novel approach to design of the supersonic aircraft outer mold line (OML) by optimizing the A-weighted loudness of sonic boom signature predicted on the ground. The optimization process uses the sensitivity information obtained by coupling the discrete adjoint formulations for the augmented Burgers Equation and Computational Fluid Dynamics (CFD) equations. This coupled formulation links the loudness of the ground boom signature to the aircraft geometry thus allowing efficient shape optimization for the purpose of minimizing the impact of loudness. The accuracy of the adjoint-based sensitivities is verified against sensitivities obtained using an independent complex-variable approach. The adjoint based optimization methodology is applied to a configuration previously optimized using alternative state of the art optimization methods and produces additional loudness reduction. The results of the optimizations are reported and discussed.

Deterministic Local Sensitivity Analysis of Augmented Systems - II: Applications to the QUENCH-04 Experiment Using the RELAP5/MOD3.2 Code System

DOE Office of Scientific and Technical Information (OSTI.GOV)

Ionescu-Bujor, Mihaela; Jin Xuezhou; Cacuci, Dan G.

2005-09-15

The adjoint sensitivity analysis procedure for augmented systems for application to the RELAP5/MOD3.2 code system is illustrated. Specifically, the adjoint sensitivity model corresponding to the heat structure models in RELAP5/MOD3.2 is derived and subsequently augmented to the two-fluid adjoint sensitivity model (ASM-REL/TF). The end product, called ASM-REL/TFH, comprises the complete adjoint sensitivity model for the coupled fluid dynamics/heat structure packages of the large-scale simulation code RELAP5/MOD3.2. The ASM-REL/TFH model is validated by computing sensitivities to the initial conditions for various time-dependent temperatures in the test bundle of the Quench-04 reactor safety experiment. This experiment simulates the reflooding with water ofmore » uncovered, degraded fuel rods, clad with material (Zircaloy-4) that has the same composition and size as that used in typical pressurized water reactors. The most important response for the Quench-04 experiment is the time evolution of the cladding temperature of heated fuel rods. The ASM-REL/TFH model is subsequently used to perform an illustrative sensitivity analysis of this and other time-dependent temperatures within the bundle. The results computed by using the augmented adjoint sensitivity system, ASM-REL/TFH, highlight the reliability, efficiency, and usefulness of the adjoint sensitivity analysis procedure for computing time-dependent sensitivities.« less

Partition function of free conformal fields in 3-plet representation

NASA Astrophysics Data System (ADS)

Beccaria, Matteo; Tseytlin, Arkady A.

2017-05-01

Simplest examples of AdS/CFT duality correspond to free CFTs in d dimensions with fields in vector or adjoint representation of an internal symmetry group dual in the large N limit to a theory of massless or massless plus massive higher spins in AdS d+1. One may also study generalizations when conformal fields belong to higher dimensional representations, i.e. carry more than two internal symmetry indices. Here we consider the case of the 3-fundamental ("3-plet") representation. One motivation is a conjectured connection to multiple M5-brane theory: heuristic arguments suggest that it may be related to an (interacting) CFT of 6d (2,0) tensor multiplets in 3-plet representation of large N symmetry group that has an AdS7 dual. We compute the singlet partition function Z on S 1 × S d-1 for a free field in 3-plet representation of U( N) and analyse its novel large N behaviour. The large N limit of the low temperature expansion of Z which is convergent in the vector and adjoint cases here is only asymptotic, reflecting the much faster growth of the number of singlet operators with dimension, indicating a phase transition at very low temperature. Indeed, while the critical temperatures in the vector ( T c ˜ N γ , γ > 0) and adjoint ( T c ˜ 1) cases are finite, we find that in the 3-plet case T c ˜ (log N)-1, i.e. it approaches zero at large N. We discuss some details of large N solution for the eigenvalue distribution. Similar conclusions apply to higher p-plet representations of U( N) or O( N) and also to the free p-tensor theories invariant under [U( N)] p or [ O( N)] p with p ≥ 3.

Profiling the robustness, efficiency and limits of the forward-adjoint method for 3-D mantle convection modelling

NASA Astrophysics Data System (ADS)

Price, M. G.; Davies, J. H.

2018-02-01

Knowledge of Earth's past mantle structure is inherently unknown. This lack of knowledge presents problems in many areas of Earth science, including in mantle circulation modelling (MCM). As a mathematical model of mantle convection, MCMs require boundary and initial conditions. While boundary conditions are readily available from sources such as plate reconstructions for the upper surface, and as free slip at the core-mantle boundary, the initial condition is not known. MCMs have historically `created' an initial condition using long `spin up' processes using the oldest available plate reconstruction period available. While these do yield good results when models are run to present day, it is difficult to infer with confidence results from early in a model's history. Techniques to overcome this problem are now being studied in geodynamics, such as by assimilating the known internal structure (e.g. from seismic tomography) of Earth at present day backwards in time. One such method is to use an iterative process known as the forward-adjoint method. While this is an efficient means of solving this inverse problem, it still strains all but the most cutting edge computational systems. In this study we endeavour to profile the effectiveness of this method using synthetic test cases as our known data source. We conclude that savings in terms of computational expense for forward-adjoint models can be achieved by streamlining the time-stepping of the calculation, as well as determining the most efficient method of updating initial conditions in the iterative scheme. Furthermore, we observe that in the models presented, there exists an upper limit on the time interval over which solutions will practically converge, although this limit is likely to be linked to Rayleigh number.

Adjoint tomography of the crust and upper mantle structure beneath the Kanto region using broadband seismograms

NASA Astrophysics Data System (ADS)

Miyoshi, Takayuki; Obayashi, Masayuki; Peter, Daniel; Tono, Yoko; Tsuboi, Seiji

2017-12-01

A three-dimensional seismic wave speed model in the Kanto region of Japan was developed using adjoint tomography for application in the effective reproduction of observed waveforms. Starting with a model based on previous travel time tomographic results, we inverted the waveforms obtained at seismic broadband stations from 140 local earthquakes in the Kanto region to obtain the P- and S-wave speeds V p and V s . Additionally, all centroid times of the source solutions were determined before the structural inversion. The synthetic displacements were calculated using the spectral-element method (SEM) in which the Kanto region was parameterized using 16 million grid points. The model parameters V p and V s were updated iteratively by Newton's method using the misfit and Hessian kernels until the misfit between the observed and synthetic waveforms was minimized. Computations of the forward and adjoint simulations were conducted on the K computer in Japan. The optimized SEM code required a total of 6720 simulations using approximately 62,000 node hours to obtain the final model after 16 iterations. The proposed model reveals several anomalous areas with extremely low- V s values in comparison with those of the initial model. These anomalies were found to correspond to geological features, earthquake sources, and volcanic regions with good data coverage and resolution. The synthetic waveforms obtained using the newly proposed model for the selected earthquakes showed better fit than the initial model to the observed waveforms in different period ranges within 5-30 s. This result indicates that the model can accurately predict actual waveforms. [Figure not available: see fulltext.

Design optimization using adjoint of Long-time LES for the trailing edge of a transonic turbine vane

NASA Astrophysics Data System (ADS)

Talnikar, Chaitanya; Wang, Qiqi

2017-11-01

Adjoint-based design optimization methods have been applied to low-fidelity simulation methods like Reynolds Averaged Navier-Stokes (RANS) and are useful for designing fluid machinery components. But to reliably capture the complex flow phenomena involved in turbomachinery, high fidelity simulations like large eddy simulation (LES) are required. Unfortunately due to the chaotic dynamics of turbulence, the unsteady adjoint method for LES diverges and produces incorrect gradients. Using a viscosity stabilized unsteady adjoint method developed for LES, the gradient can be obtained with reasonable accuracy. In this paper, design of the trailing edge of a gas turbine inlet guide vane is performed with the objective to reduce stagnation pressure loss and heat transfer over the surface of the vane. Slight changes in the shape of trailing edge can significantly impact these quantities by altering the boundary layer development process and separation points. The trailing edge is parameterized using a linear combination of 5 convex designs. Bayesian optimization is used as a global optimizer with the objective function evaluated from the LES and gradients obtained using the viscosity adjoint method. Results from the optimization, performed on the supercomputer Mira, are presented.

Linear stability analysis of scramjet unstart

NASA Astrophysics Data System (ADS)

Jang, Ik; Nichols, Joseph; Moin, Parviz

2015-11-01

We investigate the bifurcation structure of unstart and restart events in a dual-mode scramjet using the Reynolds-averaged Navier-Stokes equations. The scramjet of interest (HyShot II, Laurence et al., AIAA2011-2310) operates at a free-stream Mach number of approximately 8, and the length of the combustor chamber is 300mm. A heat-release model is applied to mimic the combustion process. Pseudo-arclength continuation with Newton-Raphson iteration is used to calculate multiple solution branches. Stability analysis based on linearized dynamics about the solution curves reveals a metric that optimally forewarns unstart. By combining direct and adjoint eigenmodes, structural sensitivity analysis suggests strategies for unstart mitigation, including changing the isolator length. This work is supported by DOE/NNSA and AFOSR.

Optimal Harvesting in a Periodic Food Chain Model with Size Structures in Predators

DOE Office of Scientific and Technical Information (OSTI.GOV)

Zhang, Feng-Qin, E-mail: zhafq@263.net; Liu, Rong; Chen, Yuming, E-mail: ychen@wlu.ca

In this paper, we investigate a periodic food chain model with harvesting, where the predators have size structures and are described by first-order partial differential equations. First, we establish the existence of a unique non-negative solution by using the Banach fixed point theorem. Then, we provide optimality conditions by means of normal cone and adjoint system. Finally, we derive the existence of an optimal strategy by means of Ekeland’s variational principle. Here the objective functional represents the net economic benefit yielded from harvesting.

An Adjoint-Based Approach to Study a Flexible Flapping Wing in Pitching-Rolling Motion

NASA Astrophysics Data System (ADS)

Jia, Kun; Wei, Mingjun; Xu, Min; Li, Chengyu; Dong, Haibo

2017-11-01

Flapping-wing aerodynamics, with advantages in agility, efficiency, and hovering capability, has been the choice of many flyers in nature. However, the study of bio-inspired flapping-wing propulsion is often hindered by the problem's large control space with different wing kinematics and deformation. The adjoint-based approach reduces largely the computational cost to a feasible level by solving an inverse problem. Facing the complication from moving boundaries, non-cylindrical calculus provides an easy extension of traditional adjoint-based approach to handle the optimization involving moving boundaries. The improved adjoint method with non-cylindrical calculus for boundary treatment is first applied on a rigid pitching-rolling plate, then extended to a flexible one with active deformation to further increase its propulsion efficiency. The comparison of flow dynamics with the initial and optimal kinematics and deformation provides a unique opportunity to understand the flapping-wing mechanism. Supported by AFOSR and ARL.

Adjoint-Baed Optimal Control on the Pitch Angle of a Single-Bladed Vertical-Axis Wind Turbine

NASA Astrophysics Data System (ADS)

Tsai, Hsieh-Chen; Colonius, Tim

2017-11-01

Optimal control on the pitch angle of a NACA0018 single-bladed vertical-axis wind turbine (VAWT) is numerically investigated at a low Reynolds number of 1500. With fixed tip-speed ratio, the input power is minimized and mean tangential force is maximized over a specific time horizon. The immersed boundary method is used to simulate the two-dimensional, incompressible flow around a horizontal cross section of the VAWT. The problem is formulated as a PDE constrained optimization problem and an iterative solution is obtained using adjoint-based conjugate gradient methods. By the end of the longest control horizon examined, two controls end up with time-invariant pitch angles of about the same magnitude but with the opposite signs. The results show that both cases lead to a reduction in the input power but not necessarily an enhancement in the mean tangential force. These reductions in input power are due to the removal of a power-damaging phenomenon that occurs when a vortex pair is captured by the blade in the upwind-half region of a cycle. This project was supported by Caltech FLOWE center/Gordon and Betty Moore Foundation.

Geodesic regression for image time-series.

PubMed

Niethammer, Marc; Huang, Yang; Vialard, François-Xavier

2011-01-01

Registration of image-time series has so far been accomplished (i) by concatenating registrations between image pairs, (ii) by solving a joint estimation problem resulting in piecewise geodesic paths between image pairs, (iii) by kernel based local averaging or (iv) by augmenting the joint estimation with additional temporal irregularity penalties. Here, we propose a generative model extending least squares linear regression to the space of images by using a second-order dynamic formulation for image registration. Unlike previous approaches, the formulation allows for a compact representation of an approximation to the full spatio-temporal trajectory through its initial values. The method also opens up possibilities to design image-based approximation algorithms. The resulting optimization problem is solved using an adjoint method.

Self-Consistent Sources for Integrable Equations Via Deformations of Binary Darboux Transformations

NASA Astrophysics Data System (ADS)

Chvartatskyi, Oleksandr; Dimakis, Aristophanes; Müller-Hoissen, Folkert

2016-08-01

We reveal the origin and structure of self-consistent source extensions of integrable equations from the perspective of binary Darboux transformations. They arise via a deformation of the potential that is central in this method. As examples, we obtain in particular matrix versions of self-consistent source extensions of the KdV, Boussinesq, sine-Gordon, nonlinear Schrödinger, KP, Davey-Stewartson, two-dimensional Toda lattice and discrete KP equation. We also recover a (2+1)-dimensional version of the Yajima-Oikawa system from a deformation of the pKP hierarchy. By construction, these systems are accompanied by a hetero binary Darboux transformation, which generates solutions of such a system from a solution of the source-free system and additionally solutions of an associated linear system and its adjoint. The essence of all this is encoded in universal equations in the framework of bidifferential calculus.

Radiative transfer equation accounting for rotational Raman scattering and its solution by the discrete-ordinates method

NASA Astrophysics Data System (ADS)

Rozanov, Vladimir V.; Vountas, Marco

2014-01-01

Rotational Raman scattering of solar light in Earth's atmosphere leads to the filling-in of Fraunhofer and telluric lines observed in the reflected spectrum. The phenomenological derivation of the inelastic radiative transfer equation including rotational Raman scattering is presented. The different forms of the approximate radiative transfer equation with first-order rotational Raman scattering terms are obtained employing the Cabannes, Rayleigh, and Cabannes-Rayleigh scattering models. The solution of these equations is considered in the framework of the discrete-ordinates method using rigorous and approximate approaches to derive particular integrals. An alternative forward-adjoint technique is suggested as well. A detailed description of the model including the exact spectral matching and a binning scheme that significantly speeds up the calculations is given. The considered solution techniques are implemented in the radiative transfer software package SCIATRAN and a specified benchmark setup is presented to enable readers to compare with own results transparently.

Applied Mathematics in EM Studies with Special Emphasis on an Uncertainty Quantification and 3-D Integral Equation Modelling

NASA Astrophysics Data System (ADS)

Pankratov, Oleg; Kuvshinov, Alexey

2016-01-01

Despite impressive progress in the development and application of electromagnetic (EM) deterministic inverse schemes to map the 3-D distribution of electrical conductivity within the Earth, there is one question which remains poorly addressed—uncertainty quantification of the recovered conductivity models. Apparently, only an inversion based on a statistical approach provides a systematic framework to quantify such uncertainties. The Metropolis-Hastings (M-H) algorithm is the most popular technique for sampling the posterior probability distribution that describes the solution of the statistical inverse problem. However, all statistical inverse schemes require an enormous amount of forward simulations and thus appear to be extremely demanding computationally, if not prohibitive, if a 3-D set up is invoked. This urges development of fast and scalable 3-D modelling codes which can run large-scale 3-D models of practical interest for fractions of a second on high-performance multi-core platforms. But, even with these codes, the challenge for M-H methods is to construct proposal functions that simultaneously provide a good approximation of the target density function while being inexpensive to be sampled. In this paper we address both of these issues. First we introduce a variant of the M-H method which uses information about the local gradient and Hessian of the penalty function. This, in particular, allows us to exploit adjoint-based machinery that has been instrumental for the fast solution of deterministic inverse problems. We explain why this modification of M-H significantly accelerates sampling of the posterior probability distribution. In addition we show how Hessian handling (inverse, square root) can be made practicable by a low-rank approximation using the Lanczos algorithm. Ultimately we discuss uncertainty analysis based on stochastic inversion results. In addition, we demonstrate how this analysis can be performed within a deterministic approach. In the second part, we summarize modern trends in the development of efficient 3-D EM forward modelling schemes with special emphasis on recent advances in the integral equation approach.

A Reduced-Order Successive Linear Estimator for Geostatistical Inversion and its Application in Hydraulic Tomography

NASA Astrophysics Data System (ADS)

Zha, Yuanyuan; Yeh, Tian-Chyi J.; Illman, Walter A.; Zeng, Wenzhi; Zhang, Yonggen; Sun, Fangqiang; Shi, Liangsheng

2018-03-01

Hydraulic tomography (HT) is a recently developed technology for characterizing high-resolution, site-specific heterogeneity using hydraulic data (nd) from a series of cross-hole pumping tests. To properly account for the subsurface heterogeneity and to flexibly incorporate additional information, geostatistical inverse models, which permit a large number of spatially correlated unknowns (ny), are frequently used to interpret the collected data. However, the memory storage requirements for the covariance of the unknowns (ny × ny) in these models are prodigious for large-scale 3-D problems. Moreover, the sensitivity evaluation is often computationally intensive using traditional difference method (ny forward runs). Although employment of the adjoint method can reduce the cost to nd forward runs, the adjoint model requires intrusive coding effort. In order to resolve these issues, this paper presents a Reduced-Order Successive Linear Estimator (ROSLE) for analyzing HT data. This new estimator approximates the covariance of the unknowns using Karhunen-Loeve Expansion (KLE) truncated to nkl order, and it calculates the directional sensitivities (in the directions of nkl eigenvectors) to form the covariance and cross-covariance used in the Successive Linear Estimator (SLE). In addition, the covariance of unknowns is updated every iteration by updating the eigenvalues and eigenfunctions. The computational advantages of the proposed algorithm are demonstrated through numerical experiments and a 3-D transient HT analysis of data from a highly heterogeneous field site.

A New Method for Computing Three-Dimensional Capture Fraction in Heterogeneous Regional Systems using the MODFLOW Adjoint Code

NASA Astrophysics Data System (ADS)

Clemo, T. M.; Ramarao, B.; Kelly, V. A.; Lavenue, M.

2011-12-01

Capture is a measure of the impact of groundwater pumping upon groundwater and surface water systems. The computation of capture through analytical or numerical methods has been the subject of articles in the literature for several decades (Bredehoeft et al., 1982). Most recently Leake et al. (2010) described a systematic way to produce capture maps in three-dimensional systems using a numerical perturbation approach in which capture from streams was computed using unit rate pumping at many locations within a MODFLOW model. The Leake et al. (2010) method advances the current state of computing capture. A limitation stems from the computational demand required by the perturbation approach wherein days or weeks of computational time might be required to obtain a robust measure of capture. In this paper, we present an efficient method to compute capture in three-dimensional systems based upon adjoint states. The efficiency of the adjoint method will enable uncertainty analysis to be conducted on capture calculations. The USGS and INTERA have collaborated to extend the MODFLOW Adjoint code (Clemo, 2007) to include stream-aquifer interaction and have applied it to one of the examples used in Leake et al. (2010), the San Pedro Basin MODFLOW model. With five layers and 140,800 grid blocks per layer, the San Pedro Basin model, provided an ideal example data set to compare the capture computed from the perturbation and the adjoint methods. The capture fraction map produced from the perturbation method for the San Pedro Basin model required significant computational time to compute and therefore the locations for the pumping wells were limited to 1530 locations in layer 4. The 1530 direct simulations of capture require approximately 76 CPU hours. Had capture been simulated in each grid block in each layer, as is done in the adjoint method, the CPU time would have been on the order of 4 years. The MODFLOW-Adjoint produced the capture fraction map of the San Pedro Basin model at 704,000 grid blocks (140,800 grid blocks x 5 layers) in just 6 minutes. The capture fraction maps from the perturbation and adjoint methods agree closely. The results of this study indicate that the adjoint capture method and its associated computational efficiency will enable scientists and engineers facing water resource management decisions to evaluate the sensitivity and uncertainty of impacts to regional water resource systems as part of groundwater supply strategies. Bredehoeft, J.D., S.S. Papadopulos, and H.H. Cooper Jr, Groundwater: The water budget myth. In Scientific Basis of Water-Resources Management, ed. National Research Council (U.S.), Geophysical Study Committee, 51-57. Washington D.C.: National Academy Press, 1982. Clemo, Tom, MODFLOW-2005 Ground-Water Model-Users Guide to Adjoint State based Sensitivity Process (ADJ), BSU CGISS 07-01, Center for the Geophysical Investigation of the Shallow Subsurface, Boise State University, 2007. Leake, S.A., H.W. Reeves, and J.E. Dickinson, A New Capture Fraction Method to Map How Pumpage Affects Surface Water Flow, Ground Water, 48(5), 670-700, 2010.

Efficient combination of a 3D Quasi-Newton inversion algorithm and a vector dual-primal finite element tearing and interconnecting method

NASA Astrophysics Data System (ADS)

Voznyuk, I.; Litman, A.; Tortel, H.

2015-08-01

A Quasi-Newton method for reconstructing the constitutive parameters of three-dimensional (3D) penetrable scatterers from scattered field measurements is presented. This method is adapted for handling large-scale electromagnetic problems while keeping the memory requirement and the time flexibility as low as possible. The forward scattering problem is solved by applying the finite-element tearing and interconnecting full-dual-primal (FETI-FDP2) method which shares the same spirit as the domain decomposition methods for finite element methods. The idea is to split the computational domain into smaller non-overlapping sub-domains in order to simultaneously solve local sub-problems. Various strategies are proposed in order to efficiently couple the inversion algorithm with the FETI-FDP2 method: a separation into permanent and non-permanent subdomains is performed, iterative solvers are favorized for resolving the interface problem and a marching-on-in-anything initial guess selection further accelerates the process. The computational burden is also reduced by applying the adjoint state vector methodology. Finally, the inversion algorithm is confronted to measurements extracted from the 3D Fresnel database.

Scalable posterior approximations for large-scale Bayesian inverse problems via likelihood-informed parameter and state reduction

NASA Astrophysics Data System (ADS)

Cui, Tiangang; Marzouk, Youssef; Willcox, Karen

2016-06-01

Two major bottlenecks to the solution of large-scale Bayesian inverse problems are the scaling of posterior sampling algorithms to high-dimensional parameter spaces and the computational cost of forward model evaluations. Yet incomplete or noisy data, the state variation and parameter dependence of the forward model, and correlations in the prior collectively provide useful structure that can be exploited for dimension reduction in this setting-both in the parameter space of the inverse problem and in the state space of the forward model. To this end, we show how to jointly construct low-dimensional subspaces of the parameter space and the state space in order to accelerate the Bayesian solution of the inverse problem. As a byproduct of state dimension reduction, we also show how to identify low-dimensional subspaces of the data in problems with high-dimensional observations. These subspaces enable approximation of the posterior as a product of two factors: (i) a projection of the posterior onto a low-dimensional parameter subspace, wherein the original likelihood is replaced by an approximation involving a reduced model; and (ii) the marginal prior distribution on the high-dimensional complement of the parameter subspace. We present and compare several strategies for constructing these subspaces using only a limited number of forward and adjoint model simulations. The resulting posterior approximations can rapidly be characterized using standard sampling techniques, e.g., Markov chain Monte Carlo. Two numerical examples demonstrate the accuracy and efficiency of our approach: inversion of an integral equation in atmospheric remote sensing, where the data dimension is very high; and the inference of a heterogeneous transmissivity field in a groundwater system, which involves a partial differential equation forward model with high dimensional state and parameters.

CERENA: ChEmical REaction Network Analyzer--A Toolbox for the Simulation and Analysis of Stochastic Chemical Kinetics.

PubMed

Kazeroonian, Atefeh; Fröhlich, Fabian; Raue, Andreas; Theis, Fabian J; Hasenauer, Jan

2016-01-01

Gene expression, signal transduction and many other cellular processes are subject to stochastic fluctuations. The analysis of these stochastic chemical kinetics is important for understanding cell-to-cell variability and its functional implications, but it is also challenging. A multitude of exact and approximate descriptions of stochastic chemical kinetics have been developed, however, tools to automatically generate the descriptions and compare their accuracy and computational efficiency are missing. In this manuscript we introduced CERENA, a toolbox for the analysis of stochastic chemical kinetics using Approximations of the Chemical Master Equation solution statistics. CERENA implements stochastic simulation algorithms and the finite state projection for microscopic descriptions of processes, the system size expansion and moment equations for meso- and macroscopic descriptions, as well as the novel conditional moment equations for a hybrid description. This unique collection of descriptions in a single toolbox facilitates the selection of appropriate modeling approaches. Unlike other software packages, the implementation of CERENA is completely general and allows, e.g., for time-dependent propensities and non-mass action kinetics. By providing SBML import, symbolic model generation and simulation using MEX-files, CERENA is user-friendly and computationally efficient. The availability of forward and adjoint sensitivity analyses allows for further studies such as parameter estimation and uncertainty analysis. The MATLAB code implementing CERENA is freely available from http://cerenadevelopers.github.io/CERENA/.

Development of a High-Order Space-Time Matrix-Free Adjoint Solver

NASA Technical Reports Server (NTRS)

Ceze, Marco A.; Diosady, Laslo T.; Murman, Scott M.

2016-01-01

The growth in computational power and algorithm development in the past few decades has granted the science and engineering community the ability to simulate flows over complex geometries, thus making Computational Fluid Dynamics (CFD) tools indispensable in analysis and design. Currently, one of the pacing items limiting the utility of CFD for general problems is the prediction of unsteady turbulent ows.1{3 Reynolds-averaged Navier-Stokes (RANS) methods, which predict a time-invariant mean flowfield, struggle to provide consistent predictions when encountering even mild separation, such as the side-of-body separation at a wing-body junction. NASA's Transformative Tools and Technologies project is developing both numerical methods and physical modeling approaches to improve the prediction of separated flows. A major focus of this e ort is efficient methods for resolving the unsteady fluctuations occurring in these flows to provide valuable engineering data of the time-accurate flow field for buffet analysis, vortex shedding, etc. This approach encompasses unsteady RANS (URANS), large-eddy simulations (LES), and hybrid LES-RANS approaches such as Detached Eddy Simulations (DES). These unsteady approaches are inherently more expensive than traditional engineering RANS approaches, hence every e ort to mitigate this cost must be leveraged. Arguably, the most cost-effective approach to improve the efficiency of unsteady methods is the optimal placement of the spatial and temporal degrees of freedom (DOF) using solution-adaptive methods.

CERENA: ChEmical REaction Network Analyzer—A Toolbox for the Simulation and Analysis of Stochastic Chemical Kinetics

PubMed Central

Kazeroonian, Atefeh; Fröhlich, Fabian; Raue, Andreas; Theis, Fabian J.; Hasenauer, Jan

2016-01-01

Gene expression, signal transduction and many other cellular processes are subject to stochastic fluctuations. The analysis of these stochastic chemical kinetics is important for understanding cell-to-cell variability and its functional implications, but it is also challenging. A multitude of exact and approximate descriptions of stochastic chemical kinetics have been developed, however, tools to automatically generate the descriptions and compare their accuracy and computational efficiency are missing. In this manuscript we introduced CERENA, a toolbox for the analysis of stochastic chemical kinetics using Approximations of the Chemical Master Equation solution statistics. CERENA implements stochastic simulation algorithms and the finite state projection for microscopic descriptions of processes, the system size expansion and moment equations for meso- and macroscopic descriptions, as well as the novel conditional moment equations for a hybrid description. This unique collection of descriptions in a single toolbox facilitates the selection of appropriate modeling approaches. Unlike other software packages, the implementation of CERENA is completely general and allows, e.g., for time-dependent propensities and non-mass action kinetics. By providing SBML import, symbolic model generation and simulation using MEX-files, CERENA is user-friendly and computationally efficient. The availability of forward and adjoint sensitivity analyses allows for further studies such as parameter estimation and uncertainty analysis. The MATLAB code implementing CERENA is freely available from http://cerenadevelopers.github.io/CERENA/. PMID:26807911

Assimilating Remote Ammonia Observations with a Refined Aerosol Thermodynamics Adjoint"

EPA Science Inventory

Ammonia emissions parameters in North America can be refined in order to improve the evaluation of modeled concentrations against observations. Here, we seek to do so by developing and applying the GEOS-Chem adjoint nested over North America to conductassimilation of observations...

Adjoint-Based Sensitivity and Uncertainty Analysis for Density and Composition: A User’s Guide

DOE PAGES

Favorite, Jeffrey A.; Perko, Zoltan; Kiedrowski, Brian C.; ...

2017-03-01

The ability to perform sensitivity analyses using adjoint-based first-order sensitivity theory has existed for decades. This paper provides guidance on how adjoint sensitivity methods can be used to predict the effect of material density and composition uncertainties in critical experiments, including when these uncertain parameters are correlated or constrained. Two widely used Monte Carlo codes, MCNP6 (Ref. 2) and SCALE 6.2 (Ref. 3), are both capable of computing isotopic density sensitivities in continuous energy and angle. Additionally, Perkó et al. have shown how individual isotope density sensitivities, easily computed using adjoint methods, can be combined to compute constrained first-order sensitivitiesmore » that may be used in the uncertainty analysis. This paper provides details on how the codes are used to compute first-order sensitivities and how the sensitivities are used in an uncertainty analysis. Constrained first-order sensitivities are computed in a simple example problem.« less

Extending the Binomial Checkpointing Technique for Resilience

DOE Office of Scientific and Technical Information (OSTI.GOV)

Walther, Andrea; Narayanan, Sri Hari Krishna

In terms of computing time, adjoint methods offer a very attractive alternative to compute gradient information, re- quired, e.g., for optimization purposes. However, together with this very favorable temporal complexity result comes a memory requirement that is in essence proportional with the operation count of the underlying function, e.g., if algo- rithmic differentiation is used to provide the adjoints. For this reason, checkpointing approaches in many variants have become popular. This paper analyzes an extension of the so-called binomial approach to cover also possible failures of the computing systems. Such a measure of precaution is of special interest for massivemore » parallel simulations and adjoint calculations where the mean time between failure of the large scale computing system is smaller than the time needed to complete the calculation of the adjoint information. We de- scribe the extensions of standard checkpointing approaches required for such resilience, provide a corresponding imple- mentation and discuss numerical results.« less

Adjoint Sensitivity Analysis for Scale-Resolving Turbulent Flow Solvers

NASA Astrophysics Data System (ADS)

Blonigan, Patrick; Garai, Anirban; Diosady, Laslo; Murman, Scott

2017-11-01

Adjoint-based sensitivity analysis methods are powerful design tools for engineers who use computational fluid dynamics. In recent years, these engineers have started to use scale-resolving simulations like large-eddy simulations (LES) and direct numerical simulations (DNS), which resolve more scales in complex flows with unsteady separation and jets than the widely-used Reynolds-averaged Navier-Stokes (RANS) methods. However, the conventional adjoint method computes large, unusable sensitivities for scale-resolving simulations, which unlike RANS simulations exhibit the chaotic dynamics inherent in turbulent flows. Sensitivity analysis based on least-squares shadowing (LSS) avoids the issues encountered by conventional adjoint methods, but has a high computational cost even for relatively small simulations. The following talk discusses a more computationally efficient formulation of LSS, ``non-intrusive'' LSS, and its application to turbulent flows simulated with a discontinuous-Galkerin spectral-element-method LES/DNS solver. Results are presented for the minimal flow unit, a turbulent channel flow with a limited streamwise and spanwise domain.

Neural Network Training by Integration of Adjoint Systems of Equations Forward in Time

NASA Technical Reports Server (NTRS)

Toomarian, Nikzad (Inventor); Barhen, Jacob (Inventor)

1999-01-01

A method and apparatus for supervised neural learning of time dependent trajectories exploits the concepts of adjoint operators to enable computation of the gradient of an objective functional with respect to the various parameters of the network architecture in a highly efficient manner. Specifically. it combines the advantage of dramatic reductions in computational complexity inherent in adjoint methods with the ability to solve two adjoint systems of equations together forward in time. Not only is a large amount of computation and storage saved. but the handling of real-time applications becomes also possible. The invention has been applied it to two examples of representative complexity which have recently been analyzed in the open literature and demonstrated that a circular trajectory can be learned in approximately 200 iterations compared to the 12000 reported in the literature. A figure eight trajectory was achieved in under 500 iterations compared to 20000 previously required. Tbc trajectories computed using our new method are much closer to the target trajectories than was reported in previous studies.

Neural network training by integration of adjoint systems of equations forward in time

NASA Technical Reports Server (NTRS)

Toomarian, Nikzad (Inventor); Barhen, Jacob (Inventor)

1992-01-01

A method and apparatus for supervised neural learning of time dependent trajectories exploits the concepts of adjoint operators to enable computation of the gradient of an objective functional with respect to the various parameters of the network architecture in a highly efficient manner. Specifically, it combines the advantage of dramatic reductions in computational complexity inherent in adjoint methods with the ability to solve two adjoint systems of equations together forward in time. Not only is a large amount of computation and storage saved, but the handling of real-time applications becomes also possible. The invention has been applied it to two examples of representative complexity which have recently been analyzed in the open literature and demonstrated that a circular trajectory can be learned in approximately 200 iterations compared to the 12000 reported in the literature. A figure eight trajectory was achieved in under 500 iterations compared to 20000 previously required. The trajectories computed using our new method are much closer to the target trajectories than was reported in previous studies.

Reduction by symmetries in singular quantum-mechanical problems: General scheme and application to Aharonov-Bohm model

DOE Office of Scientific and Technical Information (OSTI.GOV)

Smirnov, A. G., E-mail: smirnov@lpi.ru

2015-12-15

We develop a general technique for finding self-adjoint extensions of a symmetric operator that respects a given set of its symmetries. Problems of this type naturally arise when considering two- and three-dimensional Schrödinger operators with singular potentials. The approach is based on constructing a unitary transformation diagonalizing the symmetries and reducing the initial operator to the direct integral of a suitable family of partial operators. We prove that symmetry preserving self-adjoint extensions of the initial operator are in a one-to-one correspondence with measurable families of self-adjoint extensions of partial operators obtained by reduction. The general scheme is applied to themore » three-dimensional Aharonov-Bohm Hamiltonian describing the electron in the magnetic field of an infinitely thin solenoid. We construct all self-adjoint extensions of this Hamiltonian, invariant under translations along the solenoid and rotations around it, and explicitly find their eigenfunction expansions.« less

Adjoint-Based Uncertainty Quantification with MCNP

DOE Office of Scientific and Technical Information (OSTI.GOV)

Seifried, Jeffrey E.

2011-09-01

This work serves to quantify the instantaneous uncertainties in neutron transport simulations born from nuclear data and statistical counting uncertainties. Perturbation and adjoint theories are used to derive implicit sensitivity expressions. These expressions are transformed into forms that are convenient for construction with MCNP6, creating the ability to perform adjoint-based uncertainty quantification with MCNP6. These new tools are exercised on the depleted-uranium hybrid LIFE blanket, quantifying its sensitivities and uncertainties to important figures of merit. Overall, these uncertainty estimates are small (< 2%). Having quantified the sensitivities and uncertainties, physical understanding of the system is gained and some confidence inmore » the simulation is acquired.« less

Stratospheric Water Vapor and the Asian Monsoon: An Adjoint Model Investigation

NASA Technical Reports Server (NTRS)

Olsen, Mark A.; Andrews, Arlyn E.

2003-01-01

A new adjoint model of the Goddard Parameterized Chemistry and Transport Model is used to investigate the role that the Asian monsoon plays in transporting water to the stratosphere. The adjoint model provides a unique perspective compared to non-diffusive and non-mixing Lagrangian trajectory analysis. The quantity of water vapor transported from the monsoon and the pathways into the stratosphere are examined. The emphasis is on the amount of water originating from the monsoon that contributes to the tropical tape recorder signal. The cross-tropopause flux of water from the monsoon to the midlatitude lower stratosphere will also be discussed.

On universal knot polynomials

NASA Astrophysics Data System (ADS)

Mironov, A.; Mkrtchyan, R.; Morozov, A.

2016-02-01

We present a universal knot polynomials for 2- and 3-strand torus knots in adjoint representation, by universalization of appropriate Rosso-Jones formula. According to universality, these polynomials coincide with adjoined colored HOMFLY and Kauffman polynomials at SL and SO/Sp lines on Vogel's plane, respectively and give their exceptional group's counterparts on exceptional line. We demonstrate that [m,n]=[n,m] topological invariance, when applicable, take place on the entire Vogel's plane. We also suggest the universal form of invariant of figure eight knot in adjoint representation, and suggest existence of such universalization for any knot in adjoint and its descendant representations. Properties of universal polynomials and applications of these results are discussed.

Adjoint Sensitivity Analysis of Radiative Transfer Equation: Temperature and Gas Mixing Ratio Weighting Functions for Remote Sensing of Scattering Atmospheres in Thermal IR

NASA Technical Reports Server (NTRS)

Ustinov, E.

1999-01-01

Sensitivity analysis based on using of the adjoint equation of radiative transfer is applied to the case of atmospheric remote sensing in the thermal spectral region with non-negligeable atmospheric scattering.

The development of global GRAPES 4DVAR

NASA Astrophysics Data System (ADS)

Liu, Yongzhu

2017-04-01

Four-dimensional variation data assimilation (4DVAR) has given a great contribution to the improvement of NWP system over the past twenty years. Therefore, our strategy is to develop an operational global 4D-Var system from the outset. The aim at the paper is to introduce the development of the global GRAPES four-dimensional variation data assimilation (4DVAR) using incremental analysis schemes and to presents results of a comparison between 4DVAR using 6-hour assimilation window and simplified physics during the minimization with three-dimensional variation data assimilation (3DVAR). The dynamical cores of the tangent-linear and adjoint models are developed directly based on the non-hydrostatic forecast model. In addition, the standard correctness checks have been performed. As well as the development adjoint codes, most of our work is focused on improving the computational efficiency since the bulk of the computational cost of 4D-Var is in the integration of the tangent-linear and adjoint models. In terms of tangent-linear model, the wall-clock time is reduced to about 1.2 times as much as one of nonlinear model through the optimizing of the software framework. The significant computational cost savings on adjoint model result from the removing the redundant recompilations of model trajectories. It is encouraging that the wall-clock time of adjoint model is less than 1.5 times as much as one of nonlinear model. The current difficulty is that the numerical scheme used within the linear model is based on strategically on the numeric of the corresponding nonlinear model. Further computational acceleration should be expected from the improvement on nonlinear numerical algorithm. A series of linearized physical parameterization schemes has been developed to improve the representation of perturbed fields in the linear model. It consists of horizontal and vertical diffusion, sub-grid scale orographic gravity wave drag, large-scale condensation and cumulus convection schemes. We also found the straightforward linearization based on the nonlinear physical scheme might lead to significant growing of spurious unstable perturbations. It is essential to simplify the linear physics with respect to the non-linear schemes. The improvement on the perturbed fields in the tangent-linear model is visible with the linear physics included, especially at the low level. GRAPES variation data assimilation system adopts the incremental approach. The work is ongoing to develop a pre-operational 4DVAR suite with 0.25° outer loop resolution and multiple outer-loops configurations. One 4DVAR analysis using 6-hour assimilation windows can be finished within 40-minutes when using the available conventional and satellite data. In summary, it was found that the analysis over the northern, southern hemispheres, tropical region and East Asian area of GRAPES 4DVAR performed better than GRAPES 3DVAR for one month experiments. Moreover, the forecast results show that northern and southern extra-tropical scores for GRAPES 4DVAR are already better than GRAPES 3DVAR, but the tropical performance needs further investigations. Therefore, the subsequent main improvements will aim to enhance its computational efficiency and accuracy in 2017. The global GRAPES 4DVAR is planned for operation in 2018.

An Adjoint-Based Analysis of the Sampling Footprints of Tall Tower, Aircraft and Potential Future Lidar Observations of CO2

NASA Technical Reports Server (NTRS)

Andrews, Arlyn; Kawa, Randy; Zhu, Zhengxin; Burris, John; Abshire, Jim

2004-01-01

A detailed mechanistic understanding of the sources and sinks of CO2 will be required to reliably predict future CO2 levels and climate. A commonly used technique for deriving information about CO2 exchange with surface reservoirs is to solve an 'inverse problem', where CO2 observations are used with an atmospheric transport model to find the optimal distribution of sources and sinks. Synthesis inversion methods are powerful tools for addressing this question, but the results are disturbingly sensitive to the details of the calculation. Studies done using different atmospheric transport models and combinations of surface station data have produced substantially different distributions of surface fluxes. Adjoint methods are now being developed that will more effectively incorporate diverse datasets in estimates of surface fluxes of CO2. In an adjoint framework, it will be possible to combine CO2 concentration data from longterm surface and aircraft monitoring stations with data from intensive field campaigns and with proposed future satellite observations. We have recently developed an adjoint for the GSFC 3-D Parameterized Chemistry and Transport Model (PCTM). Here, we will present results from a PCTM Adjoint study comparing the sampling footprints of tall tower, aircraft and potential future lidar observations of CO2. The vertical resolution and extent of the profiles and the observation frequency will be considered for several sites in North America.

Double-Difference Global Adjoint Tomography

NASA Astrophysics Data System (ADS)

Orsvuran, R.; Bozdag, E.; Lei, W.; Tromp, J.

2017-12-01

The adjoint method allows us to incorporate full waveform simulations in inverse problems. Misfit functions play an important role in extracting the relevant information from seismic waveforms. In this study, our goal is to apply the Double-Difference (DD) methodology proposed by Yuan et al. (2016) to global adjoint tomography. Dense seismic networks, such as USArray, lead to higher-resolution seismic images underneath continents. However, the imbalanced distribution of stations and sources poses challenges in global ray coverage. We adapt double-difference multitaper measurements to global adjoint tomography. We normalize each DD measurement by its number of pairs, and if a measurement has no pair, as may frequently happen for data recorded at oceanic stations, classical multitaper measurements are used. As a result, the differential measurements and pair-wise weighting strategy help balance uneven global kernel coverage. Our initial experiments with minor- and major-arc surface waves show promising results, revealing more pronounced structure near dense networks while reducing the prominence of paths towards cluster of stations. We have started using this new measurement in global adjoint inversions, addressing azimuthal anisotropy in upper mantle. Meanwhile, we are working on combining the double-difference approach with instantaneous phase measurements to emphasize contributions of scattered waves in global inversions and extending it to body waves. We will present our results and discuss challenges and future directions in the context of global tomographic inversions.

Survey of methods for calculating sensitivity of general eigenproblems

NASA Technical Reports Server (NTRS)

Murthy, Durbha V.; Haftka, Raphael T.

1987-01-01

A survey of methods for sensitivity analysis of the algebraic eigenvalue problem for non-Hermitian matrices is presented. In addition, a modification of one method based on a better normalizing condition is proposed. Methods are classified as Direct or Adjoint and are evaluated for efficiency. Operation counts are presented in terms of matrix size, number of design variables and number of eigenvalues and eigenvectors of interest. The effect of the sparsity of the matrix and its derivatives is also considered, and typical solution times are given. General guidelines are established for the selection of the most efficient method.

Normal-mode-based analysis of electron plasma waves with second-order Hermitian formalism

DOE Office of Scientific and Technical Information (OSTI.GOV)

Ramos, J. J.; White, R. L.

The classic problem of the dynamic evolution and Landau damping of linear Langmuir electron waves in a collisionless plasma with Maxwellian background is cast as a second-order, self-adjoint problem with a continuum spectrum of real and positive squared frequencies. The corresponding complete basis of singular normal modes is obtained, along with their orthogonality relation. This yields easily the general expression of the time-reversal-invariant solution for any initial-value problem. Examples are then given for specific initial conditions that illustrate different behaviors of the Landau-damped macroscopic moments of the perturbations.

Normal-mode-based analysis of electron plasma waves with second-order Hermitian formalism

DOE PAGES

Ramos, J. J.; White, R. L.

2018-03-01

The classic problem of the dynamic evolution and Landau damping of linear Langmuir electron waves in a collisionless plasma with Maxwellian background is cast as a second-order, self-adjoint problem with a continuum spectrum of real and positive squared frequencies. The corresponding complete basis of singular normal modes is obtained, along with their orthogonality relation. This yields easily the general expression of the time-reversal-invariant solution for any initial-value problem. Examples are then given for specific initial conditions that illustrate different behaviors of the Landau-damped macroscopic moments of the perturbations.

Numerical studies of the thermal design sensitivity calculation for a reaction-diffusion system with discontinuous derivatives

NASA Technical Reports Server (NTRS)

Hou, Jean W.; Sheen, Jeen S.

1987-01-01

The aim of this study is to find a reliable numerical algorithm to calculate thermal design sensitivities of a transient problem with discontinuous derivatives. The thermal system of interest is a transient heat conduction problem related to the curing process of a composite laminate. A logical function which can smoothly approximate the discontinuity is introduced to modify the system equation. Two commonly used methods, the adjoint variable method and the direct differentiation method, are then applied to find the design derivatives of the modified system. The comparisons of numerical results obtained by these two methods demonstrate that the direct differentiation method is a better choice to be used in calculating thermal design sensitivity.

Self-adjoint realisations of the Dirac-Coulomb Hamiltonian for heavy nuclei

NASA Astrophysics Data System (ADS)

Gallone, Matteo; Michelangeli, Alessandro

2018-02-01

We derive a classification of the self-adjoint extensions of the three-dimensional Dirac-Coulomb operator in the critical regime of the Coulomb coupling. Our approach is solely based upon the Kreĭn-Višik-Birman extension scheme, or also on Grubb's universal classification theory, as opposite to previous works within the standard von Neumann framework. This let the boundary condition of self-adjointness emerge, neatly and intrinsically, as a multiplicative constraint between regular and singular part of the functions in the domain of the extension, the multiplicative constant giving also immediate information on the invertibility property and on the resolvent and spectral gap of the extension.

Application of Adjoint Methodology to Supersonic Aircraft Design Using Reversed Equivalent Areas

NASA Technical Reports Server (NTRS)

Rallabhandi, Sriram K.

2013-01-01

This paper presents an approach to shape an aircraft to equivalent area based objectives using the discrete adjoint approach. Equivalent areas can be obtained either using reversed augmented Burgers equation or direct conversion of off-body pressures into equivalent area. Formal coupling with CFD allows computation of sensitivities of equivalent area objectives with respect to aircraft shape parameters. The exactness of the adjoint sensitivities is verified against derivatives obtained using the complex step approach. This methodology has the benefit of using designer-friendly equivalent areas in the shape design of low-boom aircraft. Shape optimization results with equivalent area cost functionals are discussed and further refined using ground loudness based objectives.

Dynamic analysis of a hollow cylinder subject to a dual traveling force imposed on its inner surface

NASA Astrophysics Data System (ADS)

Lee, Sooyoung; Seok, Jongwon

2015-03-01

The dynamic behavior of a hollow cylinder under a dual traveling force applied to the inner surface is investigated in this study. The cylinder is constrained at both the top and bottom surfaces not to move in the length direction but free in other directions. And a dual force travels at a constant velocity along the length direction on the inner surface of the hollow cylinder. The resulting governing field equations and the associated boundary conditions are ruled by the general Hooke's law. Due to the nature of the field equations, proper adjoint system of equations and biorthogonality conditions were derived in a precise and detailed manner. To solve these field equations in this study, the method of separation of variable is used and the method of Fro¨benius is employed for the differential equations in the radial direction. Using the field equations, the eigenanalyses on both the original and its adjoint system were performed with great care, which results in the eigenfunction sets of both systems. The biorthogonality conditions were applied to the field equations to obtain the discretized equation for each mode. Using the solutions of the discretized equations that account for the boundary forcing terms, the critical speed for a dual traveling force for each mode could be computed.

Full Seismic Waveform Tomography of the Japan region using Adjoint Methods

NASA Astrophysics Data System (ADS)

Steptoe, Hamish; Fichtner, Andreas; Rickers, Florian; Trampert, Jeannot

2013-04-01

We present a full-waveform tomographic model of the Japan region based on spectral-element wave propagation, adjoint techniques and seismic data from dense station networks. This model is intended to further our understanding of both the complex regional tectonics and the finite rupture processes of large earthquakes. The shallow Earth structure of the Japan region has been the subject of considerable tomographic investigation. The islands of Japan exist in an area of significant plate complexity: subduction related to the Pacific and Philippine Sea plates is responsible for the majority of seismicity and volcanism of Japan, whilst smaller micro-plates in the region, including the Okhotsk, and Okinawa and Amur, part of the larger North America and Eurasia plates respectively, contribute significant local intricacy. In response to the need to monitor and understand the motion of these plates and their associated faults, numerous seismograph networks have been established, including the 768 station high-sensitivity Hi-net network, 84 station broadband F-net and the strong-motion seismograph networks K-net and KiK-net in Japan. We also include the 55 station BATS network of Taiwan. We use this exceptional coverage to construct a high-resolution model of the Japan region from the full-waveform inversion of over 15,000 individual component seismograms from 53 events that occurred between 1997 and 2012. We model these data using spectral-element simulations of seismic wave propagation at a regional scale over an area from 120°-150°E and 20°-50°N to a depth of around 500 km. We quantify differences between observed and synthetic waveforms using time-frequency misfits allowing us to separate both phase and amplitude measurements whilst exploiting the complete waveform at periods of 15-60 seconds. Fréchet kernels for these misfits are calculated via the adjoint method and subsequently used in an iterative non-linear conjugate-gradient optimization. Finally, we employ custom smoothing algorithms to remove the singularities of the Fréchet kernels and artifacts introduced by the heterogeneous coverage in oceanic regions of the model.

Parameter Optimization for Turbulent Reacting Flows Using Adjoints

NASA Astrophysics Data System (ADS)

Lapointe, Caelan; Hamlington, Peter E.

2017-11-01

The formulation of a new adjoint solver for topology optimization of turbulent reacting flows is presented. This solver provides novel configurations (e.g., geometries and operating conditions) based on desired system outcomes (i.e., objective functions) for complex reacting flow problems of practical interest. For many such problems, it would be desirable to know optimal values of design parameters (e.g., physical dimensions, fuel-oxidizer ratios, and inflow-outflow conditions) prior to real-world manufacture and testing, which can be expensive, time-consuming, and dangerous. However, computational optimization of these problems is made difficult by the complexity of most reacting flows, necessitating the use of gradient-based optimization techniques in order to explore a wide design space at manageable computational cost. The adjoint method is an attractive way to obtain the required gradients, because the cost of the method is determined by the dimension of the objective function rather than the size of the design space. Here, the formulation of a novel solver is outlined that enables gradient-based parameter optimization of turbulent reacting flows using the discrete adjoint method. Initial results and an outlook for future research directions are provided.

Unsteady Adjoint Approach for Design Optimization of Flapping Airfoils

NASA Technical Reports Server (NTRS)

Lee, Byung Joon; Liou, Meng-Sing

2012-01-01

This paper describes the work for optimizing the propulsive efficiency of flapping airfoils, i.e., improving the thrust under constraining aerodynamic work during the flapping flights by changing their shape and trajectory of motion with the unsteady discrete adjoint approach. For unsteady problems, it is essential to properly resolving time scales of motion under consideration and it must be compatible with the objective sought after. We include both the instantaneous and time-averaged (periodic) formulations in this study. For the design optimization with shape parameters or motion parameters, the time-averaged objective function is found to be more useful, while the instantaneous one is more suitable for flow control. The instantaneous objective function is operationally straightforward. On the other hand, the time-averaged objective function requires additional steps in the adjoint approach; the unsteady discrete adjoint equations for a periodic flow must be reformulated and the corresponding system of equations solved iteratively. We compare the design results from shape and trajectory optimizations and investigate the physical relevance of design variables to the flapping motion at on- and off-design conditions.

An adjoint-based simultaneous estimation method of the asthenosphere's viscosity and afterslip using a fast and scalable finite-element adjoint solver

NASA Astrophysics Data System (ADS)

Agata, Ryoichiro; Ichimura, Tsuyoshi; Hori, Takane; Hirahara, Kazuro; Hashimoto, Chihiro; Hori, Muneo

2018-04-01

The simultaneous estimation of the asthenosphere's viscosity and coseismic slip/afterslip is expected to improve largely the consistency of the estimation results to observation data of crustal deformation collected in widely spread observation points, compared to estimations of slips only. Such an estimate can be formulated as a non-linear inverse problem of material properties of viscosity and input force that is equivalent to fault slips based on large-scale finite-element (FE) modeling of crustal deformation, in which the degree of freedom is in the order of 109. We formulated and developed a computationally efficient adjoint-based estimation method for this inverse problem, together with a fast and scalable FE solver for the associated forward and adjoint problems. In a numerical experiment that imitates the 2011 Tohoku-Oki earthquake, the advantage of the proposed method is confirmed by comparing the estimated results with those obtained using simplified estimation methods. The computational cost required for the optimization shows that the proposed method enabled the targeted estimation to be completed with moderate amount of computational resources.

Adjoint-based sensitivity analysis of low-order thermoacoustic networks using a wave-based approach

NASA Astrophysics Data System (ADS)

Aguilar, José G.; Magri, Luca; Juniper, Matthew P.

2017-07-01

Strict pollutant emission regulations are pushing gas turbine manufacturers to develop devices that operate in lean conditions, with the downside that combustion instabilities are more likely to occur. Methods to predict and control unstable modes inside combustion chambers have been developed in the last decades but, in some cases, they are computationally expensive. Sensitivity analysis aided by adjoint methods provides valuable sensitivity information at a low computational cost. This paper introduces adjoint methods and their application in wave-based low order network models, which are used as industrial tools, to predict and control thermoacoustic oscillations. Two thermoacoustic models of interest are analyzed. First, in the zero Mach number limit, a nonlinear eigenvalue problem is derived, and continuous and discrete adjoint methods are used to obtain the sensitivities of the system to small modifications. Sensitivities to base-state modification and feedback devices are presented. Second, a more general case with non-zero Mach number, a moving flame front and choked outlet, is presented. The influence of the entropy waves on the computed sensitivities is shown.

Identification of unknown spatial load distributions in a vibrating Euler-Bernoulli beam from limited measured data

NASA Astrophysics Data System (ADS)

Hasanov, Alemdar; Kawano, Alexandre

2016-05-01

Two types of inverse source problems of identifying asynchronously distributed spatial loads governed by the Euler-Bernoulli beam equation ρ (x){w}{tt}+μ (x){w}t+{({EI}(x){w}{xx})}{xx}-{T}r{u}{xx}={\\sum }m=1M{g}m(t){f}m(x), (x,t)\\in {{{Ω }}}T := (0,l)× (0,T), with hinged-clamped ends (w(0,t)={w}{xx}(0,t)=0,w(l,t) = {w}x(l,t)=0,t\\in (0,T)), are studied. Here {g}m(t) are linearly independent functions, describing an asynchronous temporal loading, and {f}m(x) are the spatial load distributions. In the first identification problem the values {ν }k(t),k=\\bar{1,K}, of the deflection w(x,t), are assumed to be known, as measured output data, in a neighbourhood of the finite set of points P:= \\{{x}k\\in (0,l),k=\\bar{1,K}\\}\\subset (0,l), corresponding to the internal points of a continuous beam, for all t\\in ]0,T[. In the second identification problem the values {θ }k(t),k=\\bar{1,K}, of the slope {w}x(x,t), are assumed to be known, as measured output data in a neighbourhood of the same set of points P for all t\\in ]0,T[. These inverse source problems will be defined subsequently as the problems ISP1 and ISP2. The general purpose of this study is to develop mathematical concepts and tools that are capable of providing effective numerical algorithms for the numerical solution of the considered class of inverse problems. Note that both measured output data {ν }k(t) and {θ }k(t) contain random noise. In the first part of the study we prove that each measured output data {ν }k(t) and {θ }k(t),k=\\bar{1,K} can uniquely determine the unknown functions {f}m\\in {H}-1(]0,l[),m=\\bar{1,M}. In the second part of the study we will introduce the input-output operators {{ K }}d :{L}2(0,T)\\mapsto {L}2(0,T),({{ K }}df)(t):= w(x,t;f),x\\in P, f(x) := ({f}1(x),\\ldots ,{f}M(x)), and {{ K }}s :{L}2(0,T)\\mapsto {L}2(0,T), ({{ K }}sf)(t):= {w}x(x,t;f), x\\in P , corresponding to the problems ISP1 and ISP2, and then reformulate these problems as the operator equations: {{ K }}df=ν and {{ K }}sf=θ , where ν (t):= ({ν }1(t),\\ldots ,{ν }K(t)) and {θ }k(t):= ({θ }1(t),\\ldots ,{θ }K(t)). Since both measured output data contain random noise, we use the most prominent regularisation method, Tikhonov regularisation, introducing the regularised cost functionals {J}1α (f):= (1/2)\\parallel {{ K }}df-ν {\\parallel }{L2(0,T)}2+(1/2)α \\parallel f{\\parallel }{L2(0,T)}2 and {J}2α (f):= (1/2)\\parallel {{ K }}sf-θ {\\parallel }{L2(0,T)}2+(1/2)α \\parallel f{\\parallel }{L2(0,T)}2. Using a priori estimates for the weak solution of the direct problem and the Tikhonov regularisation method combined with the adjoint problem approach, we prove that the Fréchet gradients {J}1\\prime (f) and {J}2\\prime (f) of both cost functionals can explicitly be derived via the corresponding weak solutions of adjoint problems and the known temporal loads {g}m(t). Moreover, we show that these gradients are Lipschitz continuous, which allows the use of gradient type iteration convergent algorithms. Two applications of the proposed theory are presented. It is shown that solvability results for inverse source problems related to the synchronous loading case, with a single interior measured data, are special cases of the obtained results for asynchronously distributed spatial load cases.

Using CFD Surface Solutions to Shape Sonic Boom Signatures Propagated from Off-Body Pressure

NASA Technical Reports Server (NTRS)

Ordaz, Irian; Li, Wu

2013-01-01

The conceptual design of a low-boom and low-drag supersonic aircraft remains a challenge despite significant progress in recent years. Inverse design using reversed equivalent area and adjoint methods have been demonstrated to be effective in shaping the ground signature propagated from computational fluid dynamics (CFD) off-body pressure distributions. However, there is still a need to reduce the computational cost in the early stages of design to obtain a baseline that is feasible for low-boom shaping, and in the search for a robust low-boom design over the entire sonic boom footprint. The proposed design method addresses the need to reduce the computational cost for robust low-boom design by using surface pressure distributions from CFD solutions to shape sonic boom ground signatures propagated from CFD off-body pressure.

A new scheme of the time-domain fluorescence tomography for a semi-infinite turbid medium

NASA Astrophysics Data System (ADS)

Prieto, Kernel; Nishimura, Goro

2017-04-01

A new scheme for reconstruction of a fluorophore target embedded in a semi-infinite medium was proposed and evaluated. In this scheme, we neglected the presence of the fluorophore target for the excitation light and used an analytical solution of the time-dependent radiative transfer equation (RTE) for the excitation light in a homogeneous semi-infinite media instead of solving the RTE numerically in the forward calculation. The inverse problem for imaging the fluorophore target was solved using the Landweber-Kaczmarz method with the concept of the adjoint fields. Numerical experiments show that the proposed scheme provides acceptable results of the reconstructed shape and location of the target. The computation times of the solution of the forward problem and the whole reconstruction process were reduced by about 40 and 15%, respectively.

Time-domain seismic modeling in viscoelastic media for full waveform inversion on heterogeneous computing platforms with OpenCL

NASA Astrophysics Data System (ADS)

Fabien-Ouellet, Gabriel; Gloaguen, Erwan; Giroux, Bernard

2017-03-01

Full Waveform Inversion (FWI) aims at recovering the elastic parameters of the Earth by matching recordings of the ground motion with the direct solution of the wave equation. Modeling the wave propagation for realistic scenarios is computationally intensive, which limits the applicability of FWI. The current hardware evolution brings increasing parallel computing power that can speed up the computations in FWI. However, to take advantage of the diversity of parallel architectures presently available, new programming approaches are required. In this work, we explore the use of OpenCL to develop a portable code that can take advantage of the many parallel processor architectures now available. We present a program called SeisCL for 2D and 3D viscoelastic FWI in the time domain. The code computes the forward and adjoint wavefields using finite-difference and outputs the gradient of the misfit function given by the adjoint state method. To demonstrate the code portability on different architectures, the performance of SeisCL is tested on three different devices: Intel CPUs, NVidia GPUs and Intel Xeon PHI. Results show that the use of GPUs with OpenCL can speed up the computations by nearly two orders of magnitudes over a single threaded application on the CPU. Although OpenCL allows code portability, we show that some device-specific optimization is still required to get the best performance out of a specific architecture. Using OpenCL in conjunction with MPI allows the domain decomposition of large models on several devices located on different nodes of a cluster. For large enough models, the speedup of the domain decomposition varies quasi-linearly with the number of devices. Finally, we investigate two different approaches to compute the gradient by the adjoint state method and show the significant advantages of using OpenCL for FWI.

Intelligent earthquake data processing for global adjoint tomography

NASA Astrophysics Data System (ADS)

Chen, Y.; Hill, J.; Li, T.; Lei, W.; Ruan, Y.; Lefebvre, M. P.; Tromp, J.

2016-12-01

Due to the increased computational capability afforded by modern and future computing architectures, the seismology community is demanding a more comprehensive understanding of the full waveform information from the recorded earthquake seismograms. Global waveform tomography is a complex workflow that matches observed seismic data with synthesized seismograms by iteratively updating the earth model parameters based on the adjoint state method. This methodology allows us to compute a very accurate model of the earth's interior. The synthetic data is simulated by solving the wave equation in the entire globe using a spectral-element method. In order to ensure the inversion accuracy and stability, both the synthesized and observed seismograms must be carefully pre-processed. Because the scale of the inversion problem is extremely large and there is a very large volume of data to both be read and written, an efficient and reliable pre-processing workflow must be developed. We are investigating intelligent algorithms based on a machine-learning (ML) framework that will automatically tune parameters for the data processing chain. One straightforward application of ML in data processing is to classify all possible misfit calculation windows into usable and unusable ones, based on some intelligent ML models such as neural network, support vector machine or principle component analysis. The intelligent earthquake data processing framework will enable the seismology community to compute the global waveform tomography using seismic data from an arbitrarily large number of earthquake events in the fastest, most efficient way.

Modeling Finite Faults Using the Adjoint Wave Field

NASA Astrophysics Data System (ADS)

Hjörleifsdóttir, V.; Liu, Q.; Tromp, J.

2004-12-01

Time-reversal acoustics, a technique in which an acoustic signal is recorded by an array of transducers, time-reversed, and retransmitted, is used, e.g., in medical therapy to locate and destroy gallstones (for a review see Fink, 1997). As discussed by Tromp et al. (2004), time-reversal techniques for locating sources are closely linked to so-called `adjoint methods' (Talagrand and Courtier, 1987), which may be used to evaluate the gradient of a misfit function. Tromp et al. (2004) illustrate how a (finite) source inversion may be implemented based upon the adjoint wave field by writing the change in the misfit function, δ χ, due to a change in the moment-density tensor, δ m, as an integral of the adjoint strain field ɛ x,t) over the fault plane Σ : δ χ = ∫ 0T∫_Σ ɛ x,T-t) :δ m(x,t) d2xdt. We find that if the real fault plane is located at a distance δ h in the direction of the fault normal hat n, then to first order an additional factor of ∫ 0T∫_Σ δ h (x) ∂ n ɛ x,T-t):m(x,t) d2xdt is added to the change in the misfit function. The adjoint strain is computed by using the time-reversed difference between data and synthetics recorded at all receivers as simultaneous sources and recording the resulting strain on the fault plane. In accordance with time-reversal acoustics, all the resulting waves will constructively interfere at the position of the original source in space and time. The level of convergence will be deterimined by factors such as the source-receiver geometry, the frequency of the recorded data and synthetics, and the accuracy of the velocity structure used when back propagating the wave field. The terms ɛ x,T-t) and ∂ n ɛ x,T-t):m(x,t) can be viewed as sensitivity kernels for the moment density and the faultplane location respectively. By looking at these quantities we can make an educated choice of fault parametrization given the data in hand. The process can then be repeated to invert for the best source model, as demonstrated by Tromp et al. (2004) for the magnitude of a point force. In this presentation we explore the applicability of adjoint methods to estimating finite source parameters. Fink, M. (1997), Time reversed acoustics, Physics Today, 50(3), 34--40. Talagrand, O., and P.~Courtier (1987), Variational assimilation of meteorological observations with the adjoint vorticity equatuation. I: Theory, Q. J. R. Meteorol. Soc., 113, 1311--1328. Tromp, J., C.~Tape, and Q.~Liu (2004), Waveform tomography, adjoint methods, time reversal, and banana-doughnut kernels, Geophys. Jour. Int., in press

An all-at-once reduced Hessian SQP scheme for aerodynamic design optimization

NASA Technical Reports Server (NTRS)

Feng, Dan; Pulliam, Thomas H.

1995-01-01

This paper introduces a computational scheme for solving a class of aerodynamic design problems that can be posed as nonlinear equality constrained optimizations. The scheme treats the flow and design variables as independent variables, and solves the constrained optimization problem via reduced Hessian successive quadratic programming. It updates the design and flow variables simultaneously at each iteration and allows flow variables to be infeasible before convergence. The solution of an adjoint flow equation is never needed. In addition, a range space basis is chosen so that in a certain sense the 'cross term' ignored in reduced Hessian SQP methods is minimized. Numerical results for a nozzle design using the quasi-one-dimensional Euler equations show that this scheme is computationally efficient and robust. The computational cost of a typical nozzle design is only a fraction more than that of the corresponding analysis flow calculation. Superlinear convergence is also observed, which agrees with the theoretical properties of this scheme. All optimal solutions are obtained by starting far away from the final solution.

Topics in structural dynamics: Nonlinear unsteady transonic flows and Monte Carlo methods in acoustics

NASA Technical Reports Server (NTRS)

Haviland, J. K.

1974-01-01

The results are reported of two unrelated studies. The first was an investigation of the formulation of the equations for non-uniform unsteady flows, by perturbation of an irrotational flow to obtain the linear Green's equation. The resulting integral equation was found to contain a kernel which could be expressed as the solution of the adjoint flow equation, a linear equation for small perturbations, but with non-constant coefficients determined by the steady flow conditions. It is believed that the non-uniform flow effects may prove important in transonic flutter, and that in such cases, the use of doublet type solutions of the wave equation would then prove to be erroneous. The second task covered an initial investigation into the use of the Monte Carlo method for solution of acoustical field problems. Computed results are given for a rectangular room problem, and for a problem involving a circular duct with a source located at the closed end.

FW/CADIS-O: An Angle-Informed Hybrid Method for Neutron Transport

NASA Astrophysics Data System (ADS)

Munk, Madicken

The development of methods for deep-penetration radiation transport is of continued importance for radiation shielding, nonproliferation, nuclear threat reduction, and medical applications. As these applications become more ubiquitous, the need for transport methods that can accurately and reliably model the systems' behavior will persist. For these types of systems, hybrid methods are often the best choice to obtain a reliable answer in a short amount of time. Hybrid methods leverage the speed and uniform uncertainty distribution of a deterministic solution to bias Monte Carlo transport to reduce the variance in the solution. At present, the Consistent Adjoint-Driven Importance Sampling (CADIS) and Forward-Weighted CADIS (FW-CADIS) hybrid methods are the gold standard by which to model systems that have deeply-penetrating radiation. They use an adjoint scalar flux to generate variance reduction parameters for Monte Carlo. However, in problems where there exists strong anisotropy in the flux, CADIS and FW-CADIS are not as effective at reducing the problem variance as isotropic problems. This dissertation covers the theoretical background, implementation of, and characteri- zation of a set of angle-informed hybrid methods that can be applied to strongly anisotropic deep-penetration radiation transport problems. These methods use a forward-weighted adjoint angular flux to generate variance reduction parameters for Monte Carlo. As a result, they leverage both adjoint and contributon theory for variance reduction. They have been named CADIS-O and FW-CADIS-O. To characterize CADIS-O, several characterization problems with flux anisotropies were devised. These problems contain different physical mechanisms by which flux anisotropy is induced. Additionally, a series of novel anisotropy metrics by which to quantify flux anisotropy are used to characterize the methods beyond standard Figure of Merit (FOM) and relative error metrics. As a result, a more thorough investigation into the effects of anisotropy and the degree of anisotropy on Monte Carlo convergence is possible. The results from the characterization of CADIS-O show that it performs best in strongly anisotropic problems that have preferential particle flowpaths, but only if the flowpaths are not comprised of air. Further, the characterization of the method's sensitivity to deterministic angular discretization showed that CADIS-O has less sensitivity to discretization than CADIS for both quadrature order and PN order. However, more variation in the results were observed in response to changing quadrature order than PN order. Further, as a result of the forward-normalization in the O-methods, ray effect mitigation was observed in many of the characterization problems. The characterization of the CADIS-O-method in this dissertation serves to outline a path forward for further hybrid methods development. In particular, the response that the O-method has with changes in quadrature order, PN order, and on ray effect mitigation are strong indicators that the method is more resilient than its predecessors to strong anisotropies in the flux. With further method characterization, the full potential of the O-methods can be realized. The method can then be applied to geometrically complex, materially diverse problems and help to advance system modelling in deep-penetration radiation transport problems with strong anisotropies in the flux.

Local error estimates for discontinuous solutions of nonlinear hyperbolic equations

NASA Technical Reports Server (NTRS)

Tadmor, Eitan

1989-01-01

Let u(x,t) be the possibly discontinuous entropy solution of a nonlinear scalar conservation law with smooth initial data. Suppose u sub epsilon(x,t) is the solution of an approximate viscosity regularization, where epsilon greater than 0 is the small viscosity amplitude. It is shown that by post-processing the small viscosity approximation u sub epsilon, pointwise values of u and its derivatives can be recovered with an error as close to epsilon as desired. The analysis relies on the adjoint problem of the forward error equation, which in this case amounts to a backward linear transport with discontinuous coefficients. The novelty of this approach is to use a (generalized) E-condition of the forward problem in order to deduce a W(exp 1,infinity) energy estimate for the discontinuous backward transport equation; this, in turn, leads one to an epsilon-uniform estimate on moments of the error u(sub epsilon) - u. This approach does not follow the characteristics and, therefore, applies mutatis mutandis to other approximate solutions such as E-difference schemes.

A heuristic for suffix solutions

DOE Office of Scientific and Technical Information (OSTI.GOV)

Bilgory, A.; Gajski, D.D.

1986-01-01

The suffix problem has appeared in solutions of recurrence systems for parallel and pipelined machines and more recently in the design of gate and silicon compilers. In this paper the authors present two algorithms. The first algorithm generates parallel suffix solutions with minimum cost for a given length, time delay, availability of initial values, and fanout. This algorithm generates a minimal solution for any length n and depth range log/sub 2/ N to N. The second algorithm reduces the size of the solutions generated by the first algorithm.

Mitigation of Engine Inlet Distortion Through Adjoint-Based Design

NASA Technical Reports Server (NTRS)

Ordaz, Irian; Rallabhandi, Sriram; Nielsen, Eric J.; Diskin, Boris

2017-01-01

The adjoint-based design capability in FUN3D is extended to allow efficient gradient- based optimization and design of concepts with highly integrated aero-propulsive systems. A circumferential distortion calculation, along with the derivatives needed to perform adjoint-based design, have been implemented in FUN3D. This newly implemented distortion calculation can be used not only for design but also to drive the existing mesh adaptation process and reduce the error associated with the fan distortion calculation. The design capability is demonstrated by the shape optimization of an in-house aircraft concept equipped with an aft fuselage propulsor. The optimization objective is the minimization of flow distortion at the aerodynamic interface plane of this aft fuselage propulsor.

Mathematics of tsunami: modelling and identification

NASA Astrophysics Data System (ADS)

Krivorotko, Olga; Kabanikhin, Sergey

2015-04-01

Tsunami (long waves in the deep water) motion caused by underwater earthquakes is described by shallow water equations ( { ηtt = div (gH (x,y)-gradη), (x,y) ∈ Ω, t ∈ (0,T ); η|t=0 = q(x,y), ηt|t=0 = 0, (x,y) ∈ Ω. ( (1) Bottom relief H(x,y) characteristics and the initial perturbation data (a tsunami source q(x,y)) are required for the direct simulation of tsunamis. The main difficulty problem of tsunami modelling is a very big size of the computational domain (Ω = 500 × 1000 kilometres in space and about one hour computational time T for one meter of initial perturbation amplitude max|q|). The calculation of the function η(x,y,t) of three variables in Ω × (0,T) requires large computing resources. We construct a new algorithm to solve numerically the problem of determining the moving tsunami wave height S(x,y) which is based on kinematic-type approach and analytical representation of fundamental solution. Proposed algorithm of determining the function of two variables S(x,y) reduces the number of operations in 1.5 times than solving problem (1). If all functions does not depend on the variable y (one dimensional case), then the moving tsunami wave height satisfies of the well-known Airy-Green formula: S(x) = S(0)° --- 4H (0)/H (x). The problem of identification parameters of a tsunami source using additional measurements of a passing wave is called inverse tsunami problem. We investigate two different inverse problems of determining a tsunami source q(x,y) using two different additional data: Deep-ocean Assessment and Reporting of Tsunamis (DART) measurements and satellite altimeters wave-form images. These problems are severely ill-posed. The main idea consists of combination of two measured data to reconstruct the source parameters. We apply regularization techniques to control the degree of ill-posedness such as Fourier expansion, truncated singular value decomposition, numerical regularization. The algorithm of selecting the truncated number of singular values of an inverse problem operator which is agreed with the error level in measured data is described and analysed. In numerical experiment we used conjugate gradient method for solving inverse tsunami problems. Gradient methods are based on minimizing the corresponding misfit function. To calculate the gradient of the misfit function, the adjoint problem is solved. The conservative finite-difference schemes for solving the direct and adjoint problems in the approximation of shallow water are constructed. Results of numerical experiments of the tsunami source reconstruction are presented and discussed. We show that using a combination of two types of data allows one to increase the stability and efficiency of tsunami source reconstruction. Non-profit organization WAPMERR (World Agency of Planetary Monitoring and Earthquake Risk Reduction) in collaboration with Institute of Computational Mathematics and Mathematical Geophysics of SB RAS developed the Integrated Tsunami Research and Information System (ITRIS) to simulate tsunami waves and earthquakes, river course changes, coastal zone floods, and risk estimates for coastal constructions at wave run-ups and earthquakes. The special scientific plug-in components are embedded in a specially developed GIS-type graphic shell for easy data retrieval, visualization and processing. We demonstrate the tsunami simulation plug-in for historical tsunami events (2004 Indian Ocean tsunami, Simushir tsunami 2006 and others). This work was supported by the Ministry of Education and Science of the Russian Federation.

Analyzing Lie symmetry and constructing conservation laws for time-fractional Benny-Lin equation

NASA Astrophysics Data System (ADS)

Rashidi, Saeede; Hejazi, S. Reza

This paper investigates the invariance properties of the time fractional Benny-Lin equation with Riemann-Liouville and Caputo derivatives. This equation can be reduced to the Kawahara equation, fifth-order Kdv equation, the Kuramoto-Sivashinsky equation and Navier-Stokes equation. By using the Lie group analysis method of fractional differential equations (FDEs), we derive Lie symmetries for the Benny-Lin equation. Conservation laws for this equation are obtained with the aid of the concept of nonlinear self-adjointness and the fractional generalization of the Noether’s operators. Furthermore, by means of the invariant subspace method, exact solutions of the equation are also constructed.

Additive schemes for certain operator-differential equations

NASA Astrophysics Data System (ADS)

Vabishchevich, P. N.

2010-12-01

Unconditionally stable finite difference schemes for the time approximation of first-order operator-differential systems with self-adjoint operators are constructed. Such systems arise in many applied problems, for example, in connection with nonstationary problems for the system of Stokes (Navier-Stokes) equations. Stability conditions in the corresponding Hilbert spaces for two-level weighted operator-difference schemes are obtained. Additive (splitting) schemes are proposed that involve the solution of simple problems at each time step. The results are used to construct splitting schemes with respect to spatial variables for nonstationary Navier-Stokes equations for incompressible fluid. The capabilities of additive schemes are illustrated using a two-dimensional model problem as an example.

Formulation and Implementation of Inflow/Outflow Boundary Conditions to Simulate Propulsive Effects

NASA Technical Reports Server (NTRS)

Rodriguez, David L.; Aftosmis, Michael J.; Nemec, Marian

2018-01-01

Boundary conditions appropriate for simulating flow entering or exiting the computational domain to mimic propulsion effects have been implemented in an adaptive Cartesian simulation package. A robust iterative algorithm to control mass flow rate through an outflow boundary surface is presented, along with a formulation to explicitly specify mass flow rate through an inflow boundary surface. The boundary conditions have been applied within a mesh adaptation framework based on the method of adjoint-weighted residuals. This allows for proper adaptive mesh refinement when modeling propulsion systems. The new boundary conditions are demonstrated on several notional propulsion systems operating in flow regimes ranging from low subsonic to hypersonic. The examples show that the prescribed boundary state is more properly imposed as the mesh is refined. The mass-flowrate steering algorithm is shown to be an efficient approach in each example. To demonstrate the boundary conditions on a realistic complex aircraft geometry, two of the new boundary conditions are also applied to a modern low-boom supersonic demonstrator design with multiple flow inlets and outlets.

SU-G-TeP1-15: Toward a Novel GPU Accelerated Deterministic Solution to the Linear Boltzmann Transport Equation

DOE Office of Scientific and Technical Information (OSTI.GOV)

Yang, R; Fallone, B; Cross Cancer Institute, Edmonton, AB

Purpose: To develop a Graphic Processor Unit (GPU) accelerated deterministic solution to the Linear Boltzmann Transport Equation (LBTE) for accurate dose calculations in radiotherapy (RT). A deterministic solution yields the potential for major speed improvements due to the sparse matrix-vector and vector-vector multiplications and would thus be of benefit to RT. Methods: In order to leverage the massively parallel architecture of GPUs, the first order LBTE was reformulated as a second order self-adjoint equation using the Least Squares Finite Element Method (LSFEM). This produces a symmetric positive-definite matrix which is efficiently solved using a parallelized conjugate gradient (CG) solver. Themore » LSFEM formalism is applied in space, discrete ordinates is applied in angle, and the Multigroup method is applied in energy. The final linear system of equations produced is tightly coupled in space and angle. Our code written in CUDA-C was benchmarked on an Nvidia GeForce TITAN-X GPU against an Intel i7-6700K CPU. A spatial mesh of 30,950 tetrahedral elements was used with an S4 angular approximation. Results: To avoid repeating a full computationally intensive finite element matrix assembly at each Multigroup energy, a novel mapping algorithm was developed which minimized the operations required at each energy. Additionally, a parallelized memory mapping for the kronecker product between the sparse spatial and angular matrices, including Dirichlet boundary conditions, was created. Atomicity is preserved by graph-coloring overlapping nodes into separate kernel launches. The one-time mapping calculations for matrix assembly, kronecker product, and boundary condition application took 452±1ms on GPU. Matrix assembly for 16 energy groups took 556±3s on CPU, and 358±2ms on GPU using the mappings developed. The CG solver took 93±1s on CPU, and 468±2ms on GPU. Conclusion: Three computationally intensive subroutines in deterministically solving the LBTE have been formulated on GPU, resulting in two orders of magnitude speedup. Funding support from Natural Sciences and Engineering Research Council and Alberta Innovates Health Solutions. Dr. Fallone is a co-founder and CEO of MagnetTx Oncology Solutions (under discussions to license Alberta bi-planar linac MR for commercialization).« less

Sensitivity analysis of a model of CO2 exchange in tundra ecosystems by the adjoint method

NASA Technical Reports Server (NTRS)

Waelbroek, C.; Louis, J.-F.

1995-01-01

A model of net primary production (NPP), decomposition, and nitrogen cycling in tundra ecosystems has been developed. The adjoint technique is used to study the sensitivity of the computed annual net CO2 flux to perturbation in initial conditions, climatic inputs, and model's main parameters describing current seasonal CO2 exchange in wet sedge tundra at Barrow, Alaska. The results show that net CO2 flux is most sensitive to parameters characterizing litter chemical composition and more sensitive to decomposition parameters than to NPP parameters. This underlines the fact that in nutrient-limited ecosystems, decomposition drives net CO2 exchange by controlling mineralization of main nutrients. The results also indicate that the short-term (1 year) response of wet sedge tundra to CO2-induced warming is a significant increase in CO2 emission, creating a positive feedback to atmosphreic CO2 accumulation. However, a cloudiness increase during the same year can severely alter this response and lead to either a slight decrease or a strong increase in emitted CO2, depending on its exact timing. These results demonstrate that the adjoint method is well suited to study systems encountering regime changes, as a single run of the adjoint model provides sensitivities of the net CO2 flux to perturbations in all parameters and variables at any time of the year. Moreover, it is shown that large errors due to the presence of thresholds can be avoided by first delimiting the range of applicability of the adjoint results.

Linear Array Ambient Noise Adjoint Tomography Reveals Intense Crust-Mantle Interactions in North China Craton

NASA Astrophysics Data System (ADS)

Zhang, Chao; Yao, Huajian; Liu, Qinya; Zhang, Ping; Yuan, Yanhua O.; Feng, Jikun; Fang, Lihua

2018-01-01

We present a 2-D ambient noise adjoint tomography technique for a linear array with a significant reduction in computational cost and show its application to an array in North China. We first convert the observed data for 3-D media, i.e., surface-wave empirical Green's functions (EGFs) to the reconstructed EGFs (REGFs) for 2-D media using a 3-D/2-D transformation scheme. Different from the conventional steps of measuring phase dispersion, this technology refines 2-D shear wave speeds along the profile directly from REGFs. With an initial model based on traditional ambient noise tomography, adjoint tomography updates the model by minimizing the frequency-dependent Rayleigh wave traveltime delays between the REGFs and synthetic Green functions calculated by the spectral-element method. The multitaper traveltime difference measurement is applied in four-period bands: 20-35 s, 15-30 s, 10-20 s, and 6-15 s. The recovered model shows detailed crustal structures including pronounced low-velocity anomalies in the lower crust and a gradual crust-mantle transition zone beneath the northern Trans-North China Orogen, which suggest the possible intense thermo-chemical interactions between mantle-derived upwelling melts and the lower crust, probably associated with the magmatic underplating during the Mesozoic to Cenozoic evolution of this region. To our knowledge, it is the first time that ambient noise adjoint tomography is implemented for a 2-D medium. Compared with the intensive computational cost and storage requirement of 3-D adjoint tomography, this method offers a computationally efficient and inexpensive alternative to imaging fine-scale crustal structures beneath linear arrays.

CFD Analysis and Design Optimization Using Parallel Computers

NASA Technical Reports Server (NTRS)

Martinelli, Luigi; Alonso, Juan Jose; Jameson, Antony; Reuther, James

1997-01-01

A versatile and efficient multi-block method is presented for the simulation of both steady and unsteady flow, as well as aerodynamic design optimization of complete aircraft configurations. The compressible Euler and Reynolds Averaged Navier-Stokes (RANS) equations are discretized using a high resolution scheme on body-fitted structured meshes. An efficient multigrid implicit scheme is implemented for time-accurate flow calculations. Optimum aerodynamic shape design is achieved at very low cost using an adjoint formulation. The method is implemented on parallel computing systems using the MPI message passing interface standard to ensure portability. The results demonstrate that, by combining highly efficient algorithms with parallel computing, it is possible to perform detailed steady and unsteady analysis as well as automatic design for complex configurations using the present generation of parallel computers.

Thermodynamics of Gas Turbine Cycles with Analytic Derivatives in OpenMDAO

NASA Technical Reports Server (NTRS)

Gray, Justin; Chin, Jeffrey; Hearn, Tristan; Hendricks, Eric; Lavelle, Thomas; Martins, Joaquim R. R. A.

2016-01-01

A new equilibrium thermodynamics analysis tool was built based on the CEA method using the OpenMDAO framework. The new tool provides forward and adjoint analytic derivatives for use with gradient based optimization algorithms. The new tool was validated against the original CEA code to ensure an accurate analysis and the analytic derivatives were validated against finite-difference approximations. Performance comparisons between analytic and finite difference methods showed a significant speed advantage for the analytic methods. To further test the new analysis tool, a sample optimization was performed to find the optimal air-fuel equivalence ratio, , maximizing combustion temperature for a range of different pressures. Collectively, the results demonstrate the viability of the new tool to serve as the thermodynamic backbone for future work on a full propulsion modeling tool.

Detecting Shielded Special Nuclear Materials Using Multi-Dimensional Neutron Source and Detector Geometries

NASA Astrophysics Data System (ADS)

Santarius, John; Navarro, Marcos; Michalak, Matthew; Fancher, Aaron; Kulcinski, Gerald; Bonomo, Richard

2016-10-01

A newly initiated research project will be described that investigates methods for detecting shielded special nuclear materials by combining multi-dimensional neutron sources, forward/adjoint calculations modeling neutron and gamma transport, and sparse data analysis of detector signals. The key tasks for this project are: (1) developing a radiation transport capability for use in optimizing adaptive-geometry, inertial-electrostatic confinement (IEC) neutron source/detector configurations for neutron pulses distributed in space and/or phased in time; (2) creating distributed-geometry, gas-target, IEC fusion neutron sources; (3) applying sparse data and noise reduction algorithms, such as principal component analysis (PCA) and wavelet transform analysis, to enhance detection fidelity; and (4) educating graduate and undergraduate students. Funded by DHS DNDO Project 2015-DN-077-ARI095.

Optimization with Fuzzy Data via Evolutionary Algorithms

NASA Astrophysics Data System (ADS)

Kosiński, Witold

2010-09-01

Order fuzzy numbers (OFN) that make possible to deal with fuzzy inputs quantitatively, exactly in the same way as with real numbers, have been recently defined by the author and his 2 coworkers. The set of OFN forms a normed space and is a partially ordered ring. The case when the numbers are presented in the form of step functions, with finite resolution, simplifies all operations and the representation of defuzzification functionals. A general optimization problem with fuzzy data is formulated. Its fitness function attains fuzzy values. Since the adjoint space to the space of OFN is finite dimensional, a convex combination of all linear defuzzification functionals may be used to introduce a total order and a real-valued fitness function. Genetic operations on individuals representing fuzzy data are defined.

Spectral monodromy of non-self-adjoint operators

NASA Astrophysics Data System (ADS)

Phan, Quang Sang

2014-01-01

In the present paper, we build a combinatorial invariant, called the "spectral monodromy" from the spectrum of a single (non-self-adjoint) h-pseudodifferential operator with two degrees of freedom in the semi-classical limit. Our inspiration comes from the quantum monodromy defined for the joint spectrum of an integrable system of n commuting self-adjoint h-pseudodifferential operators, given by S. Vu Ngoc ["Quantum monodromy in integrable systems," Commun. Math. Phys. 203(2), 465-479 (1999)]. The first simple case that we treat in this work is a normal operator. In this case, the discrete spectrum can be identified with the joint spectrum of an integrable quantum system. The second more complex case we propose is a small perturbation of a self-adjoint operator with a classical integrability property. We show that the discrete spectrum (in a small band around the real axis) also has a combinatorial monodromy. The main difficulty in this case is that we do not know the description of the spectrum everywhere, but only in a Cantor type set. In addition, we also show that the corresponding monodromy can be identified with the classical monodromy, defined by J. Duistermaat ["On global action-angle coordinates," Commun. Pure Appl. Math. 33(6), 687-706 (1980)].

Waterjet and laser etching: the nonlinear inverse problem

NASA Astrophysics Data System (ADS)

Bilbao-Guillerna, A.; Axinte, D. A.; Billingham, J.; Cadot, G. B. J.

2017-07-01

In waterjet and laser milling, material is removed from a solid surface in a succession of layers to create a new shape, in a depth-controlled manner. The inverse problem consists of defining the control parameters, in particular, the two-dimensional beam path, to arrive at a prescribed freeform surface. Waterjet milling (WJM) and pulsed laser ablation (PLA) are studied in this paper, since a generic nonlinear material removal model is appropriate for both of these processes. The inverse problem is usually solved for this kind of process by simply controlling dwell time in proportion to the required depth of milling at a sequence of pixels on the surface. However, this approach is only valid when shallow surfaces are etched, since it does not take into account either the footprint of the beam or its overlapping on successive passes. A discrete adjoint algorithm is proposed in this paper to improve the solution. Nonlinear effects and non-straight passes are included in the optimization, while the calculation of the Jacobian matrix does not require large computation times. Several tests are performed to validate the proposed method and the results show that tracking error is reduced typically by a factor of two in comparison to the pixel-by-pixel approach and the classical raster path strategy with straight passes. The tracking error can be as low as 2-5% and 1-2% for WJM and PLA, respectively, depending on the complexity of the target surface.

Using field inversion to quantify functional errors in turbulence closures

NASA Astrophysics Data System (ADS)

Singh, Anand Pratap; Duraisamy, Karthik

2016-04-01

A data-informed approach is presented with the objective of quantifying errors and uncertainties in the functional forms of turbulence closure models. The approach creates modeling information from higher-fidelity simulations and experimental data. Specifically, a Bayesian formalism is adopted to infer discrepancies in the source terms of transport equations. A key enabling idea is the transformation of the functional inversion procedure (which is inherently infinite-dimensional) into a finite-dimensional problem in which the distribution of the unknown function is estimated at discrete mesh locations in the computational domain. This allows for the use of an efficient adjoint-driven inversion procedure. The output of the inversion is a full-field of discrepancy that provides hitherto inaccessible modeling information. The utility of the approach is demonstrated by applying it to a number of problems including channel flow, shock-boundary layer interactions, and flows with curvature and separation. In all these cases, the posterior model correlates well with the data. Furthermore, it is shown that even if limited data (such as surface pressures) are used, the accuracy of the inferred solution is improved over the entire computational domain. The results suggest that, by directly addressing the connection between physical data and model discrepancies, the field inversion approach materially enhances the value of computational and experimental data for model improvement. The resulting information can be used by the modeler as a guiding tool to design more accurate model forms, or serve as input to machine learning algorithms to directly replace deficient modeling terms.

Improving Vector Evaluated Particle Swarm Optimisation by Incorporating Nondominated Solutions

PubMed Central

Lim, Kian Sheng; Ibrahim, Zuwairie; Buyamin, Salinda; Ahmad, Anita; Naim, Faradila; Ghazali, Kamarul Hawari; Mokhtar, Norrima

2013-01-01

The Vector Evaluated Particle Swarm Optimisation algorithm is widely used to solve multiobjective optimisation problems. This algorithm optimises one objective using a swarm of particles where their movements are guided by the best solution found by another swarm. However, the best solution of a swarm is only updated when a newly generated solution has better fitness than the best solution at the objective function optimised by that swarm, yielding poor solutions for the multiobjective optimisation problems. Thus, an improved Vector Evaluated Particle Swarm Optimisation algorithm is introduced by incorporating the nondominated solutions as the guidance for a swarm rather than using the best solution from another swarm. In this paper, the performance of improved Vector Evaluated Particle Swarm Optimisation algorithm is investigated using performance measures such as the number of nondominated solutions found, the generational distance, the spread, and the hypervolume. The results suggest that the improved Vector Evaluated Particle Swarm Optimisation algorithm has impressive performance compared with the conventional Vector Evaluated Particle Swarm Optimisation algorithm. PMID:23737718

Improving Vector Evaluated Particle Swarm Optimisation by incorporating nondominated solutions.

PubMed

Lim, Kian Sheng; Ibrahim, Zuwairie; Buyamin, Salinda; Ahmad, Anita; Naim, Faradila; Ghazali, Kamarul Hawari; Mokhtar, Norrima

2013-01-01

The Vector Evaluated Particle Swarm Optimisation algorithm is widely used to solve multiobjective optimisation problems. This algorithm optimises one objective using a swarm of particles where their movements are guided by the best solution found by another swarm. However, the best solution of a swarm is only updated when a newly generated solution has better fitness than the best solution at the objective function optimised by that swarm, yielding poor solutions for the multiobjective optimisation problems. Thus, an improved Vector Evaluated Particle Swarm Optimisation algorithm is introduced by incorporating the nondominated solutions as the guidance for a swarm rather than using the best solution from another swarm. In this paper, the performance of improved Vector Evaluated Particle Swarm Optimisation algorithm is investigated using performance measures such as the number of nondominated solutions found, the generational distance, the spread, and the hypervolume. The results suggest that the improved Vector Evaluated Particle Swarm Optimisation algorithm has impressive performance compared with the conventional Vector Evaluated Particle Swarm Optimisation algorithm.

Adjoint Airfoil Optimization of Darrieus-Type Vertical Axis Wind Turbine

NASA Astrophysics Data System (ADS)

Fuchs, Roman; Nordborg, Henrik

2012-11-01

We present the feasibility of using an adjoint solver to optimize the torque of a Darrieus-type vertical axis wind turbine (VAWT). We start with a 2D cross section of a symmetrical airfoil and restrict us to low solidity ratios to minimize blade vortex interactions. The adjoint solver of the ANSYS FLUENT software package computes the sensitivities of airfoil surface forces based on a steady flow field. Hence, we find the torque of a full revolution using a weighted average of the sensitivities at different wind speeds and angles of attack. The weights are computed analytically, and the range of angles of attack is given by the tip speed ratio. Then the airfoil geometry is evolved, and the proposed methodology is evaluated by transient simulations.

Examination of Observation Impacts derived from OSEs and Adjoint Models

NASA Technical Reports Server (NTRS)

Gelaro, Ronald

2008-01-01

With the adjoint of a data assimilation system, the impact of any or all assimilated observations on measures of forecast skill can be estimated accurately and efficiently. The approach allows aggregation of results in terms of individual data types, channels or locations, all computed simultaneously. In this study, adjoint-based estimates of observation impact are compared with results from standard observing system experiments (OSEs) in the NASA Goddard Earth Observing System Model, Version 5 (GEOS-5) GEOS-5 system. The two approaches are shown to provide unique, but complimentary, information. Used together, they reveal both redundancies and dependencies between observing system impacts as observations are added or removed. Understanding these dependencies poses a major challenge for optimizing the use of the current observational network and defining requirements for future observing systems.

Adjoint-based constant-mass partial derivatives

DOE PAGES

Favorite, Jeffrey A.

2017-09-01

In transport theory, adjoint-based partial derivatives with respect to mass density are constant-volume derivatives. Likewise, adjoint-based partial derivatives with respect to surface locations (i.e., internal interface locations and the outer system boundary) are constant-density derivatives. This study derives the constant-mass partial derivative of a response with respect to an internal interface location or the outer system boundary and the constant-mass partial derivative of a response with respect to the mass density of a region. Numerical results are given for a multiregion two-dimensional (r-z) cylinder for three very different responses: the uncollided gamma-ray flux at an external detector point, k effmore » of the system, and the total neutron leakage. Finally, results from the derived formulas compare extremely well with direct perturbation calculations.« less

Variational Methods in Design Optimization and Sensitivity Analysis for Two-Dimensional Euler Equations

NASA Technical Reports Server (NTRS)

Ibrahim, A. H.; Tiwari, S. N.; Smith, R. E.

1997-01-01

Variational methods (VM) sensitivity analysis employed to derive the costate (adjoint) equations, the transversality conditions, and the functional sensitivity derivatives. In the derivation of the sensitivity equations, the variational methods use the generalized calculus of variations, in which the variable boundary is considered as the design function. The converged solution of the state equations together with the converged solution of the costate equations are integrated along the domain boundary to uniquely determine the functional sensitivity derivatives with respect to the design function. The application of the variational methods to aerodynamic shape optimization problems is demonstrated for internal flow problems at supersonic Mach number range. The study shows, that while maintaining the accuracy of the functional sensitivity derivatives within the reasonable range for engineering prediction purposes, the variational methods show a substantial gain in computational efficiency, i.e., computer time and memory, when compared with the finite difference sensitivity analysis.

The solution of non-linear hyperbolic equation systems by the finite element method

NASA Technical Reports Server (NTRS)

Loehner, R.; Morgan, K.; Zienkiewicz, O. C.

1984-01-01

A finite-element method for the solution of nonlinear hyperbolic systems of equations, such as those encountered in non-self-adjoint problems of transient phenomena in convection-diffusion or in the mixed representation of wave problems, is developed and demonstrated. The problem is rewritten in moving coordinates and reinterpolated to the original mesh by a Taylor expansion prior to a standard Galerkin spatial discretization, and it is shown that this procedure is equivalent to the time-discretization approach of Donea (1984). Numerical results for sample problems are presented graphically, including such shallow-water problems as the breaking of a dam, the shoaling of a wave, and the outflow of a river; compressible flows such as the isothermal flow in a nozzle and the Riemann shock-tube problem; and the two-dimensional scalar-advection, nonlinear-shallow-water, and Euler equations.

Efficient Inversion of Mult-frequency and Multi-Source Electromagnetic Data

DOE Office of Scientific and Technical Information (OSTI.GOV)

Gary D. Egbert

2007-03-22

The project covered by this report focused on development of efficient but robust non-linear inversion algorithms for electromagnetic induction data, in particular for data collected with multiple receivers, and multiple transmitters, a situation extremely common in eophysical EM subsurface imaging methods. A key observation is that for such multi-transmitter problems each step in commonly used linearized iterative limited memory search schemes such as conjugate gradients (CG) requires solution of forward and adjoint EM problems for each of the N frequencies or sources, essentially generating data sensitivities for an N dimensional data-subspace. These multiple sensitivities allow a good approximation to themore » full Jacobian of the data mapping to be built up in many fewer search steps than would be required by application of textbook optimization methods, which take no account of the multiplicity of forward problems that must be solved for each search step. We have applied this idea to a develop a hybrid inversion scheme that combines features of the iterative limited memory type methods with a Newton-type approach using a partial calculation of the Jacobian. Initial tests on 2D problems show that the new approach produces results essentially identical to a Newton type Occam minimum structure inversion, while running more rapidly than an iterative (fixed regularization parameter) CG style inversion. Memory requirements, while greater than for something like CG, are modest enough that even in 3D the scheme should allow 3D inverse problems to be solved on a common desktop PC, at least for modest (~ 100 sites, 15-20 frequencies) data sets. A secondary focus of the research has been development of a modular system for EM inversion, using an object oriented approach. This system has proven useful for more rapid prototyping of inversion algorithms, in particular allowing initial development and testing to be conducted with two-dimensional example problems, before approaching more computationally cumbersome three-dimensional problems.« less

Demonstration of Automatically-Generated Adjoint Code for Use in Aerodynamic Shape Optimization

NASA Technical Reports Server (NTRS)

Green, Lawrence; Carle, Alan; Fagan, Mike

1999-01-01

Gradient-based optimization requires accurate derivatives of the objective function and constraints. These gradients may have previously been obtained by manual differentiation of analysis codes, symbolic manipulators, finite-difference approximations, or existing automatic differentiation (AD) tools such as ADIFOR (Automatic Differentiation in FORTRAN). Each of these methods has certain deficiencies, particularly when applied to complex, coupled analyses with many design variables. Recently, a new AD tool called ADJIFOR (Automatic Adjoint Generation in FORTRAN), based upon ADIFOR, was developed and demonstrated. Whereas ADIFOR implements forward-mode (direct) differentiation throughout an analysis program to obtain exact derivatives via the chain rule of calculus, ADJIFOR implements the reverse-mode counterpart of the chain rule to obtain exact adjoint form derivatives from FORTRAN code. Automatically-generated adjoint versions of the widely-used CFL3D computational fluid dynamics (CFD) code and an algebraic wing grid generation code were obtained with just a few hours processing time using the ADJIFOR tool. The codes were verified for accuracy and were shown to compute the exact gradient of the wing lift-to-drag ratio, with respect to any number of shape parameters, in about the time required for 7 to 20 function evaluations. The codes have now been executed on various computers with typical memory and disk space for problems with up to 129 x 65 x 33 grid points, and for hundreds to thousands of independent variables. These adjoint codes are now used in a gradient-based aerodynamic shape optimization problem for a swept, tapered wing. For each design iteration, the optimization package constructs an approximate, linear optimization problem, based upon the current objective function, constraints, and gradient values. The optimizer subroutines are called within a design loop employing the approximate linear problem until an optimum shape is found, the design loop limit is reached, or no further design improvement is possible due to active design variable bounds and/or constraints. The resulting shape parameters are then used by the grid generation code to define a new wing surface and computational grid. The lift-to-drag ratio and its gradient are computed for the new design by the automatically-generated adjoint codes. Several optimization iterations may be required to find an optimum wing shape. Results from two sample cases will be discussed. The reader should note that this work primarily represents a demonstration of use of automatically- generated adjoint code within an aerodynamic shape optimization. As such, little significance is placed upon the actual optimization results, relative to the method for obtaining the results.

Kidney-inspired algorithm for optimization problems

NASA Astrophysics Data System (ADS)

Jaddi, Najmeh Sadat; Alvankarian, Jafar; Abdullah, Salwani

2017-01-01

In this paper, a population-based algorithm inspired by the kidney process in the human body is proposed. In this algorithm the solutions are filtered in a rate that is calculated based on the mean of objective functions of all solutions in the current population of each iteration. The filtered solutions as the better solutions are moved to filtered blood and the rest are transferred to waste representing the worse solutions. This is a simulation of the glomerular filtration process in the kidney. The waste solutions are reconsidered in the iterations if after applying a defined movement operator they satisfy the filtration rate, otherwise it is expelled from the waste solutions, simulating the reabsorption and excretion functions of the kidney. In addition, a solution assigned as better solution is secreted if it is not better than the worst solutions simulating the secreting process of blood in the kidney. After placement of all the solutions in the population, the best of them is ranked, the waste and filtered blood are merged to become a new population and the filtration rate is updated. Filtration provides the required exploitation while generating a new solution and reabsorption gives the necessary exploration for the algorithm. The algorithm is assessed by applying it on eight well-known benchmark test functions and compares the results with other algorithms in the literature. The performance of the proposed algorithm is better on seven out of eight test functions when it is compared with the most recent researches in literature. The proposed kidney-inspired algorithm is able to find the global optimum with less function evaluations on six out of eight test functions. A statistical analysis further confirms the ability of this algorithm to produce good-quality results.

Investigating Sensitivity to Saharan Dust in Tropical Cyclone Formation Using Nasa's Adjoint Model

NASA Technical Reports Server (NTRS)

Holdaway, Daniel

2015-01-01

As tropical cyclones develop from easterly waves coming of the coast of Africa they interact with dust from the Sahara desert. There is a long standing debate over whether this dust inhibits or advances the developing storm and how much influence it has. Dust can surround the storm and absorb incoming solar radiation, cooling the air below. As a result an energy source for the system is potentially diminished, inhibiting growth of the storm. Alternatively dust may interact with clouds through micro-physical processes, for example by causing more moisture to condense, potentially increasing the strength. As a result of climate change, concentrations and amount of dust in the atmosphere will likely change. It it is important to properly understand its effect on tropical storm formation. The adjoint of an atmospheric general circulation model provides a very powerful tool for investigating sensitivity to initial conditions. The National Aeronautics and Space Administration (NASA) has recently developed an adjoint version of the Goddard Earth Observing System version 5 (GEOS-5) dynamical core, convection scheme, cloud model and radiation schemes. This is extended so that the interaction between dust and radiation is also accounted for in the adjoint model. This provides a framework for examining the sensitivity to dust in the initial conditions. Specifically the set up allows for an investigation into the extent to which dust affects cyclone strength through absorption of radiation. In this work we investigate the validity of using an adjoint model for examining sensitivity to dust in hurricane formation. We present sensitivity results for a number of systems that developed during the Atlantic hurricane season of 2006. During this period there was a significant outbreak of Saharan dust and it is has been argued that this outbreak was responsible for the relatively calm season. This period was also covered by an extensive observation campaign. It is shown that the adjoint can provide insight into the sensitivity and reveals a relatively low sensitivity to dust compared to, for example, the thermodynamic variables. However a secondary sensitivity though moisture is seen. If dust dries the air it can significantly reduce the cyclone intensity through the moisture.

Investigating sensitivity to Saharan dust in tropical cyclone formation using NASA's adjoint model

NASA Astrophysics Data System (ADS)

Holdaway, Daniel

2015-04-01

As tropical cyclones develop from easterly waves coming off the coast of Africa they interact with dust from the Sahara desert. There is a long standing debate over whether this dust inhibits or advances the developing storm and how much influence it has. Dust can surround the storm and absorb incoming solar radiation, cooling the air below. As a result an energy source for the system is potentially diminished, inhibiting growth of the storm. Alternatively dust may interact with clouds through micro-physical processes, for example by causing more moisture to condense, potentially increasing the strength. As a result of climate change, concentrations and amount of dust in the atmosphere will likely change. It it is important to properly understand its effect on tropical storm formation. The adjoint of an atmospheric general circulation model provides a very powerful tool for investigating sensitivity to initial conditions. The National Aeronautics and Space Administration (NASA) has recently developed an adjoint version of the Goddard Earth Observing System version 5 (GEOS-5) dynamical core, convection scheme, cloud model and radiation schemes. This is extended so that the interaction between dust and radiation is also accounted for in the adjoint model. This provides a framework for examining the sensitivity to dust in the initial conditions. Specifically the set up allows for an investigation into the extent to which dust affects cyclone strength through absorption of radiation. In this work we investigate the validity of using an adjoint model for examining sensitivity to dust in hurricane formation. We present sensitivity results for a number of systems that developed during the Atlantic hurricane season of 2006. During this period there was a significant outbreak of Saharan dust and it is has been argued that this outbreak was responsible for the relatively calm season. This period was also covered by an extensive observation campaign. It is shown that the adjoint can provide insight into the sensitivity and reveals a relatively low sensitivity to dust compared to, for example, the thermodynamic variables. However a secondary sensitivity though moisture is seen. If dust dries the air it can significantly reduce the cyclone intensity through the moisture.

Efficient Computation of Info-Gap Robustness for Finite Element Models

DOE Office of Scientific and Technical Information (OSTI.GOV)

Stull, Christopher J.; Hemez, Francois M.; Williams, Brian J.

2012-07-05

A recent research effort at LANL proposed info-gap decision theory as a framework by which to measure the predictive maturity of numerical models. Info-gap theory explores the trade-offs between accuracy, that is, the extent to which predictions reproduce the physical measurements, and robustness, that is, the extent to which predictions are insensitive to modeling assumptions. Both accuracy and robustness are necessary to demonstrate predictive maturity. However, conducting an info-gap analysis can present a formidable challenge, from the standpoint of the required computational resources. This is because a robustness function requires the resolution of multiple optimization problems. This report offers anmore » alternative, adjoint methodology to assess the info-gap robustness of Ax = b-like numerical models solved for a solution x. Two situations that can arise in structural analysis and design are briefly described and contextualized within the info-gap decision theory framework. The treatments of the info-gap problems, using the adjoint methodology are outlined in detail, and the latter problem is solved for four separate finite element models. As compared to statistical sampling, the proposed methodology offers highly accurate approximations of info-gap robustness functions for the finite element models considered in the report, at a small fraction of the computational cost. It is noted that this report considers only linear systems; a natural follow-on study would extend the methodologies described herein to include nonlinear systems.« less

Seismic Window Selection and Misfit Measurements for Global Adjoint Tomography

NASA Astrophysics Data System (ADS)

Lei, W.; Bozdag, E.; Lefebvre, M.; Podhorszki, N.; Smith, J. A.; Tromp, J.

2013-12-01

Global Adjoint Tomography requires fast parallel processing of large datasets. After obtaing the preprocessed observed and synthetic seismograms, we use the open source software packages FLEXWIN (Maggi et al. 2007) to select time windows and MEASURE_ADJ to make measurements. These measurements define adjoint sources for data assimilation. Previous versions of these tools work on a pair of SAC files---observed and synthetic seismic data for the same component and station, and loop over all seismic records associated with one earthquake. Given the large number of stations and earthquakes, the frequent read and write operations create severe I/O bottlenecks on modern computing platforms. We present new versions of these tools utilizing a new seismic data format, namely the Adaptive Seismic Data Format(ASDF). This new format shows superior scalability for applications on high-performance computers and accommodates various types of data, including earthquake, industry and seismic interferometry datasets. ASDF also provides user-friendly APIs, which can be easily integrated into the adjoint tomography workflow and combined with other data processing tools. In addition to solving the I/O bottleneck, we are making several improvements to these tools. For example, FLEXWIN is tuned to select windows for different types of earthquakes. To capture their distinct features, we categorize earthquakes by their depths and frequency bands. Moreover, instead of only picking phases between the first P arrival and the surface-wave arrivals, our aim is to select and assimilate many other later prominent phases in adjoint tomography. For example, in the body-wave band (17 s - 60 s), we include SKS, sSKS and their multiple, while in the surface-wave band (60 s - 120 s) we incorporate major-arc surface waves.

An approach to computing discrete adjoints for MPI-parallelized models applied to Ice Sheet System Model 4.11

NASA Astrophysics Data System (ADS)

Larour, Eric; Utke, Jean; Bovin, Anton; Morlighem, Mathieu; Perez, Gilberto

2016-11-01

Within the framework of sea-level rise projections, there is a strong need for hindcast validation of the evolution of polar ice sheets in a way that tightly matches observational records (from radar, gravity, and altimetry observations mainly). However, the computational requirements for making hindcast reconstructions possible are severe and rely mainly on the evaluation of the adjoint state of transient ice-flow models. Here, we look at the computation of adjoints in the context of the NASA/JPL/UCI Ice Sheet System Model (ISSM), written in C++ and designed for parallel execution with MPI. We present the adaptations required in the way the software is designed and written, but also generic adaptations in the tools facilitating the adjoint computations. We concentrate on the use of operator overloading coupled with the AdjoinableMPI library to achieve the adjoint computation of the ISSM. We present a comprehensive approach to (1) carry out type changing through the ISSM, hence facilitating operator overloading, (2) bind to external solvers such as MUMPS and GSL-LU, and (3) handle MPI-based parallelism to scale the capability. We demonstrate the success of the approach by computing sensitivities of hindcast metrics such as the misfit to observed records of surface altimetry on the northeastern Greenland Ice Stream, or the misfit to observed records of surface velocities on Upernavik Glacier, central West Greenland. We also provide metrics for the scalability of the approach, and the expected performance. This approach has the potential to enable a new generation of hindcast-validated projections that make full use of the wealth of datasets currently being collected, or already collected, in Greenland and Antarctica.

An Approach to Computing Discrete Adjoints for MPI-Parallelized Models Applied to the Ice Sheet System Model}

NASA Astrophysics Data System (ADS)

Perez, G. L.; Larour, E. Y.; Morlighem, M.

2016-12-01

Within the framework of sea-level rise projections, there is a strong need for hindcast validation of the evolution of polar ice sheets in a way that tightly matches observational records (from radar and altimetry observations mainly). However, the computational requirements for making hindcast reconstructions possible are severe and rely mainly on the evaluation of the adjoint state of transient ice-flow models. Here, we look at the computation of adjoints in the context of the NASA/JPL/UCI Ice Sheet System Model, written in C++ and designed for parallel execution with MPI. We present the adaptations required in the way the software is designed and written but also generic adaptations in the tools facilitating the adjoint computations. We concentrate on the use of operator overloading coupled with the AdjoinableMPI library to achieve the adjoint computation of ISSM. We present a comprehensive approach to 1) carry out type changing through ISSM, hence facilitating operator overloading, 2) bind to external solvers such as MUMPS and GSL-LU and 3) handle MPI-based parallelism to scale the capability. We demonstrate the success of the approach by computing sensitivities of hindcast metrics such as the misfit to observed records of surface altimetry on the North-East Greenland Ice Stream, or the misfit to observed records of surface velocities on Upernavik Glacier, Central West Greenland. We also provide metrics for the scalability of the approach, and the expected performance. This approach has the potential of enabling a new generation of hindcast-validated projections that make full use of the wealth of datasets currently being collected, or alreay collected in Greenland and Antarctica, such as surface altimetry, surface velocities, and/or gravity measurements.

Adjoint-Based Design of Rotors Using the Navier-Stokes Equations in a Noninertial Reference Frame

NASA Technical Reports Server (NTRS)

Nielsen, Eric J.; Lee-Rausch, Elizabeth M.; Jones, William T.

2010-01-01

Optimization of rotorcraft flowfields using an adjoint method generally requires a time-dependent implementation of the equations. The current study examines an intermediate approach in which a subset of rotor flowfields are cast as steady problems in a noninertial reference frame. This technique permits the use of an existing steady-state adjoint formulation with minor modifications to perform sensitivity analyses. The formulation is valid for isolated rigid rotors in hover or where the freestream velocity is aligned with the axis of rotation. Discrete consistency of the implementation is demonstrated by using comparisons with a complex-variable technique, and a number of single- and multipoint optimizations for the rotorcraft figure of merit function are shown for varying blade collective angles. Design trends are shown to remain consistent as the grid is refined.

Adjoint-Based Design of Rotors using the Navier-Stokes Equations in a Noninertial Reference Frame

NASA Technical Reports Server (NTRS)

Nielsen, Eric J.; Lee-Rausch, Elizabeth M.; Jones, William T.

2009-01-01

Optimization of rotorcraft flowfields using an adjoint method generally requires a time-dependent implementation of the equations. The current study examines an intermediate approach in which a subset of rotor flowfields are cast as steady problems in a noninertial reference frame. This technique permits the use of an existing steady-state adjoint formulation with minor modifications to perform sensitivity analyses. The formulation is valid for isolated rigid rotors in hover or where the freestream velocity is aligned with the axis of rotation. Discrete consistency of the implementation is demonstrated using comparisons with a complex-variable technique, and a number of single- and multi-point optimizations for the rotorcraft figure of merit function are shown for varying blade collective angles. Design trends are shown to remain consistent as the grid is refined.

Determination of the self-adjoint matrix Schrödinger operators without the bound state data

NASA Astrophysics Data System (ADS)

Xu, Xiao-Chuan; Yang, Chuan-Fu

2018-06-01

(i) For the matrix Schrödinger operator on the half line, it is shown that the scattering data, which consists of the scattering matrix and the bound state data, uniquely determines the potential and the boundary condition. It is also shown that only the scattering matrix uniquely determines the self-adjoint potential and the boundary condition if either the potential exponentially decreases fast enough or the potential is known a priori on (), where a is an any fixed positive number. (ii) For the matrix Schrödinger operator on the full line, it is shown that the left (or right) reflection coefficient uniquely determine the self-adjoint potential if either the potential exponentially decreases fast enough or the potential is known a priori on (or ()), where b is an any fixed number.

Time-periodic solutions of the Benjamin-Ono equation

DOE Office of Scientific and Technical Information (OSTI.GOV)

Ambrose , D.M.; Wilkening, Jon

2008-04-01

We present a spectrally accurate numerical method for finding non-trivial time-periodic solutions of non-linear partial differential equations. The method is based on minimizing a functional (of the initial condition and the period) that is positive unless the solution is periodic, in which case it is zero. We solve an adjoint PDE to compute the gradient of this functional with respect to the initial condition. We include additional terms in the functional to specify the free parameters, which, in the case of the Benjamin-Ono equation, are the mean, a spatial phase, a temporal phase and the real part of one ofmore » the Fourier modes at t = 0. We use our method to study global paths of non-trivial time-periodic solutions connecting stationary and traveling waves of the Benjamin-Ono equation. As a starting guess for each path, we compute periodic solutions of the linearized problem by solving an infinite dimensional eigenvalue problem in closed form. We then use our numerical method to continue these solutions beyond the realm of linear theory until another traveling wave is reached (or until the solution blows up). By experimentation with data fitting, we identify the analytical form of the solutions on the path connecting the one-hump stationary solution to the two-hump traveling wave. We then derive exact formulas for these solutions by explicitly solving the system of ODE's governing the evolution of solitons using the ansatz suggested by the numerical simulations.« less

Crypto-Unitary Forms of Quantum Evolution Operators

NASA Astrophysics Data System (ADS)

Znojil, Miloslav

2013-06-01

The description of quantum evolution using unitary operator {u}(t)=exp(-i{h}t) requires that the underlying self-adjoint quantum Hamiltonian {h} remains time-independent. In a way extending the so called {PT}-symmetric quantum mechanics to the models with manifestly time-dependent "charge" {C}(t) we propose and describe an extension of such an exponential-operator approach to evolution to the manifestly time-dependent self-adjoint quantum Hamiltonians {h}(t).

Discrete Adjoint Sensitivity Analysis of Hybrid Dynamical Systems With Switching [Discrete Adjoint Sensitivity Analysis of Hybrid Dynamical Systems

DOE PAGES

Zhang, Hong; Abhyankar, Shrirang; Constantinescu, Emil; ...

2017-01-24

Sensitivity analysis is an important tool for describing power system dynamic behavior in response to parameter variations. It is a central component in preventive and corrective control applications. The existing approaches for sensitivity calculations, namely, finite-difference and forward sensitivity analysis, require a computational effort that increases linearly with the number of sensitivity parameters. In this paper, we investigate, implement, and test a discrete adjoint sensitivity approach whose computational effort is effectively independent of the number of sensitivity parameters. The proposed approach is highly efficient for calculating sensitivities of larger systems and is consistent, within machine precision, with the function whosemore » sensitivity we are seeking. This is an essential feature for use in optimization applications. Moreover, our approach includes a consistent treatment of systems with switching, such as dc exciters, by deriving and implementing the adjoint jump conditions that arise from state-dependent and time-dependent switchings. The accuracy and the computational efficiency of the proposed approach are demonstrated in comparison with the forward sensitivity analysis approach. In conclusion, this paper focuses primarily on the power system dynamics, but the approach is general and can be applied to hybrid dynamical systems in a broader range of fields.« less

Discrete Adjoint Sensitivity Analysis of Hybrid Dynamical Systems With Switching [Discrete Adjoint Sensitivity Analysis of Hybrid Dynamical Systems

DOE Office of Scientific and Technical Information (OSTI.GOV)

Zhang, Hong; Abhyankar, Shrirang; Constantinescu, Emil

Sensitivity analysis is an important tool for describing power system dynamic behavior in response to parameter variations. It is a central component in preventive and corrective control applications. The existing approaches for sensitivity calculations, namely, finite-difference and forward sensitivity analysis, require a computational effort that increases linearly with the number of sensitivity parameters. In this paper, we investigate, implement, and test a discrete adjoint sensitivity approach whose computational effort is effectively independent of the number of sensitivity parameters. The proposed approach is highly efficient for calculating sensitivities of larger systems and is consistent, within machine precision, with the function whosemore » sensitivity we are seeking. This is an essential feature for use in optimization applications. Moreover, our approach includes a consistent treatment of systems with switching, such as dc exciters, by deriving and implementing the adjoint jump conditions that arise from state-dependent and time-dependent switchings. The accuracy and the computational efficiency of the proposed approach are demonstrated in comparison with the forward sensitivity analysis approach. In conclusion, this paper focuses primarily on the power system dynamics, but the approach is general and can be applied to hybrid dynamical systems in a broader range of fields.« less

On the spin- 1/2 Aharonov–Bohm problem in conical space: Bound states, scattering and helicity nonconservation

DOE Office of Scientific and Technical Information (OSTI.GOV)

Andrade, F.M., E-mail: fmandrade@uepg.br; Silva, E.O., E-mail: edilbertoo@gmail.com; Pereira, M., E-mail: marciano@uepg.br

2013-12-15

In this work the bound state and scattering problems for a spin- 1/2 particle undergone to an Aharonov–Bohm potential in a conical space in the nonrelativistic limit are considered. The presence of a δ-function singularity, which comes from the Zeeman spin interaction with the magnetic flux tube, is addressed by the self-adjoint extension method. One of the advantages of the present approach is the determination of the self-adjoint extension parameter in terms of physics of the problem. Expressions for the energy bound states, phase-shift and S matrix are determined in terms of the self-adjoint extension parameter, which is explicitly determinedmore » in terms of the parameters of the problem. The relation between the bound state and zero modes and the failure of helicity conservation in the scattering problem and its relation with the gyromagnetic ratio g are discussed. Also, as an application, we consider the spin- 1/2 Aharonov–Bohm problem in conical space plus a two-dimensional isotropic harmonic oscillator. -- Highlights: •Planar dynamics of a spin- 1/2 neutral particle. •Bound state for Aharonov–Bohm systems. •Aharonov–Bohm scattering. •Helicity nonconservation. •Determination of the self-adjoint extension parameter.« less

Computational mechanics analysis tools for parallel-vector supercomputers

NASA Technical Reports Server (NTRS)

Storaasli, O. O.; Nguyen, D. T.; Baddourah, M. A.; Qin, J.

1993-01-01

Computational algorithms for structural analysis on parallel-vector supercomputers are reviewed. These parallel algorithms, developed by the authors, are for the assembly of structural equations, 'out-of-core' strategies for linear equation solution, massively distributed-memory equation solution, unsymmetric equation solution, general eigen-solution, geometrically nonlinear finite element analysis, design sensitivity analysis for structural dynamics, optimization algorithm and domain decomposition. The source code for many of these algorithms is available from NASA Langley.

Orbit Determination and Maneuver Detection Using Event Representation with Thrust-Fourier-Coefficients

NASA Astrophysics Data System (ADS)

Lubey, D.; Ko, H.; Scheeres, D.

The classical orbit determination (OD) method of dealing with unknown maneuvers is to restart the OD process with post-maneuver observations. However, it is also possible to continue the OD process through such unknown maneuvers by representing those unknown maneuvers with an appropriate event representation. It has been shown in previous work (Ko & Scheeres, JGCD 2014) that any maneuver performed by a satellite transitioning between two arbitrary orbital states can be represented as an equivalent maneuver connecting those two states using Thrust-Fourier-Coefficients (TFCs). Event representation using TFCs rigorously provides a unique control law that can generate the desired secular behavior for a given unknown maneuver. This paper presents applications of this representation approach to orbit prediction and maneuver detection problem across unknown maneuvers. The TFCs are appended to a sequential filter as an adjoint state to compensate unknown perturbing accelerations and the modified filter estimates the satellite state and thrust coefficients by processing OD across the time of an unknown maneuver. This modified sequential filter with TFCs is capable of fitting tracking data and maintaining an OD solution in the presence of unknown maneuvers. Also, the modified filter is found effective in detecting a sudden change in TFC values which indicates a maneuver. In order to illustrate that the event representation approach with TFCs is robust and sufficiently general to be easily adjustable, different types of measurement data are processed with the filter in a realistic LEO setting. Further, cases with mis-modeling of non-gravitational force are included in our study to verify the versatility and efficiency of our presented algorithm. Simulation results show that the modified sequential filter with TFCs can detect and estimate the orbit and thrust parameters in the presence of unknown maneuvers with or without measurement data during maneuvers. With no measurement data during maneuvers, the modified filter with TFCs uses an existing pre-maneuver orbit solution to compute a post-maneuver orbit solution by forcing TFCs to compensate for an unknown maneuver. With observation data available during maneuvers, maneuver start time and stop time is determined

A posteriori error estimation for multi-stage Runge–Kutta IMEX schemes

DOE Office of Scientific and Technical Information (OSTI.GOV)

Chaudhry, Jehanzeb H.; Collins, J. B.; Shadid, John N.

Implicit–Explicit (IMEX) schemes are widely used for time integration methods for approximating solutions to a large class of problems. In this work, we develop accurate a posteriori error estimates of a quantity-of-interest for approximations obtained from multi-stage IMEX schemes. This is done by first defining a finite element method that is nodally equivalent to an IMEX scheme, then using typical methods for adjoint-based error estimation. Furthermore, the use of a nodally equivalent finite element method allows a decomposition of the error into multiple components, each describing the effect of a different portion of the method on the total error inmore » a quantity-of-interest.« less

Enabling Predictive Simulation and UQ of Complex Multiphysics PDE Systems by the Development of Goal-Oriented Variational Sensitivity Analysis and a-Posteriori Error Estimation Methods

DOE Office of Scientific and Technical Information (OSTI.GOV)

Estep, Donald

2015-11-30

This project addressed the challenge of predictive computational analysis of strongly coupled, highly nonlinear multiphysics systems characterized by multiple physical phenomena that span a large range of length- and time-scales. Specifically, the project was focused on computational estimation of numerical error and sensitivity analysis of computational solutions with respect to variations in parameters and data. In addition, the project investigated the use of accurate computational estimates to guide efficient adaptive discretization. The project developed, analyzed and evaluated new variational adjoint-based techniques for integration, model, and data error estimation/control and sensitivity analysis, in evolutionary multiphysics multiscale simulations.

A theoretical approach for analyzing the restabilization of wakes

NASA Astrophysics Data System (ADS)

Hill, D. C.

1992-04-01

Recently reported experimental results demonstrate that restabilization of the low-Reynolds-number flow past a circular cylinder can be achieved by the placement of a smaller cylinder in the wake of the first at particular locations. Traditional numerical procedures for modeling such phenomena are computationally expensive. An approach is presented here in which the properties of the adjoint solutions to the linearized equations of motion are exploited to map quickly the best positions for the small cylinder's placement. Comparisons with experiment and previous computations are favorable. The approach is shown to be applicable to general flows, illustrating how strongly control mechanisms that involve sources of momentum couple to unstable (or stable) modes of the system.

A posteriori error estimation for multi-stage Runge–Kutta IMEX schemes

DOE PAGES

Chaudhry, Jehanzeb H.; Collins, J. B.; Shadid, John N.

2017-02-05

Implicit–Explicit (IMEX) schemes are widely used for time integration methods for approximating solutions to a large class of problems. In this work, we develop accurate a posteriori error estimates of a quantity-of-interest for approximations obtained from multi-stage IMEX schemes. This is done by first defining a finite element method that is nodally equivalent to an IMEX scheme, then using typical methods for adjoint-based error estimation. Furthermore, the use of a nodally equivalent finite element method allows a decomposition of the error into multiple components, each describing the effect of a different portion of the method on the total error inmore » a quantity-of-interest.« less

Control of the transition between regular and mach reflection of shock waves

NASA Astrophysics Data System (ADS)

Alekseev, A. K.

2012-06-01

A control problem was considered that makes it possible to switch the flow between stationary Mach and regular reflection of shock waves within the dual solution domain. The sensitivity of the flow was computed by solving adjoint equations. A control disturbance was sought by applying gradient optimization methods. According to the computational results, the transition from regular to Mach reflection can be executed by raising the temperature. The transition from Mach to regular reflection can be achieved by lowering the temperature at moderate Mach numbers and is impossible at large numbers. The reliability of the numerical results was confirmed by verifying them with the help of a posteriori analysis.

Comparison of four stable numerical methods for Abel's integral equation

NASA Technical Reports Server (NTRS)

Murio, Diego A.; Mejia, Carlos E.

1991-01-01

The 3-D image reconstruction from cone-beam projections in computerized tomography leads naturally, in the case of radial symmetry, to the study of Abel-type integral equations. If the experimental information is obtained from measured data, on a discrete set of points, special methods are needed in order to restore continuity with respect to the data. A new combined Regularized-Adjoint-Conjugate Gradient algorithm, together with two different implementations of the Mollification Method (one based on a data filtering technique and the other on the mollification of the kernal function) and a regularization by truncation method (initially proposed for 2-D ray sample schemes and more recently extended to 3-D cone-beam image reconstruction) are extensively tested and compared for accuracy and numerical stability as functions of the level of noise in the data.

Topology optimization of finite strain viscoplastic systems under transient loads

DOE PAGES

Ivarsson, Niklas; Wallin, Mathias; Tortorelli, Daniel

2018-02-08

In this paper, a transient finite strain viscoplastic model is implemented in a gradient-based topology optimization framework to design impact mitigating structures. The model's kinematics relies on the multiplicative split of the deformation gradient, and the constitutive response is based on isotropic hardening viscoplasticity. To solve the mechanical balance laws, the implicit Newmark-beta method is used together with a total Lagrangian finite element formulation. The optimization problem is regularized using a partial differential equation filter and solved using the method of moving asymptotes. Sensitivities required to solve the optimization problem are derived using the adjoint method. To demonstrate the capabilitymore » of the algorithm, several protective systems are designed, in which the absorbed viscoplastic energy is maximized. Finally, the numerical examples demonstrate that transient finite strain viscoplastic effects can successfully be combined with topology optimization.« less

One-loop corrections from higher dimensional tree amplitudes

DOE Office of Scientific and Technical Information (OSTI.GOV)

Cachazo, Freddy; He, Song; Yuan, Ellis Ye

We show how one-loop corrections to scattering amplitudes of scalars and gauge bosons can be obtained from tree amplitudes in one higher dimension. Starting with a complete tree-level scattering amplitude of n + 2 particles in five dimensions, one assumes that two of them cannot be “detected” and therefore an integration over their LIPS is carried out. The resulting object, function of the remaining n particles, is taken to be four-dimensional by restricting the corresponding momenta. We perform this procedure in the context of the tree-level CHY formulation of amplitudes. The scattering equations obtained in the procedure coincide with thosemore » derived by Geyer et al. from ambitwistor constructions and recently studied by two of the authors for bi-adjoint scalars. They have two sectors of solutions: regular and singular. We prove that the contribution from regular solutions generically gives rise to unphysical poles. However, using a BCFW argument we prove that the unphysical contributions are always homogeneous functions of the loop momentum and can be discarded. We also show that the contribution from singular solutions turns out to be homogeneous as well.« less

One-loop corrections from higher dimensional tree amplitudes

DOE PAGES

Cachazo, Freddy; He, Song; Yuan, Ellis Ye

2016-08-01

We show how one-loop corrections to scattering amplitudes of scalars and gauge bosons can be obtained from tree amplitudes in one higher dimension. Starting with a complete tree-level scattering amplitude of n + 2 particles in five dimensions, one assumes that two of them cannot be “detected” and therefore an integration over their LIPS is carried out. The resulting object, function of the remaining n particles, is taken to be four-dimensional by restricting the corresponding momenta. We perform this procedure in the context of the tree-level CHY formulation of amplitudes. The scattering equations obtained in the procedure coincide with thosemore » derived by Geyer et al. from ambitwistor constructions and recently studied by two of the authors for bi-adjoint scalars. They have two sectors of solutions: regular and singular. We prove that the contribution from regular solutions generically gives rise to unphysical poles. However, using a BCFW argument we prove that the unphysical contributions are always homogeneous functions of the loop momentum and can be discarded. We also show that the contribution from singular solutions turns out to be homogeneous as well.« less

Multiple shooting algorithms for jump-discontinuous problems in optimal control and estimation

NASA Technical Reports Server (NTRS)

Mook, D. J.; Lew, Jiann-Shiun

1991-01-01

Multiple shooting algorithms are developed for jump-discontinuous two-point boundary value problems arising in optimal control and optimal estimation. Examples illustrating the origin of such problems are given to motivate the development of the solution algorithms. The algorithms convert the necessary conditions, consisting of differential equations and transversality conditions, into algebraic equations. The solution of the algebraic equations provides exact solutions for linear problems. The existence and uniqueness of the solution are proved.

One-dimensional swarm algorithm packaging

NASA Astrophysics Data System (ADS)

Lebedev, Boris K.; Lebedev, Oleg B.; Lebedeva, Ekaterina O.

2018-05-01

The paper considers an algorithm for solving the problem of onedimensional packaging based on the adaptive behavior model of an ant colony. The key role in the development of the ant algorithm is the choice of representation (interpretation) of the solution. The structure of the solution search graph, the procedure for finding solutions on the graph, the methods of deposition and evaporation of pheromone are described. Unlike the canonical paradigm of an ant algorithm, an ant on the solution search graph generates sets of elements distributed across blocks. Experimental studies were conducted on IBM PC. Compared with the existing algorithms, the results are improved.

Naval Research Laboratory Multiscale Targeting Guidance for T-PARC and TCS-08

DTIC Science & Technology

2010-01-01

Naval Research Laboratory Multiscale Targeting Guidance for T- PARC and TCS-08 CAROLYN A. REYNOLDS AND JAMES D. DOYLE Marine Meteorology Division...of The Observing System Research and Predictability Experiment (THORPEX) Pacific Asian Regional Campaign (T- PARC ) and the Office of Naval Research’s...These products were produced with 24-, 36-, and 48-h lead times. The nonhydrostatic adjoint system used during T- PARC /TCS-08 contains an exact adjoint to

Self-adjoint elliptic operators with boundary conditions on not closed hypersurfaces

NASA Astrophysics Data System (ADS)

Mantile, Andrea; Posilicano, Andrea; Sini, Mourad

2016-07-01

The theory of self-adjoint extensions of symmetric operators is used to construct self-adjoint realizations of a second-order elliptic differential operator on Rn with linear boundary conditions on (a relatively open part of) a compact hypersurface. Our approach allows to obtain Kreĭn-like resolvent formulae where the reference operator coincides with the ;free; operator with domain H2 (Rn); this provides an useful tool for the scattering problem from a hypersurface. Concrete examples of this construction are developed in connection with the standard boundary conditions, Dirichlet, Neumann, Robin, δ and δ‧-type, assigned either on a (n - 1) dimensional compact boundary Γ = ∂ Ω or on a relatively open part Σ ⊂ Γ. Schatten-von Neumann estimates for the difference of the powers of resolvents of the free and the perturbed operators are also proven; these give existence and completeness of the wave operators of the associated scattering systems.

Data-based adjoint and H2 optimal control of the Ginzburg-Landau equation

NASA Astrophysics Data System (ADS)

Banks, Michael; Bodony, Daniel

2017-11-01

Equation-free, reduced-order methods of control are desirable when the governing system of interest is of very high dimension or the control is to be applied to a physical experiment. Two-phase flow optimal control problems, our target application, fit these criteria. Dynamic Mode Decomposition (DMD) is a data-driven method for model reduction that can be used to resolve the dynamics of very high dimensional systems and project the dynamics onto a smaller, more manageable basis. We evaluate the effectiveness of DMD-based forward and adjoint operator estimation when applied to H2 optimal control approaches applied to the linear and nonlinear Ginzburg-Landau equation. Perspectives on applying the data-driven adjoint to two phase flow control will be given. Office of Naval Research (ONR) as part of the Multidisciplinary University Research Initiatives (MURI) Program, under Grant Number N00014-16-1-2617.

Efficient Construction of Discrete Adjoint Operators on Unstructured Grids by Using Complex Variables

NASA Technical Reports Server (NTRS)

Nielsen, Eric J.; Kleb, William L.

2005-01-01

A methodology is developed and implemented to mitigate the lengthy software development cycle typically associated with constructing a discrete adjoint solver for aerodynamic simulations. The approach is based on a complex-variable formulation that enables straightforward differentiation of complicated real-valued functions. An automated scripting process is used to create the complex-variable form of the set of discrete equations. An efficient method for assembling the residual and cost function linearizations is developed. The accuracy of the implementation is verified through comparisons with a discrete direct method as well as a previously developed handcoded discrete adjoint approach. Comparisons are also shown for a large-scale configuration to establish the computational efficiency of the present scheme. To ultimately demonstrate the power of the approach, the implementation is extended to high temperature gas flows in chemical nonequilibrium. Finally, several fruitful research and development avenues enabled by the current work are suggested.

Efficient Construction of Discrete Adjoint Operators on Unstructured Grids Using Complex Variables

NASA Technical Reports Server (NTRS)

Nielsen, Eric J.; Kleb, William L.

2005-01-01

A methodology is developed and implemented to mitigate the lengthy software development cycle typically associated with constructing a discrete adjoint solver for aerodynamic simulations. The approach is based on a complex-variable formulation that enables straightforward differentiation of complicated real-valued functions. An automated scripting process is used to create the complex-variable form of the set of discrete equations. An efficient method for assembling the residual and cost function linearizations is developed. The accuracy of the implementation is verified through comparisons with a discrete direct method as well as a previously developed handcoded discrete adjoint approach. Comparisons are also shown for a large-scale configuration to establish the computational efficiency of the present scheme. To ultimately demonstrate the power of the approach, the implementation is extended to high temperature gas flows in chemical nonequilibrium. Finally, several fruitful research and development avenues enabled by the current work are suggested.

Adjoint Fokker-Planck equation and runaway electron dynamics

DOE Office of Scientific and Technical Information (OSTI.GOV)

Liu, Chang; Brennan, Dylan P.; Bhattacharjee, Amitava

2016-01-15

The adjoint Fokker-Planck equation method is applied to study the runaway probability function and the expected slowing-down time for highly relativistic runaway electrons, including the loss of energy due to synchrotron radiation. In direct correspondence to Monte Carlo simulation methods, the runaway probability function has a smooth transition across the runaway separatrix, which can be attributed to effect of the pitch angle scattering term in the kinetic equation. However, for the same numerical accuracy, the adjoint method is more efficient than the Monte Carlo method. The expected slowing-down time gives a novel method to estimate the runaway current decay timemore » in experiments. A new result from this work is that the decay rate of high energy electrons is very slow when E is close to the critical electric field. This effect contributes further to a hysteresis previously found in the runaway electron population.« less

Sensitivity Kernels for the Cross-Convolution Measure: Eliminate the Source in Waveform Tomography

NASA Astrophysics Data System (ADS)

Menke, W. H.

2017-12-01

We use the adjoint method to derive sensitivity kernels for the cross-convolution measure, a goodness-of-fit criterion that is applicable to seismic data containing closely-spaced multiple arrivals, such as reverberating compressional waves and split shear waves. In addition to a general formulation, specific expressions for sensitivity with respect to density, Lamé parameter and shear modulus are derived for a isotropic elastic solid. As is typical of adjoint methods, the kernels depend upon an adjoint field, the source of which, in this case, is the reference displacement field, pre-multiplied by a matrix of cross-correlations of components of the observed field. We use a numerical simulation to evaluate the resolving power of a topographic inversion that employs the cross-convolution measure. The estimated resolving kernel shows is point-like, indicating that the cross-convolution measure will perform well in waveform tomography settings.

A double commutant theorem for Murray–von Neumann algebras

PubMed Central

Liu, Zhe

2012-01-01

Murray–von Neumann algebras are algebras of operators affiliated with finite von Neumann algebras. In this article, we study commutativity and affiliation of self-adjoint operators (possibly unbounded). We show that a maximal abelian self-adjoint subalgebra of the Murray–von Neumann algebra associated with a finite von Neumann algebra is the Murray–von Neumann algebra , where is a maximal abelian self-adjoint subalgebra of and, in addition, is . We also prove that the Murray–von Neumann algebra with the center of is the center of the Murray–von Neumann algebra . Von Neumann’s celebrated double commutant theorem characterizes von Neumann algebras as those for which , where , the commutant of , is the set of bounded operators on the Hilbert space that commute with all operators in . At the end of this article, we present a double commutant theorem for Murray–von Neumann algebras. PMID:22543165

A block-based algorithm for the solution of compressible flows in rotor-stator combinations

NASA Technical Reports Server (NTRS)

Akay, H. U.; Ecer, A.; Beskok, A.

1990-01-01

A block-based solution algorithm is developed for the solution of compressible flows in rotor-stator combinations. The method allows concurrent solution of multiple solution blocks in parallel machines. It also allows a time averaged interaction at the stator-rotor interfaces. Numerical results are presented to illustrate the performance of the algorithm. The effect of the interaction between the stator and rotor is evaluated.

Suboptimal Scheduling in Switched Systems With Continuous-Time Dynamics: A Least Squares Approach.

PubMed

Sardarmehni, Tohid; Heydari, Ali

2018-06-01

Two approximate solutions for optimal control of switched systems with autonomous subsystems and continuous-time dynamics are presented. The first solution formulates a policy iteration (PI) algorithm for the switched systems with recursive least squares. To reduce the computational burden imposed by the PI algorithm, a second solution, called single loop PI, is presented. Online and concurrent training algorithms are discussed for implementing each solution. At last, effectiveness of the presented algorithms is evaluated through numerical simulations.

A new improved artificial bee colony algorithm for ship hull form optimization

NASA Astrophysics Data System (ADS)

Huang, Fuxin; Wang, Lijue; Yang, Chi

2016-04-01

The artificial bee colony (ABC) algorithm is a relatively new swarm intelligence-based optimization algorithm. Its simplicity of implementation, relatively few parameter settings and promising optimization capability make it widely used in different fields. However, it has problems of slow convergence due to its solution search equation. Here, a new solution search equation based on a combination of the elite solution pool and the block perturbation scheme is proposed to improve the performance of the algorithm. In addition, two different solution search equations are used by employed bees and onlooker bees to balance the exploration and exploitation of the algorithm. The developed algorithm is validated by a set of well-known numerical benchmark functions. It is then applied to optimize two ship hull forms with minimum resistance. The tested results show that the proposed new improved ABC algorithm can outperform the ABC algorithm in most of the tested problems.

Gossip-based solutions for discrete rendezvous in populations of communicating agents.

PubMed

Hollander, Christopher D; Wu, Annie S

2014-01-01

The objective of the rendezvous problem is to construct a method that enables a population of agents to agree on a spatial (and possibly temporal) meeting location. We introduce the buffered gossip algorithm as a general solution to the rendezvous problem in a discrete domain with direct communication between decentralized agents. We compare the performance of the buffered gossip algorithm against the well known uniform gossip algorithm. We believe that a buffered solution is preferable to an unbuffered solution, such as the uniform gossip algorithm, because the use of a buffer allows an agent to use multiple information sources when determining its desired rendezvous point, and that access to multiple information sources may improve agent decision making by reinforcing or contradicting an initial choice. To show that the buffered gossip algorithm is an actual solution for the rendezvous problem, we construct a theoretical proof of convergence and derive the conditions under which the buffered gossip algorithm is guaranteed to produce a consensus on rendezvous location. We use these results to verify that the uniform gossip algorithm also solves the rendezvous problem. We then use a multi-agent simulation to conduct a series of simulation experiments to compare the performance between the buffered and uniform gossip algorithms. Our results suggest that the buffered gossip algorithm can solve the rendezvous problem faster than the uniform gossip algorithm; however, the relative performance between these two solutions depends on the specific constraints of the problem and the parameters of the buffered gossip algorithm.

Gossip-Based Solutions for Discrete Rendezvous in Populations of Communicating Agents

PubMed Central

Hollander, Christopher D.; Wu, Annie S.

2014-01-01

The objective of the rendezvous problem is to construct a method that enables a population of agents to agree on a spatial (and possibly temporal) meeting location. We introduce the buffered gossip algorithm as a general solution to the rendezvous problem in a discrete domain with direct communication between decentralized agents. We compare the performance of the buffered gossip algorithm against the well known uniform gossip algorithm. We believe that a buffered solution is preferable to an unbuffered solution, such as the uniform gossip algorithm, because the use of a buffer allows an agent to use multiple information sources when determining its desired rendezvous point, and that access to multiple information sources may improve agent decision making by reinforcing or contradicting an initial choice. To show that the buffered gossip algorithm is an actual solution for the rendezvous problem, we construct a theoretical proof of convergence and derive the conditions under which the buffered gossip algorithm is guaranteed to produce a consensus on rendezvous location. We use these results to verify that the uniform gossip algorithm also solves the rendezvous problem. We then use a multi-agent simulation to conduct a series of simulation experiments to compare the performance between the buffered and uniform gossip algorithms. Our results suggest that the buffered gossip algorithm can solve the rendezvous problem faster than the uniform gossip algorithm; however, the relative performance between these two solutions depends on the specific constraints of the problem and the parameters of the buffered gossip algorithm. PMID:25397882

A hybrid method for the computation of quasi-3D seismograms.

NASA Astrophysics Data System (ADS)

Masson, Yder; Romanowicz, Barbara

2013-04-01

The development of powerful computer clusters and efficient numerical computation methods, such as the Spectral Element Method (SEM) made possible the computation of seismic wave propagation in a heterogeneous 3D earth. However, the cost of theses computations is still problematic for global scale tomography that requires hundreds of such simulations. Part of the ongoing research effort is dedicated to the development of faster modeling methods based on the spectral element method. Capdeville et al. (2002) proposed to couple SEM simulations with normal modes calculation (C-SEM). Nissen-Meyer et al. (2007) used 2D SEM simulations to compute 3D seismograms in a 1D earth model. Thanks to these developments, and for the first time, Lekic et al. (2011) developed a 3D global model of the upper mantle using SEM simulations. At the local and continental scale, adjoint tomography that is using a lot of SEM simulation can be implemented on current computers (Tape, Liu et al. 2009). Due to their smaller size, these models offer higher resolution. They provide us with images of the crust and the upper part of the mantle. In an attempt to teleport such local adjoint tomographic inversions into the deep earth, we are developing a hybrid method where SEM computation are limited to a region of interest within the earth. That region can have an arbitrary shape and size. Outside this region, the seismic wavefield is extrapolated to obtain synthetic data at the Earth's surface. A key feature of the method is the use of a time reversal mirror to inject the wavefield induced by distant seismic source into the region of interest (Robertsson and Chapman 2000). We compute synthetic seismograms as follow: Inside the region of interest, we are using regional spectral element software RegSEM to compute wave propagation in 3D. Outside this region, the wavefield is extrapolated to the surface by convolution with the Green's functions from the mirror to the seismic stations. For now, these Green's functions are computed using 2D SEM simulation in a 1D Earth model. Such seismograms account for the 3D structure inside the region of interest in a quasi-exact manner. Later we plan to extrapolate the misfit function computed from such seismograms at the stations back into the SEM region in order to compute local adjoint kernels. This opens a new path toward regional adjoint tomography into the deep Earth. Capdeville, Y., et al. (2002). "Coupling the spectral element method with a modal solution for elastic wave propagation in global Earth models." Geophysical Journal International 152(1): 34-67. Lekic, V. and B. Romanowicz (2011). "Inferring upper-mantle structure by full waveform tomography with the spectral element method." Geophysical Journal International 185(2): 799-831. Nissen-Meyer, T., et al. (2007). "A two-dimensional spectral-element method for computing spherical-earth seismograms-I. Moment-tensor source." Geophysical Journal International 168(3): 1067-1092. Robertsson, J. O. A. and C. H. Chapman (2000). "An efficient method for calculating finite-difference seismograms after model alterations." Geophysics 65(3): 907-918. Tape, C., et al. (2009). "Adjoint tomography of the southern California crust." Science 325(5943): 988-992.

Adjoint-Based Sensitivity Maps for the Nearshore

NASA Astrophysics Data System (ADS)

Orzech, Mark; Veeramony, Jay; Ngodock, Hans

2013-04-01

The wave model SWAN (Booij et al., 1999) solves the spectral action balance equation to produce nearshore wave forecasts and climatologies. It is widely used by the coastal modeling community and is part of a variety of coupled ocean-wave-atmosphere model systems. A variational data assimilation system (Orzech et al., 2013) has recently been developed for SWAN and is presently being transitioned to operational use by the U.S. Naval Oceanographic Office. This system is built around a numerical adjoint to the fully nonlinear, nonstationary SWAN code. When provided with measured or artificial "observed" spectral wave data at a location of interest on a given nearshore bathymetry, the adjoint can compute the degree to which spectral energy levels at other locations are correlated with - or "sensitive" to - variations in the observed spectrum. Adjoint output may be used to construct a sensitivity map for the entire domain, tracking correlations of spectral energy throughout the grid. When access is denied to the actual locations of interest, sensitivity maps can be used to determine optimal alternate locations for data collection by identifying regions of greatest sensitivity in the mapped domain. The present study investigates the properties of adjoint-generated sensitivity maps for nearshore wave spectra. The adjoint and forward SWAN models are first used in an idealized test case at Duck, NC, USA, to demonstrate the system's effectiveness at optimizing forecasts of shallow water wave spectra for an inaccessible surf-zone location. Then a series of simulations is conducted for a variety of different initializing conditions, to examine the effects of seasonal changes in wave climate, errors in bathymetry, and variations in size and shape of the inaccessible region of interest. Model skill is quantified using two methods: (1) a more traditional correlation of observed and modeled spectral statistics such as significant wave height, and (2) a recently developed RMS spectral skill score summed over all frequency-directional bins. The relative advantages and disadvantages of these two methods are considered. References: Booij, N., R.C. Ris, and L.H. Holthuijsen, 1999: A third-generation wave model for coastal regions: 1. Model description and validation. J. Geophys. Res. 104 (C4), 7649-7666. Orzech, M.D., J. Veeramony, and H.E. Ngodock, 2013: A variational assimilation system for nearshore wave modeling. J. Atm. & Oc. Tech., in press.

Adjoint-Based Implicit Uncertainty Analysis for Figures of Merit in a Laser Inertial Fusion Engine

DOE Office of Scientific and Technical Information (OSTI.GOV)

Seifried, J E; Fratoni, M; Kramer, K J

A primary purpose of computational models is to inform design decisions and, in order to make those decisions reliably, the confidence in the results of such models must be estimated. Monte Carlo neutron transport models are common tools for reactor designers. These types of models contain several sources of uncertainty that propagate onto the model predictions. Two uncertainties worthy of note are (1) experimental and evaluation uncertainties of nuclear data that inform all neutron transport models and (2) statistical counting precision, which all results of a Monte Carlo codes contain. Adjoint-based implicit uncertainty analyses allow for the consideration of anymore » number of uncertain input quantities and their effects upon the confidence of figures of merit with only a handful of forward and adjoint transport calculations. When considering a rich set of uncertain inputs, adjoint-based methods remain hundreds of times more computationally efficient than Direct Monte-Carlo methods. The LIFE (Laser Inertial Fusion Energy) engine is a concept being developed at Lawrence Livermore National Laboratory. Various options exist for the LIFE blanket, depending on the mission of the design. The depleted uranium hybrid LIFE blanket design strives to close the fission fuel cycle without enrichment or reprocessing, while simultaneously achieving high discharge burnups with reduced proliferation concerns. Neutron transport results that are central to the operation of the design are tritium production for fusion fuel, fission of fissile isotopes for energy multiplication, and production of fissile isotopes for sustained power. In previous work, explicit cross-sectional uncertainty analyses were performed for reaction rates related to the figures of merit for the depleted uranium hybrid LIFE blanket. Counting precision was also quantified for both the figures of merit themselves and the cross-sectional uncertainty estimates to gauge the validity of the analysis. All cross-sectional uncertainties were small (0.1-0.8%), bounded counting uncertainties, and were precise with regard to counting precision. Adjoint/importance distributions were generated for the same reaction rates. The current work leverages those adjoint distributions to transition from explicit sensitivities, in which the neutron flux is constrained, to implicit sensitivities, in which the neutron flux responds to input perturbations. This treatment vastly expands the set of data that contribute to uncertainties to produce larger, more physically accurate uncertainty estimates.« less

Adjoint Sensitivity Analyses Of Sand And Dust Storms In East Asia

NASA Astrophysics Data System (ADS)

Kay, J.; Kim, H.

2008-12-01

Sand and Dust Storm (SDS) in East Asia, so called Asian dust, is a seasonal meteorological phenomenon. Mostly in spring, dust particles blown into atmosphere in the arid area over northern China desert and Manchuria are transported to East Asia by prevailing flows. Three SDS events in East Asia from 2005 to 2008 are chosen to investigate how sensitive the SDS forecasts to the initial condition uncertainties and thence to suggest the sensitive regions for adaptive observations of the SDS events. Adaptive observations are additional observations in sensitive regions where the observations may have the most impact on the forecast by decreasing the forecast error. Three SDS events are chosen to represent different transport passes from the dust source regions to the Korean peninsula. To investigate the sensitivities to the initial condition, adjoint sensitivities that calculate gradient of the forecast aspect (i.e., response function) with respect to the initial condition are used. The forecast aspects relevant to the SDS transport are forecast error of the surface pressure, surface pressure perturbation, and steering vector of winds in the lower troposphere. Because the surface low pressure system usually plays an important role for SDS transport, the forecast error of the surface pressure and the surface pressure perturbation are chosen as the response function of the adjoint calculation. Another response function relevant to SDS transport is the steering flow over the downstream region (i.e., Korean peninsula) because direction and intensity of the prevailing winds usually determine the intensity and occurrence of the SDS events at the destination. The results show that the sensitive regions for the forecast error of the surface pressure and surface pressure perturbation are initially located in the vicinity of the trough and then propagate eastward as the low system moves eastward. The vertical structures of the adjoint sensitivities are upshear tilted structures, which are typical structures of extratropical cyclones. The adjoint sensitivities for lower tropospheric steering flow are also located near the trough, which confirms that the accurate forecast on the location and movement of the trough is essential to have better forecasts of Asian dust events. More comprehensive results and discussions of the adjoint sensitivity analyses for Asian dust events will be presented in the meeting.

Seismic waveform inversion best practices: regional, global and exploration test cases

NASA Astrophysics Data System (ADS)

Modrak, Ryan; Tromp, Jeroen

2016-09-01

Reaching the global minimum of a waveform misfit function requires careful choices about the nonlinear optimization, preconditioning and regularization methods underlying an inversion. Because waveform inversion problems are susceptible to erratic convergence associated with strong nonlinearity, one or two test cases are not enough to reliably inform such decisions. We identify best practices, instead, using four seismic near-surface problems, one regional problem and two global problems. To make meaningful quantitative comparisons between methods, we carry out hundreds of inversions, varying one aspect of the implementation at a time. Comparing nonlinear optimization algorithms, we find that limited-memory BFGS provides computational savings over nonlinear conjugate gradient methods in a wide range of test cases. Comparing preconditioners, we show that a new diagonal scaling derived from the adjoint of the forward operator provides better performance than two conventional preconditioning schemes. Comparing regularization strategies, we find that projection, convolution, Tikhonov regularization and total variation regularization are effective in different contexts. Besides questions of one strategy or another, reliability and efficiency in waveform inversion depend on close numerical attention and care. Implementation details involving the line search and restart conditions have a strong effect on computational cost, regardless of the chosen nonlinear optimization algorithm.

A novel discrete PSO algorithm for solving job shop scheduling problem to minimize makespan

NASA Astrophysics Data System (ADS)

Rameshkumar, K.; Rajendran, C.

2018-02-01

In this work, a discrete version of PSO algorithm is proposed to minimize the makespan of a job-shop. A novel schedule builder has been utilized to generate active schedules. The discrete PSO is tested using well known benchmark problems available in the literature. The solution produced by the proposed algorithms is compared with best known solution published in the literature and also compared with hybrid particle swarm algorithm and variable neighborhood search PSO algorithm. The solution construction methodology adopted in this study is found to be effective in producing good quality solutions for the various benchmark job-shop scheduling problems.

Chiral phases of fundamental and adjoint quarks

DOE Office of Scientific and Technical Information (OSTI.GOV)

Natale, A. A.; Instituto de Física Teórica - UNESP Rua Dr. Bento T. Ferraz, 271, Bl.II - 01140-070, São Paulo, SP

2016-01-22

We consider a QCD chiral symmetry breaking model where the gap equation contains an effective confining propagator and a dressed gluon propagator with a dynamically generated mass. This model is able to explain the ratios between the chiral transition and deconfinement temperatures in the case of fundamental and adjoint quarks. It also predicts the recovery of the chiral symmetry for a large number of quarks (n{sub f} ≈ 11 – 13) in agreement with lattice data.

Inverse modeling methods for indoor airborne pollutant tracking: literature review and fundamentals.

PubMed

Liu, X; Zhai, Z

2007-12-01

Reduction in indoor environment quality calls for effective control and improvement measures. Accurate and prompt identification of contaminant sources ensures that they can be quickly removed and contaminated spaces isolated and cleaned. This paper discusses the use of inverse modeling to identify potential indoor pollutant sources with limited pollutant sensor data. The study reviews various inverse modeling methods for advection-dispersion problems and summarizes the methods into three major categories: forward, backward, and probability inverse modeling methods. The adjoint probability inverse modeling method is indicated as an appropriate model for indoor air pollutant tracking because it can quickly find source location, strength and release time without prior information. The paper introduces the principles of the adjoint probability method and establishes the corresponding adjoint equations for both multi-zone airflow models and computational fluid dynamics (CFD) models. The study proposes a two-stage inverse modeling approach integrating both multi-zone and CFD models, which can provide a rapid estimate of indoor pollution status and history for a whole building. Preliminary case study results indicate that the adjoint probability method is feasible for indoor pollutant inverse modeling. The proposed method can help identify contaminant source characteristics (location and release time) with limited sensor outputs. This will ensure an effective and prompt execution of building management strategies and thus achieve a healthy and safe indoor environment. The method can also help design optimal sensor networks.

Finding all solutions of nonlinear equations using the dual simplex method

NASA Astrophysics Data System (ADS)

Yamamura, Kiyotaka; Fujioka, Tsuyoshi

2003-03-01

Recently, an efficient algorithm has been proposed for finding all solutions of systems of nonlinear equations using linear programming. This algorithm is based on a simple test (termed the LP test) for nonexistence of a solution to a system of nonlinear equations using the dual simplex method. In this letter, an improved version of the LP test algorithm is proposed. By numerical examples, it is shown that the proposed algorithm could find all solutions of a system of 300 nonlinear equations in practical computation time.

An Enhanced Artificial Bee Colony Algorithm with Solution Acceptance Rule and Probabilistic Multisearch.

PubMed

Yurtkuran, Alkın; Emel, Erdal

2016-01-01

The artificial bee colony (ABC) algorithm is a popular swarm based technique, which is inspired from the intelligent foraging behavior of honeybee swarms. This paper proposes a new variant of ABC algorithm, namely, enhanced ABC with solution acceptance rule and probabilistic multisearch (ABC-SA) to address global optimization problems. A new solution acceptance rule is proposed where, instead of greedy selection between old solution and new candidate solution, worse candidate solutions have a probability to be accepted. Additionally, the acceptance probability of worse candidates is nonlinearly decreased throughout the search process adaptively. Moreover, in order to improve the performance of the ABC and balance the intensification and diversification, a probabilistic multisearch strategy is presented. Three different search equations with distinctive characters are employed using predetermined search probabilities. By implementing a new solution acceptance rule and a probabilistic multisearch approach, the intensification and diversification performance of the ABC algorithm is improved. The proposed algorithm has been tested on well-known benchmark functions of varying dimensions by comparing against novel ABC variants, as well as several recent state-of-the-art algorithms. Computational results show that the proposed ABC-SA outperforms other ABC variants and is superior to state-of-the-art algorithms proposed in the literature.

Aerodynamic shape optimization using control theory

NASA Technical Reports Server (NTRS)

Reuther, James

1996-01-01

Aerodynamic shape design has long persisted as a difficult scientific challenge due its highly nonlinear flow physics and daunting geometric complexity. However, with the emergence of Computational Fluid Dynamics (CFD) it has become possible to make accurate predictions of flows which are not dominated by viscous effects. It is thus worthwhile to explore the extension of CFD methods for flow analysis to the treatment of aerodynamic shape design. Two new aerodynamic shape design methods are developed which combine existing CFD technology, optimal control theory, and numerical optimization techniques. Flow analysis methods for the potential flow equation and the Euler equations form the basis of the two respective design methods. In each case, optimal control theory is used to derive the adjoint differential equations, the solution of which provides the necessary gradient information to a numerical optimization method much more efficiently then by conventional finite differencing. Each technique uses a quasi-Newton numerical optimization algorithm to drive an aerodynamic objective function toward a minimum. An analytic grid perturbation method is developed to modify body fitted meshes to accommodate shape changes during the design process. Both Hicks-Henne perturbation functions and B-spline control points are explored as suitable design variables. The new methods prove to be computationally efficient and robust, and can be used for practical airfoil design including geometric and aerodynamic constraints. Objective functions are chosen to allow both inverse design to a target pressure distribution and wave drag minimization. Several design cases are presented for each method illustrating its practicality and efficiency. These include non-lifting and lifting airfoils operating at both subsonic and transonic conditions.

An Algorithm for Automatically Modifying Train Crew Schedule

NASA Astrophysics Data System (ADS)

Takahashi, Satoru; Kataoka, Kenji; Kojima, Teruhito; Asami, Masayuki

Once the break-down of the train schedule occurs, the crew schedule as well as the train schedule has to be modified as quickly as possible to restore them. In this paper, we propose an algorithm for automatically modifying a crew schedule that takes all constraints into consideration, presenting a model of the combined problem of crews and trains. The proposed algorithm builds an initial solution by relaxing some of the constraint conditions, and then uses a Taboo-search method to revise this solution in order to minimize the degree of constraint violation resulting from these relaxed conditions. Then we show not only that the algorithm can generate a constraint satisfaction solution, but also that the solution will satisfy the experts. That is, we show the proposed algorithm is capable of producing a usable solution in a short time by applying to actual cases of train-schedule break-down, and that the solution is at least as good as those produced manually, by comparing the both solutions with several point of view.

Assessing the Impact of Advanced Satellite Observations in the NASA GEOS-5 Forecast System Using the Adjoint Method

NASA Technical Reports Server (NTRS)

Gelaro, Ron; Liu, Emily; Sienkiewicz, Meta

2011-01-01

The adjoint of a data assimilation system provides a flexible and efficient tool for estimating observation impacts on short-range weather forecasts. The impacts of any or all observations can be estimated simultaneously based on a single execution of the adjoint system. The results can be easily aggregated according to data type, location, channel, etc., making this technique especially attractive for examining the impacts of new hyper-spectral satellite instruments and for conducting regular, even near-real time, monitoring of the entire observing system. In this talk, we present results from the adjoint-based observation impact monitoring tool in NASA's GEOS-5 global atmospheric data assimilation and forecast system. The tool has been running in various off-line configurations for some time, and is scheduled to run as a regular part of the real-time forecast suite beginning in autumn 20 I O. We focus on the impacts of the newest components of the satellite observing system, including AIRS, IASI and GPS. For AIRS and IASI, it is shown that the vast majority of the channels assimilated have systematic positive impacts (of varying magnitudes), although some channels degrade the forecast. Of the latter, most are moisture-sensitive or near-surface channels. The impact of GPS observations in the southern hemisphere is found to be a considerable overall benefit to the system. In addition, the spatial variability of observation impacts reveals coherent patterns of positive and negative impacts that may point to deficiencies in the use of certain observations over, for example, specific surface types. When performed in conjunction with selected observing system experiments (OSEs), the adjoint results reveal both redundancies and dependencies between observing system impacts as observations are added or removed from the assimilation system. Understanding these dependencies appears to pose a major challenge for optimizing the use of the current observational network and defining requirements for future observing systems.

Seismic wave-speed structure beneath the metropolitan area of Japan based on adjoint tomography

NASA Astrophysics Data System (ADS)

Miyoshi, T.; Obayashi, M.; Tono, Y.; Tsuboi, S.

2015-12-01

We have obtained a three-dimensional (3D) model of seismic wave-speed structure beneath the metropolitan area of Japan. We applied the spectral-element method (e.g. Komatitsch and Tromp 1999) and adjoint method (Liu and Tromp 2006) to the broadband seismograms in order to infer the 3D model. We used the travel-time tomography result (Matsubara and Obara 2011) as an initial 3D model and used broadband waveforms recorded at the NIED F-net stations. We selected 147 earthquakes with magnitude of larger than 4.5 from the F-net earthquake catalog and used their bandpass filtered seismograms between 5 and 20 second with a high S/N ratio. The 3D model used for the forward and adjoint simulations is represented as a region of approximately 500 by 450 km in horizontal and 120 km in depth. Minimum period of theoretical waveforms was 4.35 second. For the adjoint inversion, we picked up the windows of the body waves from the observed and theoretical seismograms. We used SPECFEM3D_Cartesian code (e.g. Peter et al. 2011) for the forward and adjoint simulations, and their simulations were implemented by K-computer in RIKEN. Each iteration required about 0.1 million CPU hours at least. The model parameters of Vp and Vs were updated by using the steepest descent method. We obtained the fourth iterative model (M04), which reproduced observed waveforms better than the initial model. The shear wave-speed of M04 was significantly smaller than the initial model at any depth. The model of compressional wave-speed was not improved by inversion because of small alpha kernel values. Acknowledgements: This research was partly supported by MEXT Strategic Program for Innovative Research. We thank to the NIED for providing seismological data.

Seismic imaging: From classical to adjoint tomography

NASA Astrophysics Data System (ADS)

Liu, Q.; Gu, Y. J.

2012-09-01

Seismic tomography has been a vital tool in probing the Earth's internal structure and enhancing our knowledge of dynamical processes in the Earth's crust and mantle. While various tomographic techniques differ in data types utilized (e.g., body vs. surface waves), data sensitivity (ray vs. finite-frequency approximations), and choices of model parameterization and regularization, most global mantle tomographic models agree well at long wavelengths, owing to the presence and typical dimensions of cold subducted oceanic lithospheres and hot, ascending mantle plumes (e.g., in central Pacific and Africa). Structures at relatively small length scales remain controversial, though, as will be discussed in this paper, they are becoming increasingly resolvable with the fast expanding global and regional seismic networks and improved forward modeling and inversion techniques. This review paper aims to provide an overview of classical tomography methods, key debates pertaining to the resolution of mantle tomographic models, as well as to highlight recent theoretical and computational advances in forward-modeling methods that spearheaded the developments in accurate computation of sensitivity kernels and adjoint tomography. The first part of the paper is devoted to traditional traveltime and waveform tomography. While these approaches established a firm foundation for global and regional seismic tomography, data coverage and the use of approximate sensitivity kernels remained as key limiting factors in the resolution of the targeted structures. In comparison to classical tomography, adjoint tomography takes advantage of full 3D numerical simulations in forward modeling and, in many ways, revolutionizes the seismic imaging of heterogeneous structures with strong velocity contrasts. For this reason, this review provides details of the implementation, resolution and potential challenges of adjoint tomography. Further discussions of techniques that are presently popular in seismic array analysis, such as noise correlation functions, receiver functions, inverse scattering imaging, and the adaptation of adjoint tomography to these different datasets highlight the promising future of seismic tomography.

Adjoint BFKL at finite coupling: a short-cut from the collinear limit

DOE PAGES

Basso, Benjamin; Caron-Huot, Simon; Sever, Amit

2015-01-08

In the high energy Regge limit, the six gluons scattering amplitude is controlled by the adjoint BFKL eigenvalue and impact factor. In this paper we determine these two building blocks at any value of the ’t Hooft coupling in planar N=4 SYM theory. This is achieved by means of analytic continuations from the collinear limit, where similar all loops expressions were recently established. We check our predictions against all available data at weak and strong coupling.

Coalition Interoperability Measurement Frameworks Literature Survey

DTIC Science & Technology

2011-08-01

National Defence, 2011 © Sa Majesté la Reine (en droit du Canada), telle que représentée par le ministre de la Défense nationale, 2011 Abstract...interarmées élabore actuellement ce cadre et les travaux sont parrainés par le Groupe du Sous- ministre adjoint ( Gestion de l’information). L’intérêt...Groupe du Sous-ministre adjoint ( Gestion de l’information). Le type d’interopérabilité qui nous intéresse : 1. touche les Cinq pays (Australie

Toward the Development of a Coupled COAMPS-ROMS Ensemble Kalman Filter and Adjoint with a focus on the Indian Ocean and the Intraseasonal Oscillation

DTIC Science & Technology

2015-09-30

1 Approved for public release; distribution is unlimited. Toward the Development of a Coupled COAMPS-ROMS Ensemble Kalman Filter and Adjoint...system at NCAR. (2) Compare the performance of the Ensemble Kalman Filter (EnKF) using the Data Assimilation Research Testbed (DART) and 4...undercurrent is clearly visible. Figure 2 shows the horizontal temperature structure and circulation at a depth of 50 m within the surface mixed layer

Comparison of the Adjoint and Adjoint-Free 4dVar Assimilation of the Hydrographic and Velocity Observations in the Adriatic Sea

DTIC Science & Technology

2015-11-10

of the ensemble method o the estimation of sensitivities was demonstrated in meteorological Ancell and Hakim, 2007 ; Torn and Hakim, 2008) and...to predetermined low- dimensional subspaces spanned either by the reduced-order approx- imations of the model Green’s functions ( Stammer and Wunsch...2005; Qui et al., 2007 ; Hoteit, 2008). In fact, the 4dEnVar technique pursues a similar, but more general approach, pa- rameterizing the search

Banach Synaptic Algebras

NASA Astrophysics Data System (ADS)

Foulis, David J.; Pulmannov, Sylvia

2018-04-01

Using a representation theorem of Erik Alfsen, Frederic Schultz, and Erling Størmer for special JB-algebras, we prove that a synaptic algebra is norm complete (i.e., Banach) if and only if it is isomorphic to the self-adjoint part of a Rickart C∗-algebra. Also, we give conditions on a Banach synaptic algebra that are equivalent to the condition that it is isomorphic to the self-adjoint part of an AW∗-algebra. Moreover, we study some relationships between synaptic algebras and so-called generalized Hermitian algebras.

Elementary operators on self-adjoint operators

NASA Astrophysics Data System (ADS)

Molnar, Lajos; Semrl, Peter

2007-03-01

Let H be a Hilbert space and let and be standard *-operator algebras on H. Denote by and the set of all self-adjoint operators in and , respectively. Assume that and are surjective maps such that M(AM*(B)A)=M(A)BM(A) and M*(BM(A)B)=M*(B)AM*(B) for every pair , . Then there exist an invertible bounded linear or conjugate-linear operator and a constant c[set membership, variant]{-1,1} such that M(A)=cTAT*, , and M*(B)=cT*BT, .

SCIPUFF Tangent-Linear/Adjoint Model for Release Source Location from Observational Data

DTIC Science & Technology

2011-01-18

magnitudes, times? Inverse model based on SCIPUFF (AIMS) 1/17/2011 Aerodyne Research, Inc. W911NF-06-C-0161 5 (1981), Daescu and Carmichael (2003...Defined Choice- Situations” The Journal of the Operational Research Society, Vol. 32, No. 2 (1981) Daescu, D. N., and Carmichael , G. R., “An Adjoint...Intelligence Applications to Environmental Sciences at AMS Annual Meeting, Atlanta, GA, Jan 17-21 (2010 ) N. Platt, S. Warner and S. M. Nunes, ’Plan for

A numerical solution of Duffing's equations including the prediction of jump phenomena

NASA Technical Reports Server (NTRS)

Moyer, E. T., Jr.; Ghasghai-Abdi, E.

1987-01-01

Numerical methodology for the solution of Duffing's differential equation is presented. Algorithms for the prediction of multiple equilibrium solutions and jump phenomena are developed. In addition, a filtering algorithm for producing steady state solutions is presented. The problem of a rigidly clamped circular plate subjected to cosinusoidal pressure loading is solved using the developed algorithms (the plate is assumed to be in the geometrically nonlinear range). The results accurately predict regions of solution multiplicity and jump phenomena.

Ritz method for transient response in systems having unsymmetric stiffness

NASA Technical Reports Server (NTRS)

Butler, Thomas G.

1989-01-01

The DMAP coding was automated to such an extent by using the device of bubble vectors, that it is useable for analyses in its present form. This feasibility study demonstrates that the Ritz Method is so compelling as to warrant coding its modules in FORTRAN and organizing the resulting coding into a new Rigid Format. Even though this Ritz technique was developed for unsymmetric stiffness matrices, it offers advantages to problems with symmetric stiffnesses. If used for the symmetric case the solution would be simplified to one set of modes, because the adjoint would be the same as the primary. Its advantage in either type of symmetry over a classical eigenvalue modal expansion is that information density per Ritz mode is far richer than per eigenvalue mode; thus far fewer modes would be needed for the same accuracy and every mode would actively participate in the response. Considerable economy can be realized in adapting Ritz vectors for modal solutions. This new Ritz capability now makes NASTRAN even more powerful than before.

Dissipative and nonunitary solutions of operator commutation relations

NASA Astrophysics Data System (ADS)

Makarov, K. A.; Tsekanovskii, E.

2016-01-01

We study the (generalized) semi-Weyl commutation relations UgAU* g = g(A) on Dom(A), where A is a densely defined operator and G ∋ g ↦ Ug is a unitary representation of the subgroup G of the affine group G, the group of affine orientation-preserving transformations of the real axis. If A is a symmetric operator, then the group G induces an action/flow on the operator unit ball of contracting transformations from Ker(A* - iI) to Ker(A* + iI). We establish several fixed-point theorems for this flow. In the case of one-parameter continuous subgroups of linear transformations, self-adjoint (maximal dissipative) operators associated with the fixed points of the flow yield solutions of the (restricted) generalized Weyl commutation relations. We show that in the dissipative setting, the restricted Weyl relations admit a variety of representations that are not unitarily equivalent. For deficiency indices (1, 1), the basic results can be strengthened and set in a separate case.

Optimal Control of Thermo--Fluid Phenomena in Variable Domains

NASA Astrophysics Data System (ADS)

Volkov, Oleg; Protas, Bartosz

2008-11-01

This presentation concerns our continued research on adjoint--based optimization of viscous incompressible flows (the Navier--Stokes problem) coupled with heat conduction involving change of phase (the Stefan problem), and occurring in domains with variable boundaries. This problem is motivated by optimization of advanced welding techniques used in automotive manufacturing, where the goal is to determine an optimal heat input, so as to obtain a desired shape of the weld pool surface upon solidification. We argue that computation of sensitivities (gradients) in such free--boundary problems requires the use of the shape--differential calculus as a key ingredient. We also show that, with such tools available, the computational solution of the direct and inverse (optimization) problems can in fact be achieved in a similar manner and in a comparable computational time. Our presentation will address certain mathematical and computational aspects of the method. As an illustration we will consider the two--phase Stefan problem with contact point singularities where our approach allows us to obtain a thermodynamically consistent solution.

Solving Fractional Programming Problems based on Swarm Intelligence

NASA Astrophysics Data System (ADS)

Raouf, Osama Abdel; Hezam, Ibrahim M.

2014-04-01

This paper presents a new approach to solve Fractional Programming Problems (FPPs) based on two different Swarm Intelligence (SI) algorithms. The two algorithms are: Particle Swarm Optimization, and Firefly Algorithm. The two algorithms are tested using several FPP benchmark examples and two selected industrial applications. The test aims to prove the capability of the SI algorithms to solve any type of FPPs. The solution results employing the SI algorithms are compared with a number of exact and metaheuristic solution methods used for handling FPPs. Swarm Intelligence can be denoted as an effective technique for solving linear or nonlinear, non-differentiable fractional objective functions. Problems with an optimal solution at a finite point and an unbounded constraint set, can be solved using the proposed approach. Numerical examples are given to show the feasibility, effectiveness, and robustness of the proposed algorithm. The results obtained using the two SI algorithms revealed the superiority of the proposed technique among others in computational time. A better accuracy was remarkably observed in the solution results of the industrial application problems.

A split finite element algorithm for the compressible Navier-Stokes equations

NASA Technical Reports Server (NTRS)

Baker, A. J.

1979-01-01

An accurate and efficient numerical solution algorithm is established for solution of the high Reynolds number limit of the Navier-Stokes equations governing the multidimensional flow of a compressible essentially inviscid fluid. Finite element interpolation theory is used within a dissipative formulation established using Galerkin criteria within the Method of Weighted Residuals. An implicit iterative solution algorithm is developed, employing tensor product bases within a fractional steps integration procedure, that significantly enhances solution economy concurrent with sharply reduced computer hardware demands. The algorithm is evaluated for resolution of steep field gradients and coarse grid accuracy using both linear and quadratic tensor product interpolation bases. Numerical solutions for linear and nonlinear, one, two and three dimensional examples confirm and extend the linearized theoretical analyses, and results are compared to competitive finite difference derived algorithms.

At-Least Version of the Generalized Minimum Spanning Tree Problem: Optimization Through Ant Colony System and Genetic Algorithms

NASA Technical Reports Server (NTRS)

Janich, Karl W.

2005-01-01

The At-Least version of the Generalized Minimum Spanning Tree Problem (L-GMST) is a problem in which the optimal solution connects all defined clusters of nodes in a given network at a minimum cost. The L-GMST is NPHard; therefore, metaheuristic algorithms have been used to find reasonable solutions to the problem as opposed to computationally feasible exact algorithms, which many believe do not exist for such a problem. One such metaheuristic uses a swarm-intelligent Ant Colony System (ACS) algorithm, in which agents converge on a solution through the weighing of local heuristics, such as the shortest available path and the number of agents that recently used a given path. However, in a network using a solution derived from the ACS algorithm, some nodes may move around to different clusters and cause small changes in the network makeup. Rerunning the algorithm from the start would be somewhat inefficient due to the significance of the changes, so a genetic algorithm based on the top few solutions found in the ACS algorithm is proposed to quickly and efficiently adapt the network to these small changes.

Non-destructive testing of ceramic materials using mid-infrared ultrashort-pulse laser

NASA Astrophysics Data System (ADS)

Sun, S. C.; Qi, Hong; An, X. Y.; Ren, Y. T.; Qiao, Y. B.; Ruan, Liming M.

2018-04-01

The non-destructive testing (NDT) of ceramic materials using mid-infrared ultrashort-pulse laser is investigated in this study. The discrete ordinate method is applied to solve the transient radiative transfer equation in 2D semitransparent medium and the emerging radiative intensity on boundary serves as input for the inverse analysis. The sequential quadratic programming algorithm is employed as the inverse technique to optimize objective function, in which the gradient of objective function with respect to reconstruction parameters is calculated using the adjoint model. Two reticulated porous ceramics including partially stabilized zirconia and oxide-bonded silicon carbide are tested. The retrieval results show that the main characteristics of defects such as optical properties, geometric shapes and positions can be accurately reconstructed by the present model. The proposed technique is effective and robust in NDT of ceramics even with measurement errors.

Topology optimization of finite strain viscoplastic systems under transient loads [Dynamic topology optimization based on finite strain visco-plasticity

DOE Office of Scientific and Technical Information (OSTI.GOV)

Ivarsson, Niklas; Wallin, Mathias; Tortorelli, Daniel

In this paper, a transient finite strain viscoplastic model is implemented in a gradient-based topology optimization framework to design impact mitigating structures. The model's kinematics relies on the multiplicative split of the deformation gradient, and the constitutive response is based on isotropic hardening viscoplasticity. To solve the mechanical balance laws, the implicit Newmark-beta method is used together with a total Lagrangian finite element formulation. The optimization problem is regularized using a partial differential equation filter and solved using the method of moving asymptotes. Sensitivities required to solve the optimization problem are derived using the adjoint method. To demonstrate the capabilitymore » of the algorithm, several protective systems are designed, in which the absorbed viscoplastic energy is maximized. Finally, the numerical examples demonstrate that transient finite strain viscoplastic effects can successfully be combined with topology optimization.« less

Topology optimization of finite strain viscoplastic systems under transient loads [Dynamic topology optimization based on finite strain visco-plasticity

DOE PAGES

Ivarsson, Niklas; Wallin, Mathias; Tortorelli, Daniel

2018-02-08

In this paper, a transient finite strain viscoplastic model is implemented in a gradient-based topology optimization framework to design impact mitigating structures. The model's kinematics relies on the multiplicative split of the deformation gradient, and the constitutive response is based on isotropic hardening viscoplasticity. To solve the mechanical balance laws, the implicit Newmark-beta method is used together with a total Lagrangian finite element formulation. The optimization problem is regularized using a partial differential equation filter and solved using the method of moving asymptotes. Sensitivities required to solve the optimization problem are derived using the adjoint method. To demonstrate the capabilitymore » of the algorithm, several protective systems are designed, in which the absorbed viscoplastic energy is maximized. Finally, the numerical examples demonstrate that transient finite strain viscoplastic effects can successfully be combined with topology optimization.« less

Traction patterns of tumor cells.

PubMed

Ambrosi, D; Duperray, A; Peschetola, V; Verdier, C

2009-01-01

The traction exerted by a cell on a planar deformable substrate can be indirectly obtained on the basis of the displacement field of the underlying layer. The usual methodology used to address this inverse problem is based on the exploitation of the Green tensor of the linear elasticity problem in a half space (Boussinesq problem), coupled with a minimization algorithm under force penalization. A possible alternative strategy is to exploit an adjoint equation, obtained on the basis of a suitable minimization requirement. The resulting system of coupled elliptic partial differential equations is applied here to determine the force field per unit surface generated by T24 tumor cells on a polyacrylamide substrate. The shear stress obtained by numerical integration provides quantitative insight of the traction field and is a promising tool to investigate the spatial pattern of force per unit surface generated in cell motion, particularly in the case of such cancer cells.

Numerical Study of Hydrothermal Wave Suppression in Thermocapillary Flow Using a Predictive Control Method

NASA Astrophysics Data System (ADS)

Muldoon, F. H.

2018-04-01

Hydrothermal waves in flows driven by thermocapillary and buoyancy effects are suppressed by applying a predictive control method. Hydrothermal waves arise in the manufacturing of crystals, including the "open boat" crystal growth process, and lead to undesirable impurities in crystals. The open boat process is modeled using the two-dimensional unsteady incompressible Navier-Stokes equations under the Boussinesq approximation and the linear approximation of the surface thermocapillary force. The flow is controlled by a spatially and temporally varying heat flux density through the free surface. The heat flux density is determined by a conjugate gradient optimization algorithm. The gradient of the objective function with respect to the heat flux density is found by solving adjoint equations derived from the Navier-Stokes ones in the Boussinesq approximation. Special attention is given to heat flux density distributions over small free-surface areas and to the maximum admissible heat flux density.

Boundary-Layer Receptivity and Integrated Transition Prediction

NASA Technical Reports Server (NTRS)

Chang, Chau-Lyan; Choudhari, Meelan

2005-01-01

The adjoint parabold stability equations (PSE) formulation is used to calculate the boundary layer receptivity to localized surface roughness and suction for compressible boundary layers. Receptivity efficiency functions predicted by the adjoint PSE approach agree well with results based on other nonparallel methods including linearized Navier-Stokes equations for both Tollmien-Schlichting waves and crossflow instability in swept wing boundary layers. The receptivity efficiency function can be regarded as the Green's function to the disturbance amplitude evolution in a nonparallel (growing) boundary layer. Given the Fourier transformed geometry factor distribution along the chordwise direction, the linear disturbance amplitude evolution for a finite size, distributed nonuniformity can be computed by evaluating the integral effects of both disturbance generation and linear amplification. The synergistic approach via the linear adjoint PSE for receptivity and nonlinear PSE for disturbance evolution downstream of the leading edge forms the basis for an integrated transition prediction tool. Eventually, such physics-based, high fidelity prediction methods could simulate the transition process from the disturbance generation through the nonlinear breakdown in a holistic manner.

Simulation and Optimization of an Airfoil with Leading Edge Slat

NASA Astrophysics Data System (ADS)

Schramm, Matthias; Stoevesandt, Bernhard; Peinke, Joachim

2016-09-01

A gradient-based optimization is used in order to improve the shape of a leading edge slat upstream of a DU 91-W2-250 airfoil. The simulations are performed by solving the Reynolds-Averaged Navier-Stokes equations (RANS) using the open source CFD code OpenFOAM. Gradients are computed via the adjoint approach, which is suitable to deal with many design parameters, but keeping the computational costs low. The implementation is verified by comparing the gradients from the adjoint method with gradients obtained by finite differences for a NACA 0012 airfoil. The simulations of the leading edge slat are validated against measurements from the acoustic wind tunnel of Oldenburg University at a Reynolds number of Re = 6 • 105. The shape of the slat is optimized using the adjoint approach resulting in a drag reduction of 2%. Although the optimization is done for Re = 6 • 105, the improvements also hold for a higher Reynolds number of Re = 7.9 • 106, which is more realistic at modern wind turbines.

Variational data assimilation with a semi-Lagrangian semi-implicit global shallow-water equation model and its adjoint

NASA Technical Reports Server (NTRS)

Li, Y.; Navon, I. M.; Courtier, P.; Gauthier, P.

1993-01-01

An adjoint model is developed for variational data assimilation using the 2D semi-Lagrangian semi-implicit (SLSI) shallow-water equation global model of Bates et al. with special attention being paid to the linearization of the interpolation routines. It is demonstrated that with larger time steps the limit of the validity of the tangent linear model will be curtailed due to the interpolations, especially in regions where sharp gradients in the interpolated variables coupled with strong advective wind occur, a synoptic situation common in the high latitudes. This effect is particularly evident near the pole in the Northern Hemisphere during the winter season. Variational data assimilation experiments of 'identical twin' type with observations available only at the end of the assimilation period perform well with this adjoint model. It is confirmed that the computational efficiency of the semi-Lagrangian scheme is preserved during the minimization process, related to the variational data assimilation procedure.

Theoretical aspect of suitable spatial boundary condition specified for adjoint model on limited area

NASA Astrophysics Data System (ADS)

Wang, Yuan; Wu, Rongsheng

2001-12-01

Theoretical argumentation for so-called suitable spatial condition is conducted by the aid of homotopy framework to demonstrate that the proposed boundary condition does guarantee that the over-specification boundary condition resulting from an adjoint model on a limited-area is no longer an issue, and yet preserve its well-poseness and optimal character in the boundary setting. The ill-poseness of over-specified spatial boundary condition is in a sense, inevitable from an adjoint model since data assimilation processes have to adapt prescribed observations that used to be over-specified at the spatial boundaries of the modeling domain. In the view of pragmatic implement, the theoretical framework of our proposed condition for spatial boundaries indeed can be reduced to the hybrid formulation of nudging filter, radiation condition taking account of ambient forcing, together with Dirichlet kind of compatible boundary condition to the observations prescribed in data assimilation procedure. All of these treatments, no doubt, are very familiar to mesoscale modelers.

A simple suboptimal least-squares algorithm for attitude determination with multiple sensors

NASA Technical Reports Server (NTRS)

Brozenec, Thomas F.; Bender, Douglas J.

1994-01-01

Three-axis attitude determination is equivalent to finding a coordinate transformation matrix which transforms a set of reference vectors fixed in inertial space to a set of measurement vectors fixed in the spacecraft. The attitude determination problem can be expressed as a constrained optimization problem. The constraint is that a coordinate transformation matrix must be proper, real, and orthogonal. A transformation matrix can be thought of as optimal in the least-squares sense if it maps the measurement vectors to the reference vectors with minimal 2-norm errors and meets the above constraint. This constrained optimization problem is known as Wahba's problem. Several algorithms which solve Wahba's problem exactly have been developed and used. These algorithms, while steadily improving, are all rather complicated. Furthermore, they involve such numerically unstable or sensitive operations as matrix determinant, matrix adjoint, and Newton-Raphson iterations. This paper describes an algorithm which minimizes Wahba's loss function, but without the constraint. When the constraint is ignored, the problem can be solved by a straightforward, numerically stable least-squares algorithm such as QR decomposition. Even though the algorithm does not explicitly take the constraint into account, it still yields a nearly orthogonal matrix for most practical cases; orthogonality only becomes corrupted when the sensor measurements are very noisy, on the same order of magnitude as the attitude rotations. The algorithm can be simplified if the attitude rotations are small enough so that the approximation sin(theta) approximately equals theta holds. We then compare the computational requirements for several well-known algorithms. For the general large-angle case, the QR least-squares algorithm is competitive with all other know algorithms and faster than most. If attitude rotations are small, the least-squares algorithm can be modified to run faster, and this modified algorithm is faster than all but a similarly specialized version of the QUEST algorithm. We also introduce a novel measurement averaging technique which reduces the n-measurement case to the two measurement case for our particular application, a star tracker and earth sensor mounted on an earth-pointed geosynchronous communications satellite. Using this technique, many n-measurement problems reduce to less than or equal to 3 measurements; this reduces the amount of required calculation without significant degradation in accuracy. Finally, we present the results of some tests which compare the least-squares algorithm with the QUEST and FOAM algorithms in the two-measurement case. For our example case, all three algorithms performed with similar accuracy.

Algorithms for optimization of branching gravity-driven water networks

NASA Astrophysics Data System (ADS)

Dardani, Ian; Jones, Gerard F.

2018-05-01

The design of a water network involves the selection of pipe diameters that satisfy pressure and flow requirements while considering cost. A variety of design approaches can be used to optimize for hydraulic performance or reduce costs. To help designers select an appropriate approach in the context of gravity-driven water networks (GDWNs), this work assesses three cost-minimization algorithms on six moderate-scale GDWN test cases. Two algorithms, a backtracking algorithm and a genetic algorithm, use a set of discrete pipe diameters, while a new calculus-based algorithm produces a continuous-diameter solution which is mapped onto a discrete-diameter set. The backtracking algorithm finds the global optimum for all but the largest of cases tested, for which its long runtime makes it an infeasible option. The calculus-based algorithm's discrete-diameter solution produced slightly higher-cost results but was more scalable to larger network cases. Furthermore, the new calculus-based algorithm's continuous-diameter and mapped solutions provided lower and upper bounds, respectively, on the discrete-diameter global optimum cost, where the mapped solutions were typically within one diameter size of the global optimum. The genetic algorithm produced solutions even closer to the global optimum with consistently short run times, although slightly higher solution costs were seen for the larger network cases tested. The results of this study highlight the advantages and weaknesses of each GDWN design method including closeness to the global optimum, the ability to prune the solution space of infeasible and suboptimal candidates without missing the global optimum, and algorithm run time. We also extend an existing closed-form model of Jones (2011) to include minor losses and a more comprehensive two-part cost model, which realistically applies to pipe sizes that span a broad range typical of GDWNs of interest in this work, and for smooth and commercial steel roughness values.

Toward a Comprehensive Carbon Budget for North America: Potential Applications of Adjoint Methods with Diverse Datasets

NASA Technical Reports Server (NTRS)

Andrews, A.

2002-01-01

A detailed mechanistic understanding of the sources and sinks of CO2 will be required to reliably predict future COS levels and climate. A commonly used technique for deriving information about CO2 exchange with surface reservoirs is to solve an "inverse problem," where CO2 observations are used with an atmospheric transport model to find the optimal distribution of sources and sinks. Synthesis inversion methods are powerful tools for addressing this question, but the results are disturbingly sensitive to the details of the calculation. Studies done using different atmospheric transport models and combinations of surface station data have produced substantially different distributions of surface fluxes. Adjoint methods are now being developed that will more effectively incorporate diverse datasets in estimates of surface fluxes of CO2. In an adjoint framework, it will be possible to combine CO2 concentration data from long-term surface monitoring stations with data from intensive field campaigns and with proposed future satellite observations. A major advantage of the adjoint approach is that meteorological and surface data, as well as data for other atmospheric constituents and pollutants can be efficiently included in addition to observations of CO2 mixing ratios. This presentation will provide an overview of potentially useful datasets for carbon cycle research in general with an emphasis on planning for the North American Carbon Project. Areas of overlap with ongoing and proposed work on air quality/air pollution issues will be highlighted.

Quasi-stationary states and fermion pair creation from a vacuum in supercritical Coulomb field

NASA Astrophysics Data System (ADS)

Khalilov, V. R.

2017-12-01

Creation of charged fermion pair from a vacuum in so-called supercritical Coulomb potential is examined for the case when fermions can move only in the same (one) plane. In which case, quantum dynamics of charged massive or massless fermions can be described by the two-dimensional Dirac Hamiltonians with an usual (-a/r) Coulomb potential. These Hamiltonians are singular and require the additional definition in order for them to be treated as self-adjoint quantum-mechanical operators. We construct the self-adjoint two-dimensional Dirac Hamiltonians with a Coulomb potential and determine the quantum-mechanical states for such Hamiltonians in the corresponding Hilbert spaces of square-integrable functions. We determine the scattering amplitude in which the self-adjoint extension parameter is incorporated and then obtain equations implicitly defining possible discrete energy spectra of the self-adjoint Dirac Hamiltonians with a Coulomb potential. It is shown that this quantum system becomes unstable in the presence of a supercritical Coulomb potential which manifests in the appearance of quasi-stationary states in the lower (negative) energy continuum. The energy spectrum of those states is quasi-discrete, consists of broadened levels with widths related to the inverse lifetimes of the quasi-stationary states as well as the probability of creation of charged fermion pair by a supercritical Coulomb field. Explicit analytical expressions for the creation probabilities of charged (massive or massless) fermion pair are obtained in a supercritical Coulomb field.

A new effective operator for the hybrid algorithm for solving global optimisation problems

NASA Astrophysics Data System (ADS)

Duc, Le Anh; Li, Kenli; Nguyen, Tien Trong; Yen, Vu Minh; Truong, Tung Khac

2018-04-01

Hybrid algorithms have been recently used to solve complex single-objective optimisation problems. The ultimate goal is to find an optimised global solution by using these algorithms. Based on the existing algorithms (HP_CRO, PSO, RCCRO), this study proposes a new hybrid algorithm called MPC (Mean-PSO-CRO), which utilises a new Mean-Search Operator. By employing this new operator, the proposed algorithm improves the search ability on areas of the solution space that the other operators of previous algorithms do not explore. Specifically, the Mean-Search Operator helps find the better solutions in comparison with other algorithms. Moreover, the authors have proposed two parameters for balancing local and global search and between various types of local search, as well. In addition, three versions of this operator, which use different constraints, are introduced. The experimental results on 23 benchmark functions, which are used in previous works, show that our framework can find better optimal or close-to-optimal solutions with faster convergence speed for most of the benchmark functions, especially the high-dimensional functions. Thus, the proposed algorithm is more effective in solving single-objective optimisation problems than the other existing algorithms.

Genetic Algorithm Optimizes Q-LAW Control Parameters

NASA Technical Reports Server (NTRS)

Lee, Seungwon; von Allmen, Paul; Petropoulos, Anastassios; Terrile, Richard

2008-01-01

A document discusses a multi-objective, genetic algorithm designed to optimize Lyapunov feedback control law (Q-law) parameters in order to efficiently find Pareto-optimal solutions for low-thrust trajectories for electronic propulsion systems. These would be propellant-optimal solutions for a given flight time, or flight time optimal solutions for a given propellant requirement. The approximate solutions are used as good initial solutions for high-fidelity optimization tools. When the good initial solutions are used, the high-fidelity optimization tools quickly converge to a locally optimal solution near the initial solution. Q-law control parameters are represented as real-valued genes in the genetic algorithm. The performances of the Q-law control parameters are evaluated in the multi-objective space (flight time vs. propellant mass) and sorted by the non-dominated sorting method that assigns a better fitness value to the solutions that are dominated by a fewer number of other solutions. With the ranking result, the genetic algorithm encourages the solutions with higher fitness values to participate in the reproduction process, improving the solutions in the evolution process. The population of solutions converges to the Pareto front that is permitted within the Q-law control parameter space.

Fast Combinatorial Algorithm for the Solution of Linearly Constrained Least Squares Problems

DOEpatents

Van Benthem, Mark H.; Keenan, Michael R.

2008-11-11

A fast combinatorial algorithm can significantly reduce the computational burden when solving general equality and inequality constrained least squares problems with large numbers of observation vectors. The combinatorial algorithm provides a mathematically rigorous solution and operates at great speed by reorganizing the calculations to take advantage of the combinatorial nature of the problems to be solved. The combinatorial algorithm exploits the structure that exists in large-scale problems in order to minimize the number of arithmetic operations required to obtain a solution.

SOTXTSTREAM: Density-based self-organizing clustering of text streams.

PubMed

Bryant, Avory C; Cios, Krzysztof J

2017-01-01

A streaming data clustering algorithm is presented building upon the density-based self-organizing stream clustering algorithm SOSTREAM. Many density-based clustering algorithms are limited by their inability to identify clusters with heterogeneous density. SOSTREAM addresses this limitation through the use of local (nearest neighbor-based) density determinations. Additionally, many stream clustering algorithms use a two-phase clustering approach. In the first phase, a micro-clustering solution is maintained online, while in the second phase, the micro-clustering solution is clustered offline to produce a macro solution. By performing self-organization techniques on micro-clusters in the online phase, SOSTREAM is able to maintain a macro clustering solution in a single phase. Leveraging concepts from SOSTREAM, a new density-based self-organizing text stream clustering algorithm, SOTXTSTREAM, is presented that addresses several shortcomings of SOSTREAM. Gains in clustering performance of this new algorithm are demonstrated on several real-world text stream datasets.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Dechant, Lawrence J.

Wave packet analysis provides a connection between linear small disturbance theory and subsequent nonlinear turbulent spot flow behavior. The traditional association between linear stability analysis and nonlinear wave form is developed via the method of stationary phase whereby asymptotic (simplified) mean flow solutions are used to estimate dispersion behavior and stationary phase approximation are used to invert the associated Fourier transform. The resulting process typically requires nonlinear algebraic equations inversions that can be best performed numerically, which partially mitigates the value of the approximation as compared to a more complete, e.g. DNS or linear/nonlinear adjoint methods. To obtain a simpler,more » closed-form analytical result, the complete packet solution is modeled via approximate amplitude (linear convected kinematic wave initial value problem) and local sinusoidal (wave equation) expressions. Significantly, the initial value for the kinematic wave transport expression follows from a separable variable coefficient approximation to the linearized pressure fluctuation Poisson expression. The resulting amplitude solution, while approximate in nature, nonetheless, appears to mimic many of the global features, e.g. transitional flow intermittency and pressure fluctuation magnitude behavior. A low wave number wave packet models also recover meaningful auto-correlation and low frequency spectral behaviors.« less

A solution quality assessment method for swarm intelligence optimization algorithms.

PubMed

Zhang, Zhaojun; Wang, Gai-Ge; Zou, Kuansheng; Zhang, Jianhua

2014-01-01

Nowadays, swarm intelligence optimization has become an important optimization tool and wildly used in many fields of application. In contrast to many successful applications, the theoretical foundation is rather weak. Therefore, there are still many problems to be solved. One problem is how to quantify the performance of algorithm in finite time, that is, how to evaluate the solution quality got by algorithm for practical problems. It greatly limits the application in practical problems. A solution quality assessment method for intelligent optimization is proposed in this paper. It is an experimental analysis method based on the analysis of search space and characteristic of algorithm itself. Instead of "value performance," the "ordinal performance" is used as evaluation criteria in this method. The feasible solutions were clustered according to distance to divide solution samples into several parts. Then, solution space and "good enough" set can be decomposed based on the clustering results. Last, using relative knowledge of statistics, the evaluation result can be got. To validate the proposed method, some intelligent algorithms such as ant colony optimization (ACO), particle swarm optimization (PSO), and artificial fish swarm algorithm (AFS) were taken to solve traveling salesman problem. Computational results indicate the feasibility of proposed method.

Approximating the 0-1 Multiple Knapsack Problem with Agent Decomposition and Market Negotiation

DOE Office of Scientific and Technical Information (OSTI.GOV)

Smolinski, B.

The 0-1 multiple knapsack problem appears in many domains from financial portfolio management to cargo ship stowing. Methods for solving it range from approximate algorithms, such as greedy algorithms, to exact algorithms, such as branch and bound. Approximate algorithms have no bounds on how poorly they perform and exact algorithms can suffer from exponential time and space complexities with large data sets. This paper introduces a market model based on agent decomposition and market auctions for approximating the 0-1 multiple knapsack problem, and an algorithm that implements the model (M(x)). M(x) traverses the solution space rather than getting caught inmore » a local maximum, overcoming an inherent problem of many greedy algorithms. The use of agents ensures that infeasible solutions are not considered while traversing the solution space and that traversal of the solution space is not just random, but is also directed. M(x) is compared to a bound and bound algorithm (BB) and a simple greedy algorithm with a random shuffle (G(x)). The results suggest that M(x) is a good algorithm for approximating the 0-1 Multiple Knapsack problem. M(x) almost always found solutions that were close to optimal in a fraction of the time it took BB to run and with much less memory on large test data sets. M(x) usually performed better than G(x) on hard problems with correlated data.« less

Pricing of American style options with an adjoint process correction method

NASA Astrophysics Data System (ADS)

Jaekel, Uwe

2005-07-01

Pricing of American options is a more complicated problem than pricing of European options. In this work a formula is derived that allows the computation of the early exercise premium, i.e. the price difference between these two option types in terms of an adjoint process evolving in the reversed time direction of the original process determining the evolution of the European price. We show how this equation can be utilised to improve option price estimates from numerical schemes like finite difference or Monte Carlo methods.

A Stochastic Total Least Squares Solution of Adaptive Filtering Problem

PubMed Central

Ahmad, Noor Atinah

2014-01-01

An efficient and computationally linear algorithm is derived for total least squares solution of adaptive filtering problem, when both input and output signals are contaminated by noise. The proposed total least mean squares (TLMS) algorithm is designed by recursively computing an optimal solution of adaptive TLS problem by minimizing instantaneous value of weighted cost function. Convergence analysis of the algorithm is given to show the global convergence of the proposed algorithm, provided that the stepsize parameter is appropriately chosen. The TLMS algorithm is computationally simpler than the other TLS algorithms and demonstrates a better performance as compared with the least mean square (LMS) and normalized least mean square (NLMS) algorithms. It provides minimum mean square deviation by exhibiting better convergence in misalignment for unknown system identification under noisy inputs. PMID:24688412

Evaluation of a Pair-Wise Conflict Detection and Resolution Algorithm in a Multiple Aircraft Scenario

NASA Technical Reports Server (NTRS)

Carreno, Victor A.

2002-01-01

The KB3D algorithm is a pairwise conflict detection and resolution (CD&R) algorithm. It detects and generates trajectory vectoring for an aircraft which has been predicted to be in an airspace minima violation within a given look-ahead time. It has been proven, using mechanized theorem proving techniques, that for a pair of aircraft, KB3D produces at least one vectoring solution and that all solutions produced are correct. Although solutions produced by the algorithm are mathematically correct, they might not be physically executable by an aircraft or might not solve multiple aircraft conflicts. This paper describes a simple solution selection method which assesses all solutions generated by KB3D and determines the solution to be executed. The solution selection method and KB3D are evaluated using a simulation in which N aircraft fly in a free-flight environment and each aircraft in the simulation uses KB3D to maintain separation. Specifically, the solution selection method filters KB3D solutions which are procedurally undesirable or physically not executable and uses a predetermined criteria for selection.

A multi-level solution algorithm for steady-state Markov chains

NASA Technical Reports Server (NTRS)

Horton, Graham; Leutenegger, Scott T.

1993-01-01

A new iterative algorithm, the multi-level algorithm, for the numerical solution of steady state Markov chains is presented. The method utilizes a set of recursively coarsened representations of the original system to achieve accelerated convergence. It is motivated by multigrid methods, which are widely used for fast solution of partial differential equations. Initial results of numerical experiments are reported, showing significant reductions in computation time, often an order of magnitude or more, relative to the Gauss-Seidel and optimal SOR algorithms for a variety of test problems. The multi-level method is compared and contrasted with the iterative aggregation-disaggregation algorithm of Takahashi.

Sampling solution traces for the problem of sorting permutations by signed reversals

PubMed Central

2012-01-01

Background Traditional algorithms to solve the problem of sorting by signed reversals output just one optimal solution while the space of all optimal solutions can be huge. A so-called trace represents a group of solutions which share the same set of reversals that must be applied to sort the original permutation following a partial ordering. By using traces, we therefore can represent the set of optimal solutions in a more compact way. Algorithms for enumerating the complete set of traces of solutions were developed. However, due to their exponential complexity, their practical use is limited to small permutations. A partial enumeration of traces is a sampling of the complete set of traces and can be an alternative for the study of distinct evolutionary scenarios of big permutations. Ideally, the sampling should be done uniformly from the space of all optimal solutions. This is however conjectured to be ♯P-complete. Results We propose and evaluate three algorithms for producing a sampling of the complete set of traces that instead can be shown in practice to preserve some of the characteristics of the space of all solutions. The first algorithm (RA) performs the construction of traces through a random selection of reversals on the list of optimal 1-sequences. The second algorithm (DFALT) consists in a slight modification of an algorithm that performs the complete enumeration of traces. Finally, the third algorithm (SWA) is based on a sliding window strategy to improve the enumeration of traces. All proposed algorithms were able to enumerate traces for permutations with up to 200 elements. Conclusions We analysed the distribution of the enumerated traces with respect to their height and average reversal length. Various works indicate that the reversal length can be an important aspect in genome rearrangements. The algorithms RA and SWA show a tendency to lose traces with high average reversal length. Such traces are however rare, and qualitatively our results show that, for testable-sized permutations, the algorithms DFALT and SWA produce distributions which approximate the reversal length distributions observed with a complete enumeration of the set of traces. PMID:22704580

Co-evolution for Problem Simplification

NASA Technical Reports Server (NTRS)

Haith, Gary L.; Lohn, Jason D.; Cplombano, Silvano P.; Stassinopoulos, Dimitris

1999-01-01

This paper explores a co-evolutionary approach applicable to difficult problems with limited failure/success performance feedback. Like familiar "predator-prey" frameworks this algorithm evolves two populations of individuals - the solutions (predators) and the problems (prey). The approach extends previous work by rewarding only the problems that match their difficulty to the level of solut,ion competence. In complex problem domains with limited feedback, this "tractability constraint" helps provide an adaptive fitness gradient that, effectively differentiates the candidate solutions. The algorithm generates selective pressure toward the evolution of increasingly competent solutions by rewarding solution generality and uniqueness and problem tractability and difficulty. Relative (inverse-fitness) and absolute (static objective function) approaches to evaluating problem difficulty are explored and discussed. On a simple control task, this co-evolutionary algorithm was found to have significant advantages over a genetic algorithm with either a static fitness function or a fitness function that changes on a hand-tuned schedule.

LETTER TO THE EDITOR: Constant-time solution to the global optimization problem using Brüschweiler's ensemble search algorithm

NASA Astrophysics Data System (ADS)

Protopopescu, V.; D'Helon, C.; Barhen, J.

2003-06-01

A constant-time solution of the continuous global optimization problem (GOP) is obtained by using an ensemble algorithm. We show that under certain assumptions, the solution can be guaranteed by mapping the GOP onto a discrete unsorted search problem, whereupon Brüschweiler's ensemble search algorithm is applied. For adequate sensitivities of the measurement technique, the query complexity of the ensemble search algorithm depends linearly on the size of the function's domain. Advantages and limitations of an eventual NMR implementation are discussed.

Optimal configuration of power grid sources based on optimal particle swarm algorithm

NASA Astrophysics Data System (ADS)

Wen, Yuanhua

2018-04-01

In order to optimize the distribution problem of power grid sources, an optimized particle swarm optimization algorithm is proposed. First, the concept of multi-objective optimization and the Pareto solution set are enumerated. Then, the performance of the classical genetic algorithm, the classical particle swarm optimization algorithm and the improved particle swarm optimization algorithm are analyzed. The three algorithms are simulated respectively. Compared with the test results of each algorithm, the superiority of the algorithm in convergence and optimization performance is proved, which lays the foundation for subsequent micro-grid power optimization configuration solution.

Observation Impacts for Longer Forecast Lead-Times

NASA Astrophysics Data System (ADS)

Mahajan, R.; Gelaro, R.; Todling, R.

2013-12-01

Observation impact on forecasts evaluated using adjoint-based techniques (e.g. Langland and Baker, 2004) are limited by the validity of the assumptions underlying the forecasting model adjoint. Most applications of this approach have focused on deriving observation impacts on short-range forecasts (e.g. 24-hour) in part to stay well within linearization assumptions. The most widely used measure of observation impact relies on the availability of the analysis for verifying the forecasts. As pointed out by Gelaro et al. (2007), and more recently by Todling (2013), this introduces undesirable correlations in the measure that are likely to affect the resulting assessment of the observing system. Stappers and Barkmeijer (2012) introduced a technique that, in principle, allows extending the validity of tangent linear and corresponding adjoint models to longer lead-times, thereby reducing the correlations in the measures used for observation impact assessments. The methodology provides the means to better represent linearized models by making use of Gaussian quadrature relations to handle various underlying non-linear model trajectories. The formulation is exact for particular bi-linear dynamics; it corresponds to an approximation for general-type nonlinearities and must be tested for large atmospheric models. The present work investigates the approach of Stappers and Barkmeijer (2012)in the context of NASA's Goddard Earth Observing System Version 5 (GEOS-5) atmospheric data assimilation system (ADAS). The goal is to calculate observation impacts in the GEOS-5 ADAS for forecast lead-times of at least 48 hours in order to reduce the potential for undesirable correlations that occur at shorter forecast lead times. References [1]Langland, R. H., and N. L. Baker, 2004: Estimation of observation impact using the NRL atmospheric variational data assimilation adjoint system. Tellus, 56A, 189-201. [2] Gelaro, R., Y. Zhu, and R. M. Errico, 2007: Examination of various-order adjoint-based approximations of observation impact. Meteoroloische Zeitschrift, 16, 685-692. [3]Stappers, R. J. J., and J. Barkmeijer, 2012: Optimal linearization trajectories for tangent linear models. Q. J. R. Meteorol. Soc., 138, 170-184. [4] Todling, R. 2013: Comparing two approaches for assessing observation impact. Mon. Wea. Rev., 141, 1484-1505.

Advances in Global Adjoint Tomography -- Massive Data Assimilation

NASA Astrophysics Data System (ADS)

Ruan, Y.; Lei, W.; Bozdag, E.; Lefebvre, M. P.; Smith, J. A.; Krischer, L.; Tromp, J.

2015-12-01

Azimuthal anisotropy and anelasticity are key to understanding a myriad of processes in Earth's interior. Resolving these properties requires accurate simulations of seismic wave propagation in complex 3-D Earth models and an iterative inversion strategy. In the wake of successes in regional studies(e.g., Chen et al., 2007; Tape et al., 2009, 2010; Fichtner et al., 2009, 2010; Chen et al.,2010; Zhu et al., 2012, 2013; Chen et al., 2015), we are employing adjoint tomography based on a spectral-element method (Komatitsch & Tromp 1999, 2002) on a global scale using the supercomputer ''Titan'' at Oak Ridge National Laboratory. After 15 iterations, we have obtained a high-resolution transversely isotropic Earth model (M15) using traveltime data from 253 earthquakes. To obtain higher resolution images of the emerging new features and to prepare the inversion for azimuthal anisotropy and anelasticity, we expanded the original dataset with approximately 4,220 additional global earthquakes (Mw5.5-7.0) --occurring between 1995 and 2014-- and downloaded 300-minute-long time series for all available data archived at the IRIS Data Management Center, ORFEUS, and F-net. Ocean Bottom Seismograph data from the last decade are also included to maximize data coverage. In order to handle the huge dataset and solve the I/O bottleneck in global adjoint tomography, we implemented a python-based parallel data processing workflow based on the newly developed Adaptable Seismic Data Format (ASDF). With the help of the data selection tool MUSTANG developed by IRIS, we cleaned our dataset and assembled event-based ASDF files for parallel processing. We have started Centroid Moment Tensors (CMT) inversions for all 4,220 earthquakes with the latest model M15, and selected high-quality data for measurement. We will statistically investigate each channel using synthetic seismograms calculated in M15 for updated CMTs and identify problematic channels. In addition to data screening, we also modified the conventional multi-taper method to obtain better frequency-dependent measurements of surface-wave phase and amplitude anomalies, and therefore more accurate adjoint sources, which are particularly important for anelastic tomography. We present a summary of these data culling and processing procedures for global adjoint tomography.

Time domain viscoelastic full waveform inversion

NASA Astrophysics Data System (ADS)

Fabien-Ouellet, Gabriel; Gloaguen, Erwan; Giroux, Bernard

2017-06-01

Viscous attenuation can have a strong impact on seismic wave propagation, but it is rarely taken into account in full waveform inversion (FWI). When viscoelasticity is considered in time domain FWI, the displacement formulation of the wave equation is usually used instead of the popular velocity-stress formulation. However, inversion schemes rely on the adjoint equations, which are quite different for the velocity-stress formulation than for the displacement formulation. In this paper, we apply the adjoint state method to the isotropic viscoelastic wave equation in the velocity-stress formulation based on the generalized standard linear solid rheology. By applying linear transformations to the wave equation before deriving the adjoint state equations, we obtain two symmetric sets of partial differential equations for the forward and adjoint variables. The resulting sets of equations only differ by a sign change and can be solved by the same numerical implementation. We also investigate the crosstalk between parameter classes (velocity and attenuation) of the viscoelastic equation. More specifically, we show that the attenuation levels can be used to recover the quality factors of P and S waves, but that they are very sensitive to velocity errors. Finally, we present a synthetic example of viscoelastic FWI in the context of monitoring CO2 geological sequestration. We show that FWI based on our formulation can indeed recover P- and S-wave velocities and their attenuation levels when attenuation is high enough. Both changes in velocity and attenuation levels recovered with FWI can be used to track the CO2 plume during and after injection. Further studies are required to evaluate the performance of viscoelastic FWI on real data.

Relating health and climate impacts to grid-scale emissions using adjoint sensitivity modeling for the Climate and Clean Air Coalition

NASA Astrophysics Data System (ADS)

Henze, D. K.; Lacey, F.; Seltzer, M.; Vallack, H.; Kuylenstierna, J.; Bowman, K. W.; Anenberg, S.; Sasser, E.; Lee, C. J.; Martin, R.

2013-12-01

The Climate and Clean Air Coalition (CCAC) was initiated in 2012 to develop, understand and promote measures to reduce short lived climate forcers such as aerosol, ozone and methane. The Coalition now includes over 30 nations, and as a service to these nations is committed to providing a decision support toolkit that allows member nations to explore the benefits of a range of emissions mitigation measures in terms of the combined impacts on air quality and climate and so help in the development of their National Action Plans. Here we will present recent modeling work to support the development of the CCAC National Action Plans toolkit. Adjoint sensitivity analysis is presented as a means of efficiently relating air quality, climate and crop impacts back to changes in emissions from each species, sector and location at the grid-scale resolution of typical global air quality model applications. The GEOS-Chem adjoint model is used to estimate the damages per ton of emissions of PM2.5 related mortality, the impacts of ozone precursors on crops and ozone-related health effects, and the combined impacts of these species on regional surface temperature changes. We show how the benefits-per-emission vary spatially as a function of the surrounding environment, and how this impacts the overall benefit of sector-specific control strategies. We present initial findings for Bangladesh, as well as Mexico, Ghana and Colombia, some of the first countries to join the CCAC, and discuss general issues related to adjoint-based metrics for quantifying air quality and climate co-benefits.