Sample records for accurate approximate solutions

  1. Highly Accurate Analytical Approximate Solution to a Nonlinear Pseudo-Oscillator

    NASA Astrophysics Data System (ADS)

    Wu, Baisheng; Liu, Weijia; Lim, C. W.

    2017-07-01

    A second-order Newton method is presented to construct analytical approximate solutions to a nonlinear pseudo-oscillator in which the restoring force is inversely proportional to the dependent variable. The nonlinear equation is first expressed in a specific form, and it is then solved in two steps, a predictor and a corrector step. In each step, the harmonic balance method is used in an appropriate manner to obtain a set of linear algebraic equations. With only one simple second-order Newton iteration step, a short, explicit, and highly accurate analytical approximate solution can be derived. The approximate solutions are valid for all amplitudes of the pseudo-oscillator. Furthermore, the method incorporates second-order Taylor expansion in a natural way, and it is of significant faster convergence rate.

  2. Differential equation based method for accurate approximations in optimization

    NASA Technical Reports Server (NTRS)

    Pritchard, Jocelyn I.; Adelman, Howard M.

    1990-01-01

    This paper describes a method to efficiently and accurately approximate the effect of design changes on structural response. The key to this new method is to interpret sensitivity equations as differential equations that may be solved explicitly for closed form approximations, hence, the method is denoted the Differential Equation Based (DEB) method. Approximations were developed for vibration frequencies, mode shapes and static displacements. The DEB approximation method was applied to a cantilever beam and results compared with the commonly-used linear Taylor series approximations and exact solutions. The test calculations involved perturbing the height, width, cross-sectional area, tip mass, and bending inertia of the beam. The DEB method proved to be very accurate, and in msot cases, was more accurate than the linear Taylor series approximation. The method is applicable to simultaneous perturbation of several design variables. Also, the approximations may be used to calculate other system response quantities. For example, the approximations for displacement are used to approximate bending stresses.

  3. Differential equation based method for accurate approximations in optimization

    NASA Technical Reports Server (NTRS)

    Pritchard, Jocelyn I.; Adelman, Howard M.

    1990-01-01

    A method to efficiently and accurately approximate the effect of design changes on structural response is described. The key to this method is to interpret sensitivity equations as differential equations that may be solved explicitly for closed form approximations, hence, the method is denoted the Differential Equation Based (DEB) method. Approximations were developed for vibration frequencies, mode shapes and static displacements. The DEB approximation method was applied to a cantilever beam and results compared with the commonly-used linear Taylor series approximations and exact solutions. The test calculations involved perturbing the height, width, cross-sectional area, tip mass, and bending inertia of the beam. The DEB method proved to be very accurate, and in most cases, was more accurate than the linear Taylor series approximation. The method is applicable to simultaneous perturbation of several design variables. Also, the approximations may be used to calculate other system response quantities. For example, the approximations for displacements are used to approximate bending stresses.

  4. Exact expressions and accurate approximations for the dependences of radius and index of refraction of solutions of inorganic solutes on relative humidity

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

    Lewis, E.R.; Schwartz, S.

    2010-03-15

    Light scattering by aerosols plays an important role in Earth’s radiative balance, and quantification of this phenomenon is important in understanding and accounting for anthropogenic influences on Earth’s climate. Light scattering by an aerosol particle is determined by its radius and index of refraction, and for aerosol particles that are hygroscopic, both of these quantities vary with relative humidity RH. Here exact expressions are derived for the dependences of the radius ratio (relative to the volume-equivalent dry radius) and index of refraction on RH for aqueous solutions of single solutes. Both of these quantities depend on the apparent molal volumemore » of the solute in solution and on the practical osmotic coefficient of the solution, which in turn depend on concentration and thus implicitly on RH. Simple but accurate approximations are also presented for the RH dependences of both radius ratio and index of refraction for several atmospherically important inorganic solutes over the entire range of RH values for which these substances can exist as solution drops. For all substances considered, the radius ratio is accurate to within a few percent, and the index of refraction to within ~0.02, over this range of RH. Such parameterizations will be useful in radiation transfer models and climate models.« less

  5. Strong shock implosion, approximate solution

    NASA Astrophysics Data System (ADS)

    Fujimoto, Y.; Mishkin, E. A.; Alejaldre, C.

    1983-01-01

    The self-similar, center-bound motion of a strong spherical, or cylindrical, shock wave moving through an ideal gas with a constant, γ= cp/ cv, is considered and a linearized, approximate solution is derived. An X, Y phase plane of the self-similar solution is defined and the representative curved of the system behind the shock front is replaced by a straight line connecting the mappings of the shock front with that of its tail. The reduced pressure P(ξ), density R(ξ) and velocity U1(ξ) are found in closed, quite accurate, form. Comparison with numerically obtained results, for γ= {5}/{3} and γ= {7}/{5}, is shown.

  6. Approximate Solutions for Flow with a Stretching Boundary due to Partial Slip

    PubMed Central

    Filobello-Nino, U.; Vazquez-Leal, H.; Sarmiento-Reyes, A.; Benhammouda, B.; Jimenez-Fernandez, V. M.; Pereyra-Diaz, D.; Perez-Sesma, A.; Cervantes-Perez, J.; Huerta-Chua, J.; Sanchez-Orea, J.; Contreras-Hernandez, A. D.

    2014-01-01

    The homotopy perturbation method (HPM) is coupled with versions of Laplace-Padé and Padé methods to provide an approximate solution to the nonlinear differential equation that describes the behaviour of a flow with a stretching flat boundary due to partial slip. Comparing results between approximate and numerical solutions, we concluded that our results are capable of providing an accurate solution and are extremely efficient. PMID:27433526

  7. Approximate Solutions for Ideal Dam-Break Sediment-Laden Flows on Uniform Slopes

    NASA Astrophysics Data System (ADS)

    Ni, Yufang; Cao, Zhixian; Borthwick, Alistair; Liu, Qingquan

    2018-04-01

    Shallow water hydro-sediment-morphodynamic (SHSM) models have been applied increasingly widely in hydraulic engineering and geomorphological studies over the past few decades. Analytical and approximate solutions are usually sought to verify such models and therefore confirm their credibility. Dam-break flows are often evoked because such flows normally feature shock waves and contact discontinuities that warrant refined numerical schemes to solve. While analytical and approximate solutions to clear-water dam-break flows have been available for some time, such solutions are rare for sediment transport in dam-break flows. Here we aim to derive approximate solutions for ideal dam-break sediment-laden flows resulting from the sudden release of a finite volume of frictionless, incompressible water-sediment mixture on a uniform slope. The approximate solutions are presented for three typical sediment transport scenarios, i.e., pure advection, pure sedimentation, and concurrent entrainment and deposition. Although the cases considered in this paper are not real, the approximate solutions derived facilitate suitable benchmark tests for evaluating SHSM models, especially presently when shock waves can be numerically resolved accurately with a suite of finite volume methods, while the accuracy of the numerical solutions of contact discontinuities in sediment transport remains generally poorer.

  8. Approximate Analytical Solutions for Hypersonic Flow Over Slender Power Law Bodies

    NASA Technical Reports Server (NTRS)

    Mirels, Harold

    1959-01-01

    Approximate analytical solutions are presented for two-dimensional and axisymmetric hypersonic flow over slender power law bodies. Both zero order (M approaches infinity) and first order (small but nonvanishing values of 1/(M(Delta)(sup 2) solutions are presented, where M is free-stream Mach number and Delta is a characteristic slope. These solutions are compared with exact numerical integration of the equations of motion and appear to be accurate particularly when the shock is relatively close to the body.

  9. Approximate solutions of acoustic 3D integral equation and their application to seismic modeling and full-waveform inversion

    NASA Astrophysics Data System (ADS)

    Malovichko, M.; Khokhlov, N.; Yavich, N.; Zhdanov, M.

    2017-10-01

    Over the recent decades, a number of fast approximate solutions of Lippmann-Schwinger equation, which are more accurate than classic Born and Rytov approximations, were proposed in the field of electromagnetic modeling. Those developments could be naturally extended to acoustic and elastic fields; however, until recently, they were almost unknown in seismology. This paper presents several solutions of this kind applied to acoustic modeling for both lossy and lossless media. We evaluated the numerical merits of those methods and provide an estimation of their numerical complexity. In our numerical realization we use the matrix-free implementation of the corresponding integral operator. We study the accuracy of those approximate solutions and demonstrate, that the quasi-analytical approximation is more accurate, than the Born approximation. Further, we apply the quasi-analytical approximation to the solution of the inverse problem. It is demonstrated that, this approach improves the estimation of the data gradient, comparing to the Born approximation. The developed inversion algorithm is based on the conjugate-gradient type optimization. Numerical model study demonstrates that the quasi-analytical solution significantly reduces computation time of the seismic full-waveform inversion. We also show how the quasi-analytical approximation can be extended to the case of elastic wavefield.

  10. Scaling laws and accurate small-amplitude stationary solution for the motion of a planar vortex filament in the Cartesian form of the local induction approximation.

    PubMed

    Van Gorder, Robert A

    2013-04-01

    We provide a formulation of the local induction approximation (LIA) for the motion of a vortex filament in the Cartesian reference frame (the extrinsic coordinate system) which allows for scaling of the reference coordinate. For general monotone scalings of the reference coordinate, we derive an equation for the planar solution to the derivative nonlinear Schrödinger equation governing the LIA. We proceed to solve this equation perturbatively in small amplitude through an application of multiple-scales analysis, which allows for accurate computation of the period of the planar vortex filament. The perturbation result is shown to agree strongly with numerical simulations, and we also relate this solution back to the solution obtained in the arclength reference frame (the intrinsic coordinate system). Finally, we discuss nonmonotone coordinate scalings and their application for finding self-intersections of vortex filaments. These self-intersecting vortex filaments are likely unstable and collapse into other structures or dissipate completely.

  11. Accurate analytical periodic solution of the elliptical Kepler equation using the Adomian decomposition method

    NASA Astrophysics Data System (ADS)

    Alshaery, Aisha; Ebaid, Abdelhalim

    2017-11-01

    Kepler's equation is one of the fundamental equations in orbital mechanics. It is a transcendental equation in terms of the eccentric anomaly of a planet which orbits the Sun. Determining the position of a planet in its orbit around the Sun at a given time depends upon the solution of Kepler's equation, which we will solve in this paper by the Adomian decomposition method (ADM). Several properties of the periodicity of the obtained approximate solutions have been proved in lemmas. Our calculations demonstrated a rapid convergence of the obtained approximate solutions which are displayed in tables and graphs. Also, it has been shown in this paper that only a few terms of the Adomian decomposition series are sufficient to achieve highly accurate numerical results for any number of revolutions of the Earth around the Sun as a consequence of the periodicity property. Numerically, the four-term approximate solution coincides with the Bessel-Fourier series solution in the literature up to seven decimal places at some values of the time parameter and nine decimal places at other values. Moreover, the absolute error approaches zero using the nine term approximate Adomian solution. In addition, the approximate Adomian solutions for the eccentric anomaly have been used to show the convergence of the approximate radial distances of the Earth from the Sun for any number of revolutions. The minimal distance (perihelion) and maximal distance (aphelion) approach 147 million kilometers and 152.505 million kilometers, respectively, and these coincide with the well known results in astronomical physics. Therefore, the Adomian decomposition method is validated as an effective tool to solve Kepler's equation for elliptical orbits.

  12. An efficient computational method for the approximate solution of nonlinear Lane-Emden type equations arising in astrophysics

    NASA Astrophysics Data System (ADS)

    Singh, Harendra

    2018-04-01

    The key purpose of this article is to introduce an efficient computational method for the approximate solution of the homogeneous as well as non-homogeneous nonlinear Lane-Emden type equations. Using proposed computational method given nonlinear equation is converted into a set of nonlinear algebraic equations whose solution gives the approximate solution to the Lane-Emden type equation. Various nonlinear cases of Lane-Emden type equations like standard Lane-Emden equation, the isothermal gas spheres equation and white-dwarf equation are discussed. Results are compared with some well-known numerical methods and it is observed that our results are more accurate.

  13. A Mathematica program for the approximate analytical solution to a nonlinear undamped Duffing equation by a new approximate approach

    NASA Astrophysics Data System (ADS)

    Wu, Dongmei; Wang, Zhongcheng

    2006-03-01

    , we present a new iteration algorithm to calculate the coefficients of the Fourier series. By using this new method, the iteration procedure starts with a(x)cos(ωx)+b(x)sin(ωx), and the accuracy may be improved gradually by determining new coefficients a,a,… will be produced automatically in an one-by-one manner. In all the stage of calculation, we need only to solve a cubic equation. Using this new algorithm, we develop a Mathematica program, which demonstrates following main advantages over the previous HB method: (1) it avoids solving a set of associate nonlinear equations; (2) it is easier to be implemented into a computer program, and produces a highly accurate solution with analytical expression efficiently. It is interesting to find that, generally, for a given set of parameters, a nonlinear Duffing equation can have three independent oscillation modes. For some sets of the parameters, it can have two modes with complex displacement and one with real displacement. But in some cases, it can have three modes, all of them having real displacement. Therefore, we can divide the parameters into two classes, according to the solution property: there is only one mode with real displacement and there are three modes with real displacement. This program should be useful to study the dynamically periodic behavior of a Duffing oscillator and can provide an approximate analytical solution with high-accuracy for testing the error behavior of newly developed numerical methods with a wide range of parameters. Program summaryTitle of program:AnalyDuffing.nb Catalogue identifier:ADWR_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADWR_v1_0 Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland Licensing provisions:none Computer for which the program is designed and others on which it has been tested:the program has been designed for a microcomputer and been tested on the microcomputer. Computers:IBM PC Installations:the address

  14. Accurate ω-ψ Spectral Solution of the Singular Driven Cavity Problem

    NASA Astrophysics Data System (ADS)

    Auteri, F.; Quartapelle, L.; Vigevano, L.

    2002-08-01

    This article provides accurate spectral solutions of the driven cavity problem, calculated in the vorticity-stream function representation without smoothing the corner singularities—a prima facie impossible task. As in a recent benchmark spectral calculation by primitive variables of Botella and Peyret, closed-form contributions of the singular solution for both zero and finite Reynolds numbers are subtracted from the unknown of the problem tackled here numerically in biharmonic form. The method employed is based on a split approach to the vorticity and stream function equations, a Galerkin-Legendre approximation of the problem for the perturbation, and an evaluation of the nonlinear terms by Gauss-Legendre numerical integration. Results computed for Re=0, 100, and 1000 compare well with the benchmark steady solutions provided by the aforementioned collocation-Chebyshev projection method. The validity of the proposed singularity subtraction scheme for computing time-dependent solutions is also established.

  15. An approximate solution for a penny-shaped hydraulic fracture that accounts for fracture toughness, fluid viscosity and leak-off.

    PubMed

    Dontsov, E V

    2016-12-01

    This paper develops a closed-form approximate solution for a penny-shaped hydraulic fracture whose behaviour is determined by an interplay of three competing physical processes that are associated with fluid viscosity, fracture toughness and fluid leak-off. The primary assumption that permits one to construct the solution is that the fracture behaviour is mainly determined by the three-process multiscale tip asymptotics and the global fluid volume balance. First, the developed approximation is compared with the existing solutions for all limiting regimes of propagation. Then, a solution map, which indicates applicability regions of the limiting solutions, is constructed. It is also shown that the constructed approximation accurately captures the scaling that is associated with the transition from any one limiting solution to another. The developed approximation is tested against a reference numerical solution, showing that accuracy of the fracture width and radius predictions lie within a fraction of a per cent for a wide range of parameters. As a result, the constructed approximation provides a rapid solution for a penny-shaped hydraulic fracture, which can be used for quick fracture design calculations or as a reference solution to evaluate accuracy of various hydraulic fracture simulators.

  16. An approximate solution for a penny-shaped hydraulic fracture that accounts for fracture toughness, fluid viscosity and leak-off

    NASA Astrophysics Data System (ADS)

    Dontsov, E. V.

    2016-12-01

    This paper develops a closed-form approximate solution for a penny-shaped hydraulic fracture whose behaviour is determined by an interplay of three competing physical processes that are associated with fluid viscosity, fracture toughness and fluid leak-off. The primary assumption that permits one to construct the solution is that the fracture behaviour is mainly determined by the three-process multiscale tip asymptotics and the global fluid volume balance. First, the developed approximation is compared with the existing solutions for all limiting regimes of propagation. Then, a solution map, which indicates applicability regions of the limiting solutions, is constructed. It is also shown that the constructed approximation accurately captures the scaling that is associated with the transition from any one limiting solution to another. The developed approximation is tested against a reference numerical solution, showing that accuracy of the fracture width and radius predictions lie within a fraction of a per cent for a wide range of parameters. As a result, the constructed approximation provides a rapid solution for a penny-shaped hydraulic fracture, which can be used for quick fracture design calculations or as a reference solution to evaluate accuracy of various hydraulic fracture simulators.

  17. Accurate approximation of in-ecliptic trajectories for E-sail with constant pitch angle

    NASA Astrophysics Data System (ADS)

    Huo, Mingying; Mengali, Giovanni; Quarta, Alessandro A.

    2018-05-01

    Propellantless continuous-thrust propulsion systems, such as electric solar wind sails, may be successfully used for new space missions, especially those requiring high-energy orbit transfers. When the mass-to-thrust ratio is sufficiently large, the spacecraft trajectory is characterized by long flight times with a number of revolutions around the Sun. The corresponding mission analysis, especially when addressed within an optimal context, requires a significant amount of simulation effort. Analytical trajectories are therefore useful aids in a preliminary phase of mission design, even though exact solution are very difficult to obtain. The aim of this paper is to present an accurate, analytical, approximation of the spacecraft trajectory generated by an electric solar wind sail with a constant pitch angle, using the latest mathematical model of the thrust vector. Assuming a heliocentric circular parking orbit and a two-dimensional scenario, the simulation results show that the proposed equations are able to accurately describe the actual spacecraft trajectory for a long time interval when the propulsive acceleration magnitude is sufficiently small.

  18. Higher and lowest order mixed finite element approximation of subsurface flow problems with solutions of low regularity

    NASA Astrophysics Data System (ADS)

    Bause, Markus

    2008-02-01

    In this work we study mixed finite element approximations of Richards' equation for simulating variably saturated subsurface flow and simultaneous reactive solute transport. Whereas higher order schemes have proved their ability to approximate reliably reactive solute transport (cf., e.g. [Bause M, Knabner P. Numerical simulation of contaminant biodegradation by higher order methods and adaptive time stepping. Comput Visual Sci 7;2004:61-78]), the Raviart- Thomas mixed finite element method ( RT0) with a first order accurate flux approximation is popular for computing the underlying water flow field (cf. [Bause M, Knabner P. Computation of variably saturated subsurface flow by adaptive mixed hybrid finite element methods. Adv Water Resour 27;2004:565-581, Farthing MW, Kees CE, Miller CT. Mixed finite element methods and higher order temporal approximations for variably saturated groundwater flow. Adv Water Resour 26;2003:373-394, Starke G. Least-squares mixed finite element solution of variably saturated subsurface flow problems. SIAM J Sci Comput 21;2000:1869-1885, Younes A, Mosé R, Ackerer P, Chavent G. A new formulation of the mixed finite element method for solving elliptic and parabolic PDE with triangular elements. J Comp Phys 149;1999:148-167, Woodward CS, Dawson CN. Analysis of expanded mixed finite element methods for a nonlinear parabolic equation modeling flow into variably saturated porous media. SIAM J Numer Anal 37;2000:701-724]). This combination might be non-optimal. Higher order techniques could increase the accuracy of the flow field calculation and thereby improve the prediction of the solute transport. Here, we analyse the application of the Brezzi- Douglas- Marini element ( BDM1) with a second order accurate flux approximation to elliptic, parabolic and degenerate problems whose solutions lack the regularity that is assumed in optimal order error analyses. For the flow field calculation a superiority of the BDM1 approach to the RT0 one is

  19. Accurate spectral solutions for the parabolic and elliptic partial differential equations by the ultraspherical tau method

    NASA Astrophysics Data System (ADS)

    Doha, E. H.; Abd-Elhameed, W. M.

    2005-09-01

    We present a double ultraspherical spectral methods that allow the efficient approximate solution for the parabolic partial differential equations in a square subject to the most general inhomogeneous mixed boundary conditions. The differential equations with their boundary and initial conditions are reduced to systems of ordinary differential equations for the time-dependent expansion coefficients. These systems are greatly simplified by using tensor matrix algebra, and are solved by using the step-by-step method. Numerical applications of how to use these methods are described. Numerical results obtained compare favorably with those of the analytical solutions. Accurate double ultraspherical spectral approximations for Poisson's and Helmholtz's equations are also noted. Numerical experiments show that spectral approximation based on Chebyshev polynomials of the first kind is not always better than others based on ultraspherical polynomials.

  20. Fast and accurate implementation of Fourier spectral approximations of nonlocal diffusion operators and its applications

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

    Du, Qiang, E-mail: jyanghkbu@gmail.com; Yang, Jiang, E-mail: qd2125@columbia.edu

    This work is concerned with the Fourier spectral approximation of various integral differential equations associated with some linear nonlocal diffusion and peridynamic operators under periodic boundary conditions. For radially symmetric kernels, the nonlocal operators under consideration are diagonalizable in the Fourier space so that the main computational challenge is on the accurate and fast evaluation of their eigenvalues or Fourier symbols consisting of possibly singular and highly oscillatory integrals. For a large class of fractional power-like kernels, we propose a new approach based on reformulating the Fourier symbols both as coefficients of a series expansion and solutions of some simplemore » ODE models. We then propose a hybrid algorithm that utilizes both truncated series expansions and high order Runge–Kutta ODE solvers to provide fast evaluation of Fourier symbols in both one and higher dimensional spaces. It is shown that this hybrid algorithm is robust, efficient and accurate. As applications, we combine this hybrid spectral discretization in the spatial variables and the fourth-order exponential time differencing Runge–Kutta for temporal discretization to offer high order approximations of some nonlocal gradient dynamics including nonlocal Allen–Cahn equations, nonlocal Cahn–Hilliard equations, and nonlocal phase-field crystal models. Numerical results show the accuracy and effectiveness of the fully discrete scheme and illustrate some interesting phenomena associated with the nonlocal models.« less

  1. On the accurate long-time solution of the wave equation in exterior domains: Asymptotic expansions and corrected boundary conditions

    NASA Technical Reports Server (NTRS)

    Hagstrom, Thomas; Hariharan, S. I.; Maccamy, R. C.

    1993-01-01

    We consider the solution of scattering problems for the wave equation using approximate boundary conditions at artificial boundaries. These conditions are explicitly viewed as approximations to an exact boundary condition satisfied by the solution on the unbounded domain. We study the short and long term behavior of the error. It is provided that, in two space dimensions, no local in time, constant coefficient boundary operator can lead to accurate results uniformly in time for the class of problems we consider. A variable coefficient operator is developed which attains better accuracy (uniformly in time) than is possible with constant coefficient approximations. The theory is illustrated by numerical examples. We also analyze the proposed boundary conditions using energy methods, leading to asymptotically correct error bounds.

  2. Exact Markov chain and approximate diffusion solution for haploid genetic drift with one-way mutation.

    PubMed

    Hössjer, Ola; Tyvand, Peder A; Miloh, Touvia

    2016-02-01

    The classical Kimura solution of the diffusion equation is investigated for a haploid random mating (Wright-Fisher) model, with one-way mutations and initial-value specified by the founder population. The validity of the transient diffusion solution is checked by exact Markov chain computations, using a Jordan decomposition of the transition matrix. The conclusion is that the one-way diffusion model mostly works well, although the rate of convergence depends on the initial allele frequency and the mutation rate. The diffusion approximation is poor for mutation rates so low that the non-fixation boundary is regular. When this happens we perturb the diffusion solution around the non-fixation boundary and obtain a more accurate approximation that takes quasi-fixation of the mutant allele into account. The main application is to quantify how fast a specific genetic variant of the infinite alleles model is lost. We also discuss extensions of the quasi-fixation approach to other models with small mutation rates. Copyright © 2015 Elsevier Inc. All rights reserved.

  3. A Generalization of the Karush-Kuhn-Tucker Theorem for Approximate Solutions of Mathematical Programming Problems Based on Quadratic Approximation

    NASA Astrophysics Data System (ADS)

    Voloshinov, V. V.

    2018-03-01

    In computations related to mathematical programming problems, one often has to consider approximate, rather than exact, solutions satisfying the constraints of the problem and the optimality criterion with a certain error. For determining stopping rules for iterative procedures, in the stability analysis of solutions with respect to errors in the initial data, etc., a justified characteristic of such solutions that is independent of the numerical method used to obtain them is needed. A necessary δ-optimality condition in the smooth mathematical programming problem that generalizes the Karush-Kuhn-Tucker theorem for the case of approximate solutions is obtained. The Lagrange multipliers corresponding to the approximate solution are determined by solving an approximating quadratic programming problem.

  4. Analytical approximate solutions for a general class of nonlinear delay differential equations.

    PubMed

    Căruntu, Bogdan; Bota, Constantin

    2014-01-01

    We use the polynomial least squares method (PLSM), which allows us to compute analytical approximate polynomial solutions for a very general class of strongly nonlinear delay differential equations. The method is tested by computing approximate solutions for several applications including the pantograph equations and a nonlinear time-delay model from biology. The accuracy of the method is illustrated by a comparison with approximate solutions previously computed using other methods.

  5. Legendre-tau approximations for functional differential equations

    NASA Technical Reports Server (NTRS)

    Ito, K.; Teglas, R.

    1986-01-01

    The numerical approximation of solutions to linear retarded functional differential equations are considered using the so-called Legendre-tau method. The functional differential equation is first reformulated as a partial differential equation with a nonlocal boundary condition involving time-differentiation. The approximate solution is then represented as a truncated Legendre series with time-varying coefficients which satisfy a certain system of ordinary differential equations. The method is very easy to code and yields very accurate approximations. Convergence is established, various numerical examples are presented, and comparison between the latter and cubic spline approximation is made.

  6. Legendre-Tau approximations for functional differential equations

    NASA Technical Reports Server (NTRS)

    Ito, K.; Teglas, R.

    1983-01-01

    The numerical approximation of solutions to linear functional differential equations are considered using the so called Legendre tau method. The functional differential equation is first reformulated as a partial differential equation with a nonlocal boundary condition involving time differentiation. The approximate solution is then represented as a truncated Legendre series with time varying coefficients which satisfy a certain system of ordinary differential equations. The method is very easy to code and yields very accurate approximations. Convergence is established, various numerical examples are presented, and comparison between the latter and cubic spline approximations is made.

  7. An Accurate Analytic Approximation for Light Scattering by Non-absorbing Spherical Aerosol Particles

    NASA Astrophysics Data System (ADS)

    Lewis, E. R.

    2017-12-01

    The scattering of light by particles in the atmosphere is a ubiquitous and important phenomenon, with applications to numerous fields of science and technology. The problem of scattering of electromagnetic radiation by a uniform spherical particle can be solved by the method of Mie and Debye as a series of terms depending on the size parameter, x=2πr/λ, and the complex index of refraction, m. However, this solution does not provide insight into the dependence of the scattering on the radius of the particle, the wavelength, or the index of refraction, or how the scattering varies with relative humidity. Van de Hulst demonstrated that the scattering efficiency (the scattering cross section divided by the geometric cross section) of a non-absorbing sphere, over a wide range of particle sizes of atmospheric importance, depends not on x and m separately, but on the quantity 2x(m-1); this is the basis for the anomalous diffraction approximation. Here an analytic approximation for the scattering efficiency of a non-absorbing spherical particle is presented in terms of this new quantity that is accurate over a wide range of particle sizes of atmospheric importance and which readily displays the dependences of the scattering efficiency on particle radius, index of refraction, and wavelength. For an aerosol for which the particle size distribution is parameterized as a gamma function, this approximation also yields analytical results for the scattering coefficient and for the Ångström exponent, with the dependences of scattering properties on wavelength and index of refraction clearly displayed. This approximation provides insight into the dependence of light scattering properties on factors such as relative humidity, readily enables conversion of scattering from one index of refraction to another, and demonstrates the conditions under which the aerosol index (the product of the aerosol optical depth and the Ångström exponent) is a useful proxy for the number of cloud

  8. Approximate solutions to Mathieu's equation

    NASA Astrophysics Data System (ADS)

    Wilkinson, Samuel A.; Vogt, Nicolas; Golubev, Dmitry S.; Cole, Jared H.

    2018-06-01

    Mathieu's equation has many applications throughout theoretical physics. It is especially important to the theory of Josephson junctions, where it is equivalent to Schrödinger's equation. Mathieu's equation can be easily solved numerically, however there exists no closed-form analytic solution. Here we collect various approximations which appear throughout the physics and mathematics literature and examine their accuracy and regimes of applicability. Particular attention is paid to quantities relevant to the physics of Josephson junctions, but the arguments and notation are kept general so as to be of use to the broader physics community.

  9. ACCURATE CHEMICAL MASTER EQUATION SOLUTION USING MULTI-FINITE BUFFERS

    PubMed Central

    Cao, Youfang; Terebus, Anna; Liang, Jie

    2016-01-01

    The discrete chemical master equation (dCME) provides a fundamental framework for studying stochasticity in mesoscopic networks. Because of the multi-scale nature of many networks where reaction rates have large disparity, directly solving dCMEs is intractable due to the exploding size of the state space. It is important to truncate the state space effectively with quantified errors, so accurate solutions can be computed. It is also important to know if all major probabilistic peaks have been computed. Here we introduce the Accurate CME (ACME) algorithm for obtaining direct solutions to dCMEs. With multi-finite buffers for reducing the state space by O(n!), exact steady-state and time-evolving network probability landscapes can be computed. We further describe a theoretical framework of aggregating microstates into a smaller number of macrostates by decomposing a network into independent aggregated birth and death processes, and give an a priori method for rapidly determining steady-state truncation errors. The maximal sizes of the finite buffers for a given error tolerance can also be pre-computed without costly trial solutions of dCMEs. We show exactly computed probability landscapes of three multi-scale networks, namely, a 6-node toggle switch, 11-node phage-lambda epigenetic circuit, and 16-node MAPK cascade network, the latter two with no known solutions. We also show how probabilities of rare events can be computed from first-passage times, another class of unsolved problems challenging for simulation-based techniques due to large separations in time scales. Overall, the ACME method enables accurate and efficient solutions of the dCME for a large class of networks. PMID:27761104

  10. Accurate chemical master equation solution using multi-finite buffers

    DOE PAGES

    Cao, Youfang; Terebus, Anna; Liang, Jie

    2016-06-29

    Here, the discrete chemical master equation (dCME) provides a fundamental framework for studying stochasticity in mesoscopic networks. Because of the multiscale nature of many networks where reaction rates have a large disparity, directly solving dCMEs is intractable due to the exploding size of the state space. It is important to truncate the state space effectively with quantified errors, so accurate solutions can be computed. It is also important to know if all major probabilistic peaks have been computed. Here we introduce the accurate CME (ACME) algorithm for obtaining direct solutions to dCMEs. With multifinite buffers for reducing the state spacemore » by $O(n!)$, exact steady-state and time-evolving network probability landscapes can be computed. We further describe a theoretical framework of aggregating microstates into a smaller number of macrostates by decomposing a network into independent aggregated birth and death processes and give an a priori method for rapidly determining steady-state truncation errors. The maximal sizes of the finite buffers for a given error tolerance can also be precomputed without costly trial solutions of dCMEs. We show exactly computed probability landscapes of three multiscale networks, namely, a 6-node toggle switch, 11-node phage-lambda epigenetic circuit, and 16-node MAPK cascade network, the latter two with no known solutions. We also show how probabilities of rare events can be computed from first-passage times, another class of unsolved problems challenging for simulation-based techniques due to large separations in time scales. Overall, the ACME method enables accurate and efficient solutions of the dCME for a large class of networks.« less

  11. Accurate chemical master equation solution using multi-finite buffers

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

    Cao, Youfang; Terebus, Anna; Liang, Jie

    Here, the discrete chemical master equation (dCME) provides a fundamental framework for studying stochasticity in mesoscopic networks. Because of the multiscale nature of many networks where reaction rates have a large disparity, directly solving dCMEs is intractable due to the exploding size of the state space. It is important to truncate the state space effectively with quantified errors, so accurate solutions can be computed. It is also important to know if all major probabilistic peaks have been computed. Here we introduce the accurate CME (ACME) algorithm for obtaining direct solutions to dCMEs. With multifinite buffers for reducing the state spacemore » by $O(n!)$, exact steady-state and time-evolving network probability landscapes can be computed. We further describe a theoretical framework of aggregating microstates into a smaller number of macrostates by decomposing a network into independent aggregated birth and death processes and give an a priori method for rapidly determining steady-state truncation errors. The maximal sizes of the finite buffers for a given error tolerance can also be precomputed without costly trial solutions of dCMEs. We show exactly computed probability landscapes of three multiscale networks, namely, a 6-node toggle switch, 11-node phage-lambda epigenetic circuit, and 16-node MAPK cascade network, the latter two with no known solutions. We also show how probabilities of rare events can be computed from first-passage times, another class of unsolved problems challenging for simulation-based techniques due to large separations in time scales. Overall, the ACME method enables accurate and efficient solutions of the dCME for a large class of networks.« less

  12. The convergence rate of approximate solutions for nonlinear scalar conservation laws

    NASA Technical Reports Server (NTRS)

    Nessyahu, Haim; Tadmor, Eitan

    1991-01-01

    The convergence rate is discussed of approximate solutions for the nonlinear scalar conservation law. The linear convergence theory is extended into a weak regime. The extension is based on the usual two ingredients of stability and consistency. On the one hand, the counterexamples show that one must strengthen the linearized L(sup 2)-stability requirement. It is assumed that the approximate solutions are Lip(sup +)-stable in the sense that they satisfy a one-sided Lipschitz condition, in agreement with Oleinik's E-condition for the entropy solution. On the other hand, the lack of smoothness requires to weaken the consistency requirement, which is measured in the Lip'-(semi)norm. It is proved for Lip(sup +)-stable approximate solutions, that their Lip'convergence rate to the entropy solution is of the same order as their Lip'-consistency. The Lip'-convergence rate is then converted into stronger L(sup p) convergence rate estimates.

  13. An optimization-based approach for high-order accurate discretization of conservation laws with discontinuous solutions

    NASA Astrophysics Data System (ADS)

    Zahr, M. J.; Persson, P.-O.

    2018-07-01

    This work introduces a novel discontinuity-tracking framework for resolving discontinuous solutions of conservation laws with high-order numerical discretizations that support inter-element solution discontinuities, such as discontinuous Galerkin or finite volume methods. The proposed method aims to align inter-element boundaries with discontinuities in the solution by deforming the computational mesh. A discontinuity-aligned mesh ensures the discontinuity is represented through inter-element jumps while smooth basis functions interior to elements are only used to approximate smooth regions of the solution, thereby avoiding Gibbs' phenomena that create well-known stability issues. Therefore, very coarse high-order discretizations accurately resolve the piecewise smooth solution throughout the domain, provided the discontinuity is tracked. Central to the proposed discontinuity-tracking framework is a discrete PDE-constrained optimization formulation that simultaneously aligns the computational mesh with discontinuities in the solution and solves the discretized conservation law on this mesh. The optimization objective is taken as a combination of the deviation of the finite-dimensional solution from its element-wise average and a mesh distortion metric to simultaneously penalize Gibbs' phenomena and distorted meshes. It will be shown that our objective function satisfies two critical properties that are required for this discontinuity-tracking framework to be practical: (1) possesses a local minima at a discontinuity-aligned mesh and (2) decreases monotonically to this minimum in a neighborhood of radius approximately h / 2, whereas other popular discontinuity indicators fail to satisfy the latter. Another important contribution of this work is the observation that traditional reduced space PDE-constrained optimization solvers that repeatedly solve the conservation law at various mesh configurations are not viable in this context since severe overshoot and

  14. Direct application of Padé approximant for solving nonlinear differential equations.

    PubMed

    Vazquez-Leal, Hector; Benhammouda, Brahim; Filobello-Nino, Uriel; Sarmiento-Reyes, Arturo; Jimenez-Fernandez, Victor Manuel; Garcia-Gervacio, Jose Luis; Huerta-Chua, Jesus; Morales-Mendoza, Luis Javier; Gonzalez-Lee, Mario

    2014-01-01

    This work presents a direct procedure to apply Padé method to find approximate solutions for nonlinear differential equations. Moreover, we present some cases study showing the strength of the method to generate highly accurate rational approximate solutions compared to other semi-analytical methods. The type of tested nonlinear equations are: a highly nonlinear boundary value problem, a differential-algebraic oscillator problem, and an asymptotic problem. The high accurate handy approximations obtained by the direct application of Padé method shows the high potential if the proposed scheme to approximate a wide variety of problems. What is more, the direct application of the Padé approximant aids to avoid the previous application of an approximative method like Taylor series method, homotopy perturbation method, Adomian Decomposition method, homotopy analysis method, variational iteration method, among others, as tools to obtain a power series solutions to post-treat with the Padé approximant. 34L30.

  15. Efficient solution of parabolic equations by Krylov approximation methods

    NASA Technical Reports Server (NTRS)

    Gallopoulos, E.; Saad, Y.

    1990-01-01

    Numerical techniques for solving parabolic equations by the method of lines is addressed. The main motivation for the proposed approach is the possibility of exploiting a high degree of parallelism in a simple manner. The basic idea of the method is to approximate the action of the evolution operator on a given state vector by means of a projection process onto a Krylov subspace. Thus, the resulting approximation consists of applying an evolution operator of a very small dimension to a known vector which is, in turn, computed accurately by exploiting well-known rational approximations to the exponential. Because the rational approximation is only applied to a small matrix, the only operations required with the original large matrix are matrix-by-vector multiplications, and as a result the algorithm can easily be parallelized and vectorized. Some relevant approximation and stability issues are discussed. We present some numerical experiments with the method and compare its performance with a few explicit and implicit algorithms.

  16. An approximate analytical solution for interlaminar stresses in angle-ply laminates

    NASA Technical Reports Server (NTRS)

    Rose, Cheryl A.; Herakovich, Carl T.

    1991-01-01

    An improved approximate analytical solution for interlaminar stresses in finite width, symmetric, angle-ply laminated coupons subjected to axial loading is presented. The solution is based upon statically admissible stress fields which take into consideration local property mismatch effects and global equilibrium requirements. Unknown constants in the admissible stress states are determined through minimization of the complementary energy. Typical results are presented for through-the-thickness and interlaminar stress distributions for angle-ply laminates. It is shown that the results represent an improved approximate analytical solution for interlaminar stresses.

  17. Approximated analytical solution to an Ebola optimal control problem

    NASA Astrophysics Data System (ADS)

    Hincapié-Palacio, Doracelly; Ospina, Juan; Torres, Delfim F. M.

    2016-11-01

    An analytical expression for the optimal control of an Ebola problem is obtained. The analytical solution is found as a first-order approximation to the Pontryagin Maximum Principle via the Euler-Lagrange equation. An implementation of the method is given using the computer algebra system Maple. Our analytical solutions confirm the results recently reported in the literature using numerical methods.

  18. Accurate numerical solution of the Helmholtz equation by iterative Lanczos reduction.

    PubMed

    Ratowsky, R P; Fleck, J A

    1991-06-01

    The Lanczos recursion algorithm is used to determine forward-propagating solutions for both the paraxial and Helmholtz wave equations for longitudinally invariant refractive indices. By eigenvalue analysis it is demonstrated that the method gives extremely accurate solutions to both equations.

  19. Accelerated solution of discrete ordinates approximation to the Boltzmann transport equation via model reduction

    DOE PAGES

    Tencer, John; Carlberg, Kevin; Larsen, Marvin; ...

    2017-06-17

    Radiation heat transfer is an important phenomenon in many physical systems of practical interest. When participating media is important, the radiative transfer equation (RTE) must be solved for the radiative intensity as a function of location, time, direction, and wavelength. In many heat-transfer applications, a quasi-steady assumption is valid, thereby removing time dependence. The dependence on wavelength is often treated through a weighted sum of gray gases (WSGG) approach. The discrete ordinates method (DOM) is one of the most common methods for approximating the angular (i.e., directional) dependence. The DOM exactly solves for the radiative intensity for a finite numbermore » of discrete ordinate directions and computes approximations to integrals over the angular space using a quadrature rule; the chosen ordinate directions correspond to the nodes of this quadrature rule. This paper applies a projection-based model-reduction approach to make high-order quadrature computationally feasible for the DOM for purely absorbing applications. First, the proposed approach constructs a reduced basis from (high-fidelity) solutions of the radiative intensity computed at a relatively small number of ordinate directions. Then, the method computes inexpensive approximations of the radiative intensity at the (remaining) quadrature points of a high-order quadrature using a reduced-order model constructed from the reduced basis. Finally, this results in a much more accurate solution than might have been achieved using only the ordinate directions used to compute the reduced basis. One- and three-dimensional test problems highlight the efficiency of the proposed method.« less

  20. Accelerated solution of discrete ordinates approximation to the Boltzmann transport equation via model reduction

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

    Tencer, John; Carlberg, Kevin; Larsen, Marvin

    Radiation heat transfer is an important phenomenon in many physical systems of practical interest. When participating media is important, the radiative transfer equation (RTE) must be solved for the radiative intensity as a function of location, time, direction, and wavelength. In many heat-transfer applications, a quasi-steady assumption is valid, thereby removing time dependence. The dependence on wavelength is often treated through a weighted sum of gray gases (WSGG) approach. The discrete ordinates method (DOM) is one of the most common methods for approximating the angular (i.e., directional) dependence. The DOM exactly solves for the radiative intensity for a finite numbermore » of discrete ordinate directions and computes approximations to integrals over the angular space using a quadrature rule; the chosen ordinate directions correspond to the nodes of this quadrature rule. This paper applies a projection-based model-reduction approach to make high-order quadrature computationally feasible for the DOM for purely absorbing applications. First, the proposed approach constructs a reduced basis from (high-fidelity) solutions of the radiative intensity computed at a relatively small number of ordinate directions. Then, the method computes inexpensive approximations of the radiative intensity at the (remaining) quadrature points of a high-order quadrature using a reduced-order model constructed from the reduced basis. Finally, this results in a much more accurate solution than might have been achieved using only the ordinate directions used to compute the reduced basis. One- and three-dimensional test problems highlight the efficiency of the proposed method.« less

  1. Boundary Approximation Methods for Sloving Elliptic Problems on Unbounded Domains

    NASA Astrophysics Data System (ADS)

    Li, Zi-Cai; Mathon, Rudolf

    1990-08-01

    Boundary approximation methods with partial solutions are presented for solving a complicated problem on an unbounded domain, with both a crack singularity and a corner singularity. Also an analysis of partial solutions near the singular points is provided. These methods are easy to apply, have good stability properties, and lead to highly accurate solutions. Hence, boundary approximation methods with partial solutions are recommended for the treatment of elliptic problems on unbounded domains provided that piecewise solution expansions, in particular, asymptotic solutions near the singularities and infinity, can be found.

  2. Approximate analytical solution for induction heating of solid cylinders

    DOE PAGES

    Jankowski, Todd Andrew; Pawley, Norma Helen; Gonzales, Lindsey Michal; ...

    2015-10-20

    An approximate solution to the mathematical model for induction heating of a solid cylinder in a cylindrical induction coil is presented here. The coupled multiphysics model includes equations describing the electromagnetic field in the heated object, a heat transfer simulation to determine temperature of the heated object, and an AC circuit simulation of the induction heating power supply. A multiple-scale perturbation method is used to solve the multiphysics model. The approximate analytical solution yields simple closed-form expressions for the electromagnetic field and heat generation rate in the solid cylinder, for the equivalent impedance of the associated tank circuit, and formore » the frequency response of a variable frequency power supply driving the tank circuit. The solution developed here is validated by comparing predicted power supply frequency to both experimental measurements and calculated values from finite element analysis for heating of graphite cylinders in an induction furnace. The simple expressions from the analytical solution clearly show the functional dependence of the power supply frequency on the material properties of the load and the geometrical characteristics of the furnace installation. In conclusion, the expressions developed here provide physical insight into observations made during load signature analysis of induction heating.« less

  3. Radiative transfer in falling snow: A two-stream approximation

    NASA Astrophysics Data System (ADS)

    Koh, Gary

    1989-04-01

    Light transmission measurements through falling snow have produced results unexplainable by single scattering arguments. A two-stream approximation to radiative transfer is used to derive an analytical expression that describes the effects of multiple scattering as a function of the snow optical depth and the snow asymmetry parameter. The approximate solution is simple and it may be as accurate as the exact solution for describing the transmission measurements within the limits of experimental uncertainties.

  4. Comparison of approximate solutions to the phonon Boltzmann transport equation with the relaxation time approximation: Spherical harmonics expansions and the discrete ordinates method

    NASA Astrophysics Data System (ADS)

    Christenson, J. G.; Austin, R. A.; Phillips, R. J.

    2018-05-01

    The phonon Boltzmann transport equation is used to analyze model problems in one and two spatial dimensions, under transient and steady-state conditions. New, explicit solutions are obtained by using the P1 and P3 approximations, based on expansions in spherical harmonics, and are compared with solutions from the discrete ordinates method. For steady-state energy transfer, it is shown that analytic expressions derived using the P1 and P3 approximations agree quantitatively with the discrete ordinates method, in some cases for large Knudsen numbers, and always for Knudsen numbers less than unity. However, for time-dependent energy transfer, the PN solutions differ qualitatively from converged solutions obtained by the discrete ordinates method. Although they correctly capture the wave-like behavior of energy transfer at short times, the P1 and P3 approximations rely on one or two wave velocities, respectively, yielding abrupt, step-changes in temperature profiles that are absent when the angular dependence of the phonon velocities is captured more completely. It is shown that, with the gray approximation, the P1 approximation is formally equivalent to the so-called "hyperbolic heat equation." Overall, these results support the use of the PN approximation to find solutions to the phonon Boltzmann transport equation for steady-state conditions. Such solutions can be useful in the design and analysis of devices that involve heat transfer at nanometer length scales, where continuum-scale approaches become inaccurate.

  5. Approximate analytic solutions to coupled nonlinear Dirac equations

    DOE PAGES

    Khare, Avinash; Cooper, Fred; Saxena, Avadh

    2017-01-30

    Here, we consider the coupled nonlinear Dirac equations (NLDEs) in 1+11+1 dimensions with scalar–scalar self-interactions g 1 2/2(more » $$\\bar{ψ}$$ψ) 2 + g 2 2/2($$\\bar{Φ}$$Φ) 2 + g 2 3($$\\bar{ψ}$$ψ)($$\\bar{Φ}$$Φ) as well as vector–vector interactions g 1 2/2($$\\bar{ψ}$$γμψ)($$\\bar{ψ}$$γμψ) + g 2 2/2($$\\bar{Φ}$$γμΦ)($$\\bar{Φ}$$γμΦ) + g 2 3($$\\bar{ψ}$$γμψ)($$\\bar{Φ}$$γμΦ). Writing the two components of the assumed rest frame solution of the coupled NLDE equations in the form ψ=e –iω1tR 1cosθ,R 1sinθΦ=e –iω2tR 2cosη,R 2sinη, and assuming that θ(x),η(x) have the same functional form they had when g3 = 0, which is an approximation consistent with the conservation laws, we then find approximate analytic solutions for Ri(x) which are valid for small values of g 3 2/g 2 2 and g 3 2/g 1 2. In the nonrelativistic limit we show that both of these coupled models go over to the same coupled nonlinear Schrödinger equation for which we obtain two exact pulse solutions vanishing at x → ±∞.« less

  6. Approximate analytic solutions to coupled nonlinear Dirac equations

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

    Khare, Avinash; Cooper, Fred; Saxena, Avadh

    Here, we consider the coupled nonlinear Dirac equations (NLDEs) in 1+11+1 dimensions with scalar–scalar self-interactions g 1 2/2(more » $$\\bar{ψ}$$ψ) 2 + g 2 2/2($$\\bar{Φ}$$Φ) 2 + g 2 3($$\\bar{ψ}$$ψ)($$\\bar{Φ}$$Φ) as well as vector–vector interactions g 1 2/2($$\\bar{ψ}$$γμψ)($$\\bar{ψ}$$γμψ) + g 2 2/2($$\\bar{Φ}$$γμΦ)($$\\bar{Φ}$$γμΦ) + g 2 3($$\\bar{ψ}$$γμψ)($$\\bar{Φ}$$γμΦ). Writing the two components of the assumed rest frame solution of the coupled NLDE equations in the form ψ=e –iω1tR 1cosθ,R 1sinθΦ=e –iω2tR 2cosη,R 2sinη, and assuming that θ(x),η(x) have the same functional form they had when g3 = 0, which is an approximation consistent with the conservation laws, we then find approximate analytic solutions for Ri(x) which are valid for small values of g 3 2/g 2 2 and g 3 2/g 1 2. In the nonrelativistic limit we show that both of these coupled models go over to the same coupled nonlinear Schrödinger equation for which we obtain two exact pulse solutions vanishing at x → ±∞.« less

  7. Exact and approximate solutions for transient squeezing flow

    NASA Astrophysics Data System (ADS)

    Lang, Ji; Santhanam, Sridhar; Wu, Qianhong

    2017-10-01

    In this paper, we report two novel theoretical approaches to examine a fast-developing flow in a thin fluid gap, which is widely observed in industrial applications and biological systems. The problem is featured by a very small Reynolds number and Strouhal number, making the fluid convective acceleration negligible, while its local acceleration is not. We have developed an exact solution for this problem which shows that the flow starts with an inviscid limit when the viscous effect has no time to appear and is followed by a subsequent developing flow, in which the viscous effect continues to penetrate into the entire fluid gap. An approximate solution is also developed using a boundary layer integral method. This solution precisely captures the general behavior of the transient fluid flow process and agrees very well with the exact solution. We also performed numerical simulation using Ansys-CFX. Excellent agreement between the analytical and the numerical solutions is obtained, indicating the validity of the analytical approaches. The study presented herein fills the gap in the literature and will have a broad impact on industrial and biomedical applications.

  8. Hamilton's Principle and Approximate Solutions to Problems in Classical Mechanics

    ERIC Educational Resources Information Center

    Schlitt, D. W.

    1977-01-01

    Shows how to use the Ritz method for obtaining approximate solutions to problems expressed in variational form directly from the variational equation. Application of this method to classical mechanics is given. (MLH)

  9. An approximate solution for interlaminar stresses in laminated composites: Applied mechanics program

    NASA Technical Reports Server (NTRS)

    Rose, Cheryl A.; Herakovich, Carl T.

    1992-01-01

    An approximate solution for interlaminar stresses in finite width, laminated composites subjected to uniform extensional, and bending loads is presented. The solution is based upon the principle of minimum complementary energy and an assumed, statically admissible stress state, derived by considering local material mismatch effects and global equilibrium requirements. The stresses in each layer are approximated by polynomial functions of the thickness coordinate, multiplied by combinations of exponential functions of the in-plane coordinate, expressed in terms of fourteen unknown decay parameters. Imposing the stationary condition of the laminate complementary energy with respect to the unknown variables yields a system of fourteen non-linear algebraic equations for the parameters. Newton's method is implemented to solve this system. Once the parameters are known, the stresses can be easily determined at any point in the laminate. Results are presented for through-thickness and interlaminar stress distributions for angle-ply, cross-ply (symmetric and unsymmetric laminates), and quasi-isotropic laminates subjected to uniform extension and bending. It is shown that the solution compares well with existing finite element solutions and represents an improved approximate solution for interlaminar stresses, primarily at interfaces where global equilibrium is satisfied by the in-plane stresses, but large local mismatch in properties requires the presence of interlaminar stresses.

  10. An improved thin film approximation to accurately determine the optical conductivity of graphene from infrared transmittance

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

    Weber, J. W.; Bol, A. A.; Sanden, M. C. M. van de

    2014-07-07

    This work presents an improved thin film approximation to extract the optical conductivity from infrared transmittance in a simple yet accurate way. This approximation takes into account the incoherent reflections from the backside of the substrate. These reflections are shown to have a significant effect on the extracted optical conductivity and hence on derived parameters as carrier mobility and density. By excluding the backside reflections, the error for these parameters for typical chemical vapor deposited (CVD) graphene on a silicon substrate can be as high as 17% and 45% for the carrier mobility and density, respectively. For the mid- andmore » near-infrared, the approximation can be simplified such that the real part of the optical conductivity is extracted without the need for a parameterization of the optical conductivity. This direct extraction is shown for Fourier transform infrared (FTIR) transmittance measurements of CVD graphene on silicon in the photon energy range of 370–7000 cm{sup −1}. From the real part of the optical conductivity, the carrier density, mobility, and number of graphene layers are determined but also residue, originating from the graphene transfer, is detected. FTIR transmittance analyzed with the improved thin film approximation is shown to be a non-invasive, easy, and accurate measurement and analysis method for assessing the quality of graphene and can be used for other 2-D materials.« less

  11. Padé approximant for normal stress differences in large-amplitude oscillatory shear flow

    NASA Astrophysics Data System (ADS)

    Poungthong, P.; Saengow, C.; Giacomin, A. J.; Kolitawong, C.; Merger, D.; Wilhelm, M.

    2018-04-01

    Analytical solutions for the normal stress differences in large-amplitude oscillatory shear flow (LAOS), for continuum or molecular models, normally take the inexact form of the first few terms of a series expansion in the shear rate amplitude. Here, we improve the accuracy of these truncated expansions by replacing them with rational functions called Padé approximants. The recent advent of exact solutions in LAOS presents an opportunity to identify accurate and useful Padé approximants. For this identification, we replace the truncated expansion for the corotational Jeffreys fluid with its Padé approximants for the normal stress differences. We uncover the most accurate and useful approximant, the [3,4] approximant, and then test its accuracy against the exact solution [C. Saengow and A. J. Giacomin, "Normal stress differences from Oldroyd 8-constant framework: Exact analytical solution for large-amplitude oscillatory shear flow," Phys. Fluids 29, 121601 (2017)]. We use Ewoldt grids to show the stunning accuracy of our [3,4] approximant in LAOS. We quantify this accuracy with an objective function and then map it onto the Pipkin space. Our two applications illustrate how to use our new approximant reliably. For this, we use the Spriggs relations to generalize our best approximant to multimode, and then, we compare with measurements on molten high-density polyethylene and on dissolved polyisobutylene in isobutylene oligomer.

  12. A fast algorithm for determining bounds and accurate approximate p-values of the rank product statistic for replicate experiments.

    PubMed

    Heskes, Tom; Eisinga, Rob; Breitling, Rainer

    2014-11-21

    The rank product method is a powerful statistical technique for identifying differentially expressed molecules in replicated experiments. A critical issue in molecule selection is accurate calculation of the p-value of the rank product statistic to adequately address multiple testing. Both exact calculation and permutation and gamma approximations have been proposed to determine molecule-level significance. These current approaches have serious drawbacks as they are either computationally burdensome or provide inaccurate estimates in the tail of the p-value distribution. We derive strict lower and upper bounds to the exact p-value along with an accurate approximation that can be used to assess the significance of the rank product statistic in a computationally fast manner. The bounds and the proposed approximation are shown to provide far better accuracy over existing approximate methods in determining tail probabilities, with the slightly conservative upper bound protecting against false positives. We illustrate the proposed method in the context of a recently published analysis on transcriptomic profiling performed in blood. We provide a method to determine upper bounds and accurate approximate p-values of the rank product statistic. The proposed algorithm provides an order of magnitude increase in throughput as compared with current approaches and offers the opportunity to explore new application domains with even larger multiple testing issue. The R code is published in one of the Additional files and is available at http://www.ru.nl/publish/pages/726696/rankprodbounds.zip .

  13. How accurate is the Pearson r-from-Z approximation? A Monte Carlo simulation study.

    PubMed

    Hittner, James B; May, Kim

    2012-01-01

    The Pearson r-from-Z approximation estimates the sample correlation (as an effect size measure) from the ratio of two quantities: the standard normal deviate equivalent (Z-score) corresponding to a one-tailed p-value divided by the square root of the total (pooled) sample size. The formula has utility in meta-analytic work when reports of research contain minimal statistical information. Although simple to implement, the accuracy of the Pearson r-from-Z approximation has not been empirically evaluated. To address this omission, we performed a series of Monte Carlo simulations. Results indicated that in some cases the formula did accurately estimate the sample correlation. However, when sample size was very small (N = 10) and effect sizes were small to small-moderate (ds of 0.1 and 0.3), the Pearson r-from-Z approximation was very inaccurate. Detailed figures that provide guidance as to when the Pearson r-from-Z formula will likely yield valid inferences are presented.

  14. Approximate Solution of Time-Fractional Advection-Dispersion Equation via Fractional Variational Iteration Method

    PubMed Central

    İbiş, Birol

    2014-01-01

    This paper aims to obtain the approximate solution of time-fractional advection-dispersion equation (FADE) involving Jumarie's modification of Riemann-Liouville derivative by the fractional variational iteration method (FVIM). FVIM provides an analytical approximate solution in the form of a convergent series. Some examples are given and the results indicate that the FVIM is of high accuracy, more efficient, and more convenient for solving time FADEs. PMID:24578662

  15. The exact solutions and approximate analytic solutions of the (2 + 1)-dimensional KP equation based on symmetry method.

    PubMed

    Gai, Litao; Bilige, Sudao; Jie, Yingmo

    2016-01-01

    In this paper, we successfully obtained the exact solutions and the approximate analytic solutions of the (2 + 1)-dimensional KP equation based on the Lie symmetry, the extended tanh method and the homotopy perturbation method. In first part, we obtained the symmetries of the (2 + 1)-dimensional KP equation based on the Wu-differential characteristic set algorithm and reduced it. In the second part, we constructed the abundant exact travelling wave solutions by using the extended tanh method. These solutions are expressed by the hyperbolic functions, the trigonometric functions and the rational functions respectively. It should be noted that when the parameters are taken as special values, some solitary wave solutions are derived from the hyperbolic function solutions. Finally, we apply the homotopy perturbation method to obtain the approximate analytic solutions based on four kinds of initial conditions.

  16. APPROXIMATE AND ANALYTICAL SOLUTIONS FOR SOLUTE TRANSPORT FROM AN INJECTION WELL INTO A SINGLE FRACTURE

    EPA Science Inventory

    In dealing with problems related to land-based nuclear waste management, a number of analytical and approximate solutions were developed to quantify radionuclide transport through fractures contained in the porous formation. t has been reported that by treating the radioactive de...

  17. Error Estimates for Approximate Solutions of the Riccati Equation with Real or Complex Potentials

    NASA Astrophysics Data System (ADS)

    Finster, Felix; Smoller, Joel

    2010-09-01

    A method is presented for obtaining rigorous error estimates for approximate solutions of the Riccati equation, with real or complex potentials. Our main tool is to derive invariant region estimates for complex solutions of the Riccati equation. We explain the general strategy for applying these estimates and illustrate the method in typical examples, where the approximate solutions are obtained by gluing together WKB and Airy solutions of corresponding one-dimensional Schrödinger equations. Our method is motivated by, and has applications to, the analysis of linear wave equations in the geometry of a rotating black hole.

  18. Exact and Approximate Solutions for Transient Squeezing Flow

    NASA Astrophysics Data System (ADS)

    Lang, Ji; Santhanam, Sridhar; Wu, Qianhong

    2017-11-01

    In this paper, we report two novel theoretical approaches to examine a fast-developing flow in a thin fluid gap, which is widely observed in industrial applications and biological systems. The problem is featured by a very small Reynolds number and Strouhal number, making the fluid convective acceleration is negligible, while its local acceleration is not. We have developed an exact solution for this problem which shows that the flow starts with an inviscid limit when the viscous effect has no time to appear, and is followed by a subsequent developing flow, in which the viscous effect continues to penetrate into the entire fluid gap. An approximate solution is also developed using a boundary layer integral method. This solution precisely captures the general behavior of the transient fluid flow process, and agrees very well with the exact solution. We also performed numerical simulation using Ansys-CFX. Excellent agreement between the analytical and the numerical solutions is obtained, indicating the validity of the analytical approaches. The study presented herein fills the gap in the literature, and will have a broad impact in industrial and biomedical applications. This work is supported by National Science Foundation CBET Fluid Dynamics Program under Award #1511096, and supported by the Seed Grant from The Villanova Center for the Advancement of Sustainability in Engineering (VCASE).

  19. Double power series method for approximating cosmological perturbations

    NASA Astrophysics Data System (ADS)

    Wren, Andrew J.; Malik, Karim A.

    2017-04-01

    We introduce a double power series method for finding approximate analytical solutions for systems of differential equations commonly found in cosmological perturbation theory. The method was set out, in a noncosmological context, by Feshchenko, Shkil' and Nikolenko (FSN) in 1966, and is applicable to cases where perturbations are on subhorizon scales. The FSN method is essentially an extension of the well known Wentzel-Kramers-Brillouin (WKB) method for finding approximate analytical solutions for ordinary differential equations. The FSN method we use is applicable well beyond perturbation theory to solve systems of ordinary differential equations, linear in the derivatives, that also depend on a small parameter, which here we take to be related to the inverse wave-number. We use the FSN method to find new approximate oscillating solutions in linear order cosmological perturbation theory for a flat radiation-matter universe. Together with this model's well-known growing and decaying Mészáros solutions, these oscillating modes provide a complete set of subhorizon approximations for the metric potential, radiation and matter perturbations. Comparison with numerical solutions of the perturbation equations shows that our approximations can be made accurate to within a typical error of 1%, or better. We also set out a heuristic method for error estimation. A Mathematica notebook which implements the double power series method is made available online.

  20. Beyond mean-field approximations for accurate and computationally efficient models of on-lattice chemical kinetics

    NASA Astrophysics Data System (ADS)

    Pineda, M.; Stamatakis, M.

    2017-07-01

    Modeling the kinetics of surface catalyzed reactions is essential for the design of reactors and chemical processes. The majority of microkinetic models employ mean-field approximations, which lead to an approximate description of catalytic kinetics by assuming spatially uncorrelated adsorbates. On the other hand, kinetic Monte Carlo (KMC) methods provide a discrete-space continuous-time stochastic formulation that enables an accurate treatment of spatial correlations in the adlayer, but at a significant computation cost. In this work, we use the so-called cluster mean-field approach to develop higher order approximations that systematically increase the accuracy of kinetic models by treating spatial correlations at a progressively higher level of detail. We further demonstrate our approach on a reduced model for NO oxidation incorporating first nearest-neighbor lateral interactions and construct a sequence of approximations of increasingly higher accuracy, which we compare with KMC and mean-field. The latter is found to perform rather poorly, overestimating the turnover frequency by several orders of magnitude for this system. On the other hand, our approximations, while more computationally intense than the traditional mean-field treatment, still achieve tremendous computational savings compared to KMC simulations, thereby opening the way for employing them in multiscale modeling frameworks.

  1. Approximate analytic solutions to 3D unconfined groundwater flow within regional 2D models

    NASA Astrophysics Data System (ADS)

    Luther, K.; Haitjema, H. M.

    2000-04-01

    We present methods for finding approximate analytic solutions to three-dimensional (3D) unconfined steady state groundwater flow near partially penetrating and horizontal wells, and for combining those solutions with regional two-dimensional (2D) models. The 3D solutions use distributed singularities (analytic elements) to enforce boundary conditions on the phreatic surface and seepage faces at vertical wells, and to maintain fixed-head boundary conditions, obtained from the 2D model, at the perimeter of the 3D model. The approximate 3D solutions are analytic (continuous and differentiable) everywhere, including on the phreatic surface itself. While continuity of flow is satisfied exactly in the infinite 3D flow domain, water balance errors can occur across the phreatic surface.

  2. Approximate Solution Methods for Spectral Radiative Transfer in High Refractive Index Layers

    NASA Technical Reports Server (NTRS)

    Siegel, R.; Spuckler, C. M.

    1994-01-01

    Some ceramic materials for high temperature applications are partially transparent for radiative transfer. The refractive indices of these materials can be substantially greater than one which influences internal radiative emission and reflections. Heat transfer behavior of single and laminated layers has been obtained in the literature by numerical solutions of the radiative transfer equations coupled with heat conduction and heating at the boundaries by convection and radiation. Two-flux and diffusion methods are investigated here to obtain approximate solutions using a simpler formulation than required for exact numerical solutions. Isotropic scattering is included. The two-flux method for a single layer yields excellent results for gray and two band spectral calculations. The diffusion method yields a good approximation for spectral behavior in laminated multiple layers if the overall optical thickness is larger than about ten. A hybrid spectral model is developed using the two-flux method in the optically thin bands, and radiative diffusion in bands that are optically thick.

  3. The passage of an infinite swept airfoil through an oblique gust. [approximate solution for aerodynamic response

    NASA Technical Reports Server (NTRS)

    Adamczyk, J. L.

    1974-01-01

    An approximate solution is reported for the unsteady aerodynamic response of an infinite swept wing encountering a vertical oblique gust in a compressible stream. The approximate expressions are of closed form and do not require excessive computer storage or computation time, and further, they are in good agreement with the results of exact theory. This analysis is used to predict the unsteady aerodynamic response of a helicopter rotor blade encountering the trailing vortex from a previous blade. Significant effects of three dimensionality and compressibility are evident in the results obtained. In addition, an approximate solution for the unsteady aerodynamic forces associated with the pitching or plunging motion of a two dimensional airfoil in a subsonic stream is presented. The mathematical form of this solution approaches the incompressible solution as the Mach number vanishes, the linear transonic solution as the Mach number approaches one, and the solution predicted by piston theory as the reduced frequency becomes large.

  4. Reproduction of exact solutions of Lipkin model by nonlinear higher random-phase approximation

    NASA Astrophysics Data System (ADS)

    Terasaki, J.; Smetana, A.; Šimkovic, F.; Krivoruchenko, M. I.

    2017-10-01

    It is shown that the random-phase approximation (RPA) method with its nonlinear higher generalization, which was previously considered as approximation except for a very limited case, reproduces the exact solutions of the Lipkin model. The nonlinear higher RPA is based on an equation nonlinear on eigenvectors and includes many-particle-many-hole components in the creation operator of the excited states. We demonstrate the exact character of solutions analytically for the particle number N = 2 and numerically for N = 8. This finding indicates that the nonlinear higher RPA is equivalent to the exact Schrödinger equation.

  5. Time-Accurate Solutions of Incompressible Navier-Stokes Equations for Potential Turbopump Applications

    NASA Technical Reports Server (NTRS)

    Kiris, Cetin; Kwak, Dochan

    2001-01-01

    Two numerical procedures, one based on artificial compressibility method and the other pressure projection method, are outlined for obtaining time-accurate solutions of the incompressible Navier-Stokes equations. The performance of the two method are compared by obtaining unsteady solutions for the evolution of twin vortices behind a at plate. Calculated results are compared with experimental and other numerical results. For an un- steady ow which requires small physical time step, pressure projection method was found to be computationally efficient since it does not require any subiterations procedure. It was observed that the artificial compressibility method requires a fast convergence scheme at each physical time step in order to satisfy incompressibility condition. This was obtained by using a GMRES-ILU(0) solver in our computations. When a line-relaxation scheme was used, the time accuracy was degraded and time-accurate computations became very expensive.

  6. Exact and approximate solutions to the oblique shock equations for real-time applications

    NASA Technical Reports Server (NTRS)

    Hartley, T. T.; Brandis, R.; Mossayebi, F.

    1991-01-01

    The derivation of exact solutions for determining the characteristics of an oblique shock wave in a supersonic flow is investigated. Specifically, an explicit expression for the oblique shock angle in terms of the free stream Mach number, the centerbody deflection angle, and the ratio of the specific heats, is derived. A simpler approximate solution is obtained and compared to the exact solution. The primary objectives of obtaining these solutions is to provide a fast algorithm that can run in a real time environment.

  7. Algorithms and analytical solutions for rapidly approximating long-term dispersion from line and area sources

    NASA Astrophysics Data System (ADS)

    Barrett, Steven R. H.; Britter, Rex E.

    Predicting long-term mean pollutant concentrations in the vicinity of airports, roads and other industrial sources are frequently of concern in regulatory and public health contexts. Many emissions are represented geometrically as ground-level line or area sources. Well developed modelling tools such as AERMOD and ADMS are able to model dispersion from finite (i.e. non-point) sources with considerable accuracy, drawing upon an up-to-date understanding of boundary layer behaviour. Due to mathematical difficulties associated with line and area sources, computationally expensive numerical integration schemes have been developed. For example, some models decompose area sources into a large number of line sources orthogonal to the mean wind direction, for which an analytical (Gaussian) solution exists. Models also employ a time-series approach, which involves computing mean pollutant concentrations for every hour over one or more years of meteorological data. This can give rise to computer runtimes of several days for assessment of a site. While this may be acceptable for assessment of a single industrial complex, airport, etc., this level of computational cost precludes national or international policy assessments at the level of detail available with dispersion modelling. In this paper, we extend previous work [S.R.H. Barrett, R.E. Britter, 2008. Development of algorithms and approximations for rapid operational air quality modelling. Atmospheric Environment 42 (2008) 8105-8111] to line and area sources. We introduce approximations which allow for the development of new analytical solutions for long-term mean dispersion from line and area sources, based on hypergeometric functions. We describe how these solutions can be parameterized from a single point source run from an existing advanced dispersion model, thereby accounting for all processes modelled in the more costly algorithms. The parameterization method combined with the analytical solutions for long-term mean

  8. Strong convergence and convergence rates of approximating solutions for algebraic Riccati equations in Hilbert spaces

    NASA Technical Reports Server (NTRS)

    Ito, Kazufumi

    1987-01-01

    The linear quadratic optimal control problem on infinite time interval for linear time-invariant systems defined on Hilbert spaces is considered. The optimal control is given by a feedback form in terms of solution pi to the associated algebraic Riccati equation (ARE). A Ritz type approximation is used to obtain a sequence pi sup N of finite dimensional approximations of the solution to ARE. A sufficient condition that shows pi sup N converges strongly to pi is obtained. Under this condition, a formula is derived which can be used to obtain a rate of convergence of pi sup N to pi. The results of the Galerkin approximation is demonstrated and applied for parabolic systems and the averaging approximation for hereditary differential systems.

  9. Exact and Approximate Stability of Solutions to Traveling Salesman Problems.

    PubMed

    Niendorf, Moritz; Girard, Anouck R

    2018-02-01

    This paper presents the stability analysis of an optimal tour for the symmetric traveling salesman problem (TSP) by obtaining stability regions. The stability region of an optimal tour is the set of all cost changes for which that solution remains optimal and can be understood as the margin of optimality for a solution with respect to perturbations in the problem data. It is known that it is not possible to test in polynomial time whether an optimal tour remains optimal after the cost of an arbitrary set of edges changes. Therefore, this paper develops tractable methods to obtain under and over approximations of stability regions based on neighborhoods and relaxations. The application of the results to the two-neighborhood and the minimum 1 tree (M1T) relaxation are discussed in detail. For Euclidean TSPs, stability regions with respect to vertex location perturbations and the notion of safe radii and location criticalities are introduced. Benefits of this paper include insight into robustness properties of tours, minimum spanning trees, M1Ts, and fast methods to evaluate optimality after perturbations occur. Numerical examples are given to demonstrate the methods and achievable approximation quality.

  10. Approximate analytical solutions to the double-stance dynamics of the lossy spring-loaded inverted pendulum.

    PubMed

    Shahbazi, Mohammad; Saranlı, Uluç; Babuška, Robert; Lopes, Gabriel A D

    2016-12-05

    This paper introduces approximate time-domain solutions to the otherwise non-integrable double-stance dynamics of the 'bipedal' spring-loaded inverted pendulum (B-SLIP) in the presence of non-negligible damping. We first introduce an auxiliary system whose behavior under certain conditions is approximately equivalent to the B-SLIP in double-stance. Then, we derive approximate solutions to the dynamics of the new system following two different methods: (i) updated-momentum approach that can deal with both the lossy and lossless B-SLIP models, and (ii) perturbation-based approach following which we only derive a solution to the lossless case. The prediction performance of each method is characterized via a comprehensive numerical analysis. The derived representations are computationally very efficient compared to numerical integrations, and, hence, are suitable for online planning, increasing the autonomy of walking robots. Two application examples of walking gait control are presented. The proposed solutions can serve as instrumental tools in various fields such as control in legged robotics and human motion understanding in biomechanics.

  11. Approximate analytical solutions in the analysis of thin elastic plates

    NASA Astrophysics Data System (ADS)

    Goloskokov, Dmitriy P.; Matrosov, Alexander V.

    2018-05-01

    Two approaches to the construction of approximate analytical solutions for bending of a rectangular thin plate are presented: the superposition method based on the method of initial functions (MIF) and the one built using the Green's function in the form of orthogonal series. Comparison of two approaches is carried out by analyzing a square plate clamped along its contour. Behavior of the moment and the shear force in the neighborhood of the corner points is discussed. It is shown that both solutions give identical results at all points of the plate except for the neighborhoods of the corner points. There are differences in the values of bending moments and generalized shearing forces in the neighborhoods of the corner points.

  12. Thin airfoil theory based on approximate solution of the transonic flow equation

    NASA Technical Reports Server (NTRS)

    Spreiter, John R; Alksne, Alberta Y

    1957-01-01

    A method is presented for the approximate solution of the nonlinear equations transonic flow theory. Solutions are found for two-dimensional flows at a Mach number of 1 and for purely subsonic and purely supersonic flows. Results are obtained in closed analytic form for a large and significant class of nonlifting airfoils. At a Mach number of 1 general expressions are given for the pressure distribution on an airfoil of specified geometry and for the shape of an airfoil having a prescribed pressure distribution. Extensive comparisons are made with available data, particularly for a Mach number of 1, and with existing solutions.

  13. Stresses and deformations in cross-ply composite tubes subjected to a uniform temperature change: Elasticity and Approximate Solutions

    NASA Technical Reports Server (NTRS)

    Hyer, M. W.; Cooper, D. E.; Cohen, D.

    1985-01-01

    The effects of a uniform temperature change on the stresses and deformations of composite tubes are investigated. The accuracy of an approximate solution based on the principle of complementary virtual work is determined. Interest centers on tube response away from the ends and so a planar elasticity approach is used. For the approximate solution a piecewise linear variation of stresses with the radial coordinate is assumed. The results from the approximate solution are compared with the elasticity solution. The stress predictions agree well, particularly peak interlaminar stresses. Surprisingly, the axial deformations also agree well. This, despite the fact that the deformations predicted by the approximate solution do not satisfy the interface displacement continuity conditions required by the elasticity solution. The study shows that the axial thermal expansion coefficient of tubes with a specific number of axial and circumferential layers depends on the stacking sequence. This is in contrast to classical lamination theory which predicts the expansion to be independent of the stacking arrangement. As expected, the sign and magnitude of the peak interlaminar stresses depends on stacking sequence.

  14. Fall with Linear Drag and Wien's Displacement Law: Approximate Solution and Lambert Function

    ERIC Educational Resources Information Center

    Vial, Alexandre

    2012-01-01

    We present an approximate solution for the downward time of travel in the case of a mass falling with a linear drag force. We show how a quasi-analytical solution implying the Lambert function can be found. We also show that solving the previous problem is equivalent to the search for Wien's displacement law. These results can be of interest for…

  15. A simple approximation for the current-voltage characteristics of high-power, relativistic diodes

    DOE PAGES

    Ekdahl, Carl

    2016-06-10

    A simple approximation for the current-voltage characteristics of a relativistic electron diode is presented. The approximation is accurate from non-relativistic through relativistic electron energies. Although it is empirically developed, it has many of the fundamental properties of the exact diode solutions. Lastly, the approximation is simple enough to be remembered and worked on almost any pocket calculator, so it has proven to be quite useful on the laboratory floor.

  16. Development and application of accurate analytical models for single active electron potentials

    NASA Astrophysics Data System (ADS)

    Miller, Michelle; Jaron-Becker, Agnieszka; Becker, Andreas

    2015-05-01

    The single active electron (SAE) approximation is a theoretical model frequently employed to study scenarios in which inner-shell electrons may productively be treated as frozen spectators to a physical process of interest, and accurate analytical approximations for these potentials are sought as a useful simulation tool. Density function theory is often used to construct a SAE potential, requiring that a further approximation for the exchange correlation functional be enacted. In this study, we employ the Krieger, Li, and Iafrate (KLI) modification to the optimized-effective-potential (OEP) method to reduce the complexity of the problem to the straightforward solution of a system of linear equations through simple arguments regarding the behavior of the exchange-correlation potential in regions where a single orbital dominates. We employ this method for the solution of atomic and molecular potentials, and use the resultant curve to devise a systematic construction for highly accurate and useful analytical approximations for several systems. Supported by the U.S. Department of Energy (Grant No. DE-FG02-09ER16103), and the U.S. National Science Foundation (Graduate Research Fellowship, Grants No. PHY-1125844 and No. PHY-1068706).

  17. ACCURATE SOLUTION AND GRADIENT COMPUTATION FOR ELLIPTIC INTERFACE PROBLEMS WITH VARIABLE COEFFICIENTS

    PubMed Central

    LI, ZHILIN; JI, HAIFENG; CHEN, XIAOHONG

    2016-01-01

    A new augmented method is proposed for elliptic interface problems with a piecewise variable coefficient that has a finite jump across a smooth interface. The main motivation is not only to get a second order accurate solution but also a second order accurate gradient from each side of the interface. The key of the new method is to introduce the jump in the normal derivative of the solution as an augmented variable and re-write the interface problem as a new PDE that consists of a leading Laplacian operator plus lower order derivative terms near the interface. In this way, the leading second order derivatives jump relations are independent of the jump in the coefficient that appears only in the lower order terms after the scaling. An upwind type discretization is used for the finite difference discretization at the irregular grid points near or on the interface so that the resulting coefficient matrix is an M-matrix. A multi-grid solver is used to solve the linear system of equations and the GMRES iterative method is used to solve the augmented variable. Second order convergence for the solution and the gradient from each side of the interface has also been proved in this paper. Numerical examples for general elliptic interface problems have confirmed the theoretical analysis and efficiency of the new method. PMID:28983130

  18. Fast and Accurate Approximation to Significance Tests in Genome-Wide Association Studies

    PubMed Central

    Zhang, Yu; Liu, Jun S.

    2011-01-01

    Genome-wide association studies commonly involve simultaneous tests of millions of single nucleotide polymorphisms (SNP) for disease association. The SNPs in nearby genomic regions, however, are often highly correlated due to linkage disequilibrium (LD, a genetic term for correlation). Simple Bonferonni correction for multiple comparisons is therefore too conservative. Permutation tests, which are often employed in practice, are both computationally expensive for genome-wide studies and limited in their scopes. We present an accurate and computationally efficient method, based on Poisson de-clumping heuristics, for approximating genome-wide significance of SNP associations. Compared with permutation tests and other multiple comparison adjustment approaches, our method computes the most accurate and robust p-value adjustments for millions of correlated comparisons within seconds. We demonstrate analytically that the accuracy and the efficiency of our method are nearly independent of the sample size, the number of SNPs, and the scale of p-values to be adjusted. In addition, our method can be easily adopted to estimate false discovery rate. When applied to genome-wide SNP datasets, we observed highly variable p-value adjustment results evaluated from different genomic regions. The variation in adjustments along the genome, however, are well conserved between the European and the African populations. The p-value adjustments are significantly correlated with LD among SNPs, recombination rates, and SNP densities. Given the large variability of sequence features in the genome, we further discuss a novel approach of using SNP-specific (local) thresholds to detect genome-wide significant associations. This article has supplementary material online. PMID:22140288

  19. Approximate N-Player Nonzero-Sum Game Solution for an Uncertain Continuous Nonlinear System.

    PubMed

    Johnson, Marcus; Kamalapurkar, Rushikesh; Bhasin, Shubhendu; Dixon, Warren E

    2015-08-01

    An approximate online equilibrium solution is developed for an N -player nonzero-sum game subject to continuous-time nonlinear unknown dynamics and an infinite horizon quadratic cost. A novel actor-critic-identifier structure is used, wherein a robust dynamic neural network is used to asymptotically identify the uncertain system with additive disturbances, and a set of critic and actor NNs are used to approximate the value functions and equilibrium policies, respectively. The weight update laws for the actor neural networks (NNs) are generated using a gradient-descent method, and the critic NNs are generated by least square regression, which are both based on the modified Bellman error that is independent of the system dynamics. A Lyapunov-based stability analysis shows that uniformly ultimately bounded tracking is achieved, and a convergence analysis demonstrates that the approximate control policies converge to a neighborhood of the optimal solutions. The actor, critic, and identifier structures are implemented in real time continuously and simultaneously. Simulations on two and three player games illustrate the performance of the developed method.

  20. Estimates of the absolute error and a scheme for an approximate solution to scheduling problems

    NASA Astrophysics Data System (ADS)

    Lazarev, A. A.

    2009-02-01

    An approach is proposed for estimating absolute errors and finding approximate solutions to classical NP-hard scheduling problems of minimizing the maximum lateness for one or many machines and makespan is minimized. The concept of a metric (distance) between instances of the problem is introduced. The idea behind the approach is, given the problem instance, to construct another instance for which an optimal or approximate solution can be found at the minimum distance from the initial instance in the metric introduced. Instead of solving the original problem (instance), a set of approximating polynomially/pseudopolynomially solvable problems (instances) are considered, an instance at the minimum distance from the given one is chosen, and the resulting schedule is then applied to the original instance.

  1. Using radiance predicted by the P3 approximation in a spherical geometry to predict tissue optical properties

    NASA Astrophysics Data System (ADS)

    Dickey, Dwayne J.; Moore, Ronald B.; Tulip, John

    2001-01-01

    For photodynamic therapy of solid tumors, such as prostatic carcinoma, to be achieved, an accurate model to predict tissue parameters and light dose must be found. Presently, most analytical light dosimetry models are fluence based and are not clinically viable for tissue characterization. Other methods of predicting optical properties, such as Monet Carlo, are accurate but far too time consuming for clinical application. However, radiance predicted by the P3-Approximation, an anaylitical solution to the transport equation, may be a viable and accurate alternative. The P3-Approximation accurately predicts optical parameters in intralipid/methylene blue based phantoms in a spherical geometry. The optical parameters furnished by the radiance, when introduced into fluence predicted by both P3- Approximation and Grosjean Theory, correlate well with experimental data. The P3-Approximation also predicts the optical properties of prostate tissue, agreeing with documented optical parameters. The P3-Approximation could be the clinical tool necessary to facilitate PDT of solid tumors because of the limited number of invasive measurements required and the speed in which accurate calculations can be performed.

  2. Perturbed invariant subspaces and approximate generalized functional variable separation solution for nonlinear diffusion-convection equations with weak source

    NASA Astrophysics Data System (ADS)

    Xia, Ya-Rong; Zhang, Shun-Li; Xin, Xiang-Peng

    2018-03-01

    In this paper, we propose the concept of the perturbed invariant subspaces (PISs), and study the approximate generalized functional variable separation solution for the nonlinear diffusion-convection equation with weak source by the approximate generalized conditional symmetries (AGCSs) related to the PISs. Complete classification of the perturbed equations which admit the approximate generalized functional separable solutions (AGFSSs) is obtained. As a consequence, some AGFSSs to the resulting equations are explicitly constructed by way of examples.

  3. Approximate series solution of multi-dimensional, time fractional-order (heat-like) diffusion equations using FRDTM.

    PubMed

    Singh, Brajesh K; Srivastava, Vineet K

    2015-04-01

    The main goal of this paper is to present a new approximate series solution of the multi-dimensional (heat-like) diffusion equation with time-fractional derivative in Caputo form using a semi-analytical approach: fractional-order reduced differential transform method (FRDTM). The efficiency of FRDTM is confirmed by considering four test problems of the multi-dimensional time fractional-order diffusion equation. FRDTM is a very efficient, effective and powerful mathematical tool which provides exact or very close approximate solutions for a wide range of real-world problems arising in engineering and natural sciences, modelled in terms of differential equations.

  4. Approximate series solution of multi-dimensional, time fractional-order (heat-like) diffusion equations using FRDTM

    PubMed Central

    Singh, Brajesh K.; Srivastava, Vineet K.

    2015-01-01

    The main goal of this paper is to present a new approximate series solution of the multi-dimensional (heat-like) diffusion equation with time-fractional derivative in Caputo form using a semi-analytical approach: fractional-order reduced differential transform method (FRDTM). The efficiency of FRDTM is confirmed by considering four test problems of the multi-dimensional time fractional-order diffusion equation. FRDTM is a very efficient, effective and powerful mathematical tool which provides exact or very close approximate solutions for a wide range of real-world problems arising in engineering and natural sciences, modelled in terms of differential equations. PMID:26064639

  5. Approximate solutions for diffusive fracture-matrix transfer: Application to storage of dissolved CO 2 in fractured rocks

    DOE PAGES

    Zhou, Quanlin; Oldenburg, Curtis M.; Spangler, Lee H.; ...

    2017-01-05

    Analytical solutions with infinite exponential series are available to calculate the rate of diffusive transfer between low-permeability blocks and high-permeability zones in the subsurface. Truncation of these series is often employed by neglecting the early-time regime. Here in this paper, we present unified-form approximate solutions in which the early-time and the late-time solutions are continuous at a switchover time. The early-time solutions are based on three-term polynomial functions in terms of square root of dimensionless time, with the first coefficient dependent only on the dimensionless area-to-volume ratio. The last two coefficients are either determined analytically for isotropic blocks (e.g., spheresmore » and slabs) or obtained by fitting the exact solutions, and they solely depend on the aspect ratios for rectangular columns and parallelepipeds. For the late-time solutions, only the leading exponential term is needed for isotropic blocks, while a few additional exponential terms are needed for highly anisotropic rectangular blocks. The optimal switchover time is between 0.157 and 0.229, with highest relative approximation error less than 0.2%. The solutions are used to demonstrate the storage of dissolved CO 2 in fractured reservoirs with low-permeability matrix blocks of single and multiple shapes and sizes. These approximate solutions are building blocks for development of analytical and numerical tools for hydraulic, solute, and thermal diffusion processes in low-permeability matrix blocks.« less

  6. On existence and approximate solutions for stochastic differential equations in the framework of G-Brownian motion

    NASA Astrophysics Data System (ADS)

    Ullah, Rahman; Faizullah, Faiz

    2017-10-01

    This investigation aims at studying a Euler-Maruyama (EM) approximate solutions scheme for stochastic differential equations (SDEs) in the framework of G-Brownian motion. Subject to the growth condition, it is shown that the EM solutions Z^q(t) are bounded, in particular, Z^q(t)\\in M_G^2([t_0,T];R^n) . Letting Z( t) as a unique solution to SDEs in the G-framework and utilizing the growth and Lipschitz conditions, the convergence of Z^q(t) to Z( t) is revealed. The Burkholder-Davis-Gundy (BDG) inequalities, Hölder's inequality, Gronwall's inequality and Doobs martingale's inequality are used to derive the results. In addition, without assuming a solution of the stated SDE, we have shown that the Euler-Maruyama approximation sequence {Z^q(t)} is Cauchy in M_G^2([t_0,T];R^n) thus converges to a limit which is a unique solution to SDE in the G-framework.

  7. Nonlinear Dispersive Elastic Waves in Solids: Exact, Approximate, and Numerical Solutions

    NASA Astrophysics Data System (ADS)

    Khajehtourian, Romik

    Wave motion lies at the heart of many disciplines in the physical sciences and engineering. For example, problems and applications involving light, sound, heat, or fluid flow are all likely to involve wave dynamics at some level. A particular class of problems is concerned with the propagation of elastic waves in a solid medium, such as a fiber-reinforced composite material responding to vibratory excitations, or soil and rock admitting seismic waves moments after the onset of an earthquake, or phonon transport in a semiconducting crystal like silicon. Regardless of the type of wave, the dispersion relation provides a fundamental characterization of the elastodynamic properties of the medium. The first part of the dissertation examines the propagation of a large-amplitude elastic wave in a one-dimensional homogeneous medium with a focus on the effects of inherent nonlinearities on the dispersion relation. Considering a thin rod, where the thickness is small compared to the wavelength, an exact, closed-form formulation is presented for the treatment of two types of nonlinearity in the strain-displacement gradient relation: Green-Lagrange and Hencky. The derived relation is then verified by direct time-domain simulations, examining both instantaneous dispersion (by direct observation) and short-term, pre-breaking dispersion (by Fourier transformation). A high-order perturbation analysis is also conducted yielding an explicit analytical space-time solution, which is shown to be spectrally accurate. The results establish a perfect match between theory and simulation and reveal that regardless of the strength of the nonlinearity, the dispersion relation fully embodies all information pertaining to the nonlinear harmonic generation mechanism that unfolds as an arbitrary-profiled wave evolves in the medium. In the second part of the dissertation, the analysis is extended to a continuous periodic thin rod exhibiting multiple phases or embedded local resonators. The

  8. Solute transport in aquifers: The comeback of the advection dispersion equation and the First Order Approximation

    NASA Astrophysics Data System (ADS)

    Fiori, A.; Zarlenga, A.; Jankovic, I.; Dagan, G.

    2017-12-01

    Natural gradient steady flow of mean velocity U takes place in heterogeneous aquifers of random logconductivity Y = lnK , characterized by the normal univariate PDF f(Y) and autocorrelation ρY, of variance σY2 and horizontal integral scale I. Solute transport is quantified by the Breakthrough Curve (BTC) M at planes at distance x from the injection plane. The study builds on the extensive 3D numerical simulations of flow and transport of Jankovic et al. (2017) for different conductivity structures. The present study further explores the predictive capabilities of the Advection Dispersion Equation (ADE), with macrodispersivity αL given by the First Order Approximation (FOA), by checking in a quantitative manner its applicability. After a discussion on the suitable boundary conditions for ADE, we find that the ADE-FOA solution is a sufficiently accurate predictor for applications, the many other sources of uncertainty prevailing in practice notwithstanding. We checked by least squares and by comparison of travel time of quantiles of M that indeed the analytical Inverse Gaussian M with αL =σY2 I , is able to fit well the bulk of the simulated BTCs. It tends to underestimate the late arrival time of the thin and persistent tail. The tail is better reproduced by the semi-analytical MIMSCA model, which also allows for a physical explanation of the success of the Inverse Gaussian solution. Examination of the pertinent longitudinal mass distribution shows that it is different from the commonly used Gaussian one in the analysis of field experiments, and it captures the main features of the plume measurements of the MADE experiment. The results strengthen the confidence in the applicability of the ADE and the FOA to predicting longitudinal spreading in solute transport through heterogeneous aquifers of stationary random structure.

  9. Numerical scheme approximating solution and parameters in a beam equation

    NASA Astrophysics Data System (ADS)

    Ferdinand, Robert R.

    2003-12-01

    We present a mathematical model which describes vibration in a metallic beam about its equilibrium position. This model takes the form of a nonlinear second-order (in time) and fourth-order (in space) partial differential equation with boundary and initial conditions. A finite-element Galerkin approximation scheme is used to estimate model solution. Infinite-dimensional model parameters are then estimated numerically using an inverse method procedure which involves the minimization of a least-squares cost functional. Numerical results are presented and future work to be done is discussed.

  10. Approximate Solutions for a Self-Folding Problem of Carbon Nanotubes

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

    Y Mikata

    2006-08-22

    This paper treats approximate solutions for a self-folding problem of carbon nanotubes. It has been observed in the molecular dynamics calculations [1] that a carbon nanotube with a large aspect ratio can self-fold due to van der Waals force between the parts of the same carbon nanotube. The main issue in the self-folding problem is to determine the minimum threshold length of the carbon nanotube at which it becomes possible for the carbon nanotube to self-fold due to the van der Waals force. An approximate mathematical model based on the force method is constructed for the self-folding problem of carbonmore » nanotubes, and it is solved exactly as an elastica problem using elliptic functions. Additionally, three other mathematical models are constructed based on the energy method. As a particular example, the lower and upper estimates for the critical threshold (minimum) length are determined based on both methods for the (5,5) armchair carbon nanotube.« less

  11. A stopping criterion for the iterative solution of partial differential equations

    NASA Astrophysics Data System (ADS)

    Rao, Kaustubh; Malan, Paul; Perot, J. Blair

    2018-01-01

    A stopping criterion for iterative solution methods is presented that accurately estimates the solution error using low computational overhead. The proposed criterion uses information from prior solution changes to estimate the error. When the solution changes are noisy or stagnating it reverts to a less accurate but more robust, low-cost singular value estimate to approximate the error given the residual. This estimator can also be applied to iterative linear matrix solvers such as Krylov subspace or multigrid methods. Examples of the stopping criterion's ability to accurately estimate the non-linear and linear solution error are provided for a number of different test cases in incompressible fluid dynamics.

  12. Excitation energies from particle-particle random phase approximation with accurate optimized effective potentials

    NASA Astrophysics Data System (ADS)

    Jin, Ye; Yang, Yang; Zhang, Du; Peng, Degao; Yang, Weitao

    2017-10-01

    The optimized effective potential (OEP) that gives accurate Kohn-Sham (KS) orbitals and orbital energies can be obtained from a given reference electron density. These OEP-KS orbitals and orbital energies are used here for calculating electronic excited states with the particle-particle random phase approximation (pp-RPA). Our calculations allow the examination of pp-RPA excitation energies with the exact KS density functional theory (DFT). Various input densities are investigated. Specifically, the excitation energies using the OEP with the electron densities from the coupled-cluster singles and doubles method display the lowest mean absolute error from the reference data for the low-lying excited states. This study probes into the theoretical limit of the pp-RPA excitation energies with the exact KS-DFT orbitals and orbital energies. We believe that higher-order correlation contributions beyond the pp-RPA bare Coulomb kernel are needed in order to achieve even higher accuracy in excitation energy calculations.

  13. Intermediate boundary conditions for LOD, ADI and approximate factorization methods

    NASA Technical Reports Server (NTRS)

    Leveque, R. J.

    1985-01-01

    A general approach to determining the correct intermediate boundary conditions for dimensional splitting methods is presented. The intermediate solution U is viewed as a second order accurate approximation to a modified equation. Deriving the modified equation and using the relationship between this equation and the original equation allows us to determine the correct boundary conditions for U*. This technique is illustrated by applying it to locally one dimensional (LOD) and alternating direction implicit (ADI) methods for the heat equation in two and three space dimensions. The approximate factorization method is considered in slightly more generality.

  14. Optimal approximation of harmonic growth clusters by orthogonal polynomials

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

    Teodorescu, Razvan

    2008-01-01

    Interface dynamics in two-dimensional systems with a maximal number of conservation laws gives an accurate theoreticaI model for many physical processes, from the hydrodynamics of immiscible, viscous flows (zero surface-tension limit of Hele-Shaw flows), to the granular dynamics of hard spheres, and even diffusion-limited aggregation. Although a complete solution for the continuum case exists, efficient approximations of the boundary evolution are very useful due to their practical applications. In this article, the approximation scheme based on orthogonal polynomials with a deformed Gaussian kernel is discussed, as well as relations to potential theory.

  15. Global collocation methods for approximation and the solution of partial differential equations

    NASA Technical Reports Server (NTRS)

    Solomonoff, A.; Turkel, E.

    1986-01-01

    Polynomial interpolation methods are applied both to the approximation of functions and to the numerical solutions of hyperbolic and elliptic partial differential equations. The derivative matrix for a general sequence of the collocation points is constructed. The approximate derivative is then found by a matrix times vector multiply. The effects of several factors on the performance of these methods including the effect of different collocation points are then explored. The resolution of the schemes for both smooth functions and functions with steep gradients or discontinuities in some derivative are also studied. The accuracy when the gradients occur both near the center of the region and in the vicinity of the boundary is investigated. The importance of the aliasing limit on the resolution of the approximation is investigated in detail. Also examined is the effect of boundary treatment on the stability and accuracy of the scheme.

  16. Approximate solution to the Hopf Phi equation for isotropic homogeneous fluid turbulence

    NASA Technical Reports Server (NTRS)

    Rosen, G.

    1982-01-01

    Consistent with the observed t to the -n decay laws for isotropic homogeneous turbulence and the form of the longitudinal correlation function f(r, t) for small r, the Hopf Phi equation is shown to be satisfied approximately by an asymptotic power series in t to the -n. This solution features a self-similar universal equilibrium functional which manifests Kolmogoroff-type scaling.

  17. Exponentially accurate approximations to piece-wise smooth periodic functions

    NASA Technical Reports Server (NTRS)

    Greer, James; Banerjee, Saheb

    1995-01-01

    A family of simple, periodic basis functions with 'built-in' discontinuities are introduced, and their properties are analyzed and discussed. Some of their potential usefulness is illustrated in conjunction with the Fourier series representations of functions with discontinuities. In particular, it is demonstrated how they can be used to construct a sequence of approximations which converges exponentially in the maximum norm to a piece-wise smooth function. The theory is illustrated with several examples and the results are discussed in the context of other sequences of functions which can be used to approximate discontinuous functions.

  18. An approximate analytical solution for describing surface runoff and sediment transport over hillslope

    NASA Astrophysics Data System (ADS)

    Tao, Wanghai; Wang, Quanjiu; Lin, Henry

    2018-03-01

    Soil and water loss from farmland causes land degradation and water pollution, thus continued efforts are needed to establish mathematical model for quantitative analysis of relevant processes and mechanisms. In this study, an approximate analytical solution has been developed for overland flow model and sediment transport model, offering a simple and effective means to predict overland flow and erosion under natural rainfall conditions. In the overland flow model, the flow regime was considered to be transitional with the value of parameter β (in the kinematic wave model) approximately two. The change rate of unit discharge with distance was assumed to be constant and equal to the runoff rate at the outlet of the plane. The excess rainfall was considered to be constant under uniform rainfall conditions. The overland flow model developed can be further applied to natural rainfall conditions by treating excess rainfall intensity as constant over a small time interval. For the sediment model, the recommended values of the runoff erosion calibration constant (cr) and the splash erosion calibration constant (cf) have been given in this study so that it is easier to use the model. These recommended values are 0.15 and 0.12, respectively. Comparisons with observed results were carried out to validate the proposed analytical solution. The results showed that the approximate analytical solution developed in this paper closely matches the observed data, thus providing an alternative method of predicting runoff generation and sediment yield, and offering a more convenient method of analyzing the quantitative relationships between variables. Furthermore, the model developed in this study can be used as a theoretical basis for developing runoff and erosion control methods.

  19. Application of the weighted-density approximation to the accurate description of electron-positron correlation effects in materials

    NASA Astrophysics Data System (ADS)

    Callewaert, Vincent; Saniz, Rolando; Barbiellini, Bernardo; Bansil, Arun; Partoens, Bart

    2017-08-01

    We discuss positron-annihilation lifetimes for a set of illustrative bulk materials within the framework of the weighted-density approximation (WDA). The WDA can correctly describe electron-positron correlations in strongly inhomogeneous systems, such as surfaces, where the applicability of (semi-)local approximations is limited. We analyze the WDA in detail and show that the electrons which cannot screen external charges efficiently, such as the core electrons, cannot be treated accurately via the pair correlation of the homogeneous electron gas. We discuss how this problem can be addressed by reducing the screening in the homogeneous electron gas by adding terms depending on the gradient of the electron density. Further improvements are obtained when core electrons are treated within the LDA and the valence electron using the WDA. Finally, we discuss a semiempirical WDA-based approach in which a sum rule is imposed to reproduce the experimental lifetimes.

  20. Using trees to compute approximate solutions to ordinary differential equations exactly

    NASA Technical Reports Server (NTRS)

    Grossman, Robert

    1991-01-01

    Some recent work is reviewed which relates families of trees to symbolic algorithms for the exact computation of series which approximate solutions of ordinary differential equations. It turns out that the vector space whose basis is the set of finite, rooted trees carries a natural multiplication related to the composition of differential operators, making the space of trees an algebra. This algebraic structure can be exploited to yield a variety of algorithms for manipulating vector fields and the series and algebras they generate.

  1. Low-dimensional, morphologically accurate models of subthreshold membrane potential

    PubMed Central

    Kellems, Anthony R.; Roos, Derrick; Xiao, Nan; Cox, Steven J.

    2009-01-01

    The accurate simulation of a neuron’s ability to integrate distributed synaptic input typically requires the simultaneous solution of tens of thousands of ordinary differential equations. For, in order to understand how a cell distinguishes between input patterns we apparently need a model that is biophysically accurate down to the space scale of a single spine, i.e., 1 μm. We argue here that one can retain this highly detailed input structure while dramatically reducing the overall system dimension if one is content to accurately reproduce the associated membrane potential at a small number of places, e.g., at the site of action potential initiation, under subthreshold stimulation. The latter hypothesis permits us to approximate the active cell model with an associated quasi-active model, which in turn we reduce by both time-domain (Balanced Truncation) and frequency-domain (ℋ2 approximation of the transfer function) methods. We apply and contrast these methods on a suite of typical cells, achieving up to four orders of magnitude in dimension reduction and an associated speed-up in the simulation of dendritic democratization and resonance. We also append a threshold mechanism and indicate that this reduction has the potential to deliver an accurate quasi-integrate and fire model. PMID:19172386

  2. FAST TRACK PAPER: Non-iterative multiple-attenuation methods: linear inverse solutions to non-linear inverse problems - II. BMG approximation

    NASA Astrophysics Data System (ADS)

    Ikelle, Luc T.; Osen, Are; Amundsen, Lasse; Shen, Yunqing

    2004-12-01

    The classical linear solutions to the problem of multiple attenuation, like predictive deconvolution, τ-p filtering, or F-K filtering, are generally fast, stable, and robust compared to non-linear solutions, which are generally either iterative or in the form of a series with an infinite number of terms. These qualities have made the linear solutions more attractive to seismic data-processing practitioners. However, most linear solutions, including predictive deconvolution or F-K filtering, contain severe assumptions about the model of the subsurface and the class of free-surface multiples they can attenuate. These assumptions limit their usefulness. In a recent paper, we described an exception to this assertion for OBS data. We showed in that paper that a linear and non-iterative solution to the problem of attenuating free-surface multiples which is as accurate as iterative non-linear solutions can be constructed for OBS data. We here present a similar linear and non-iterative solution for attenuating free-surface multiples in towed-streamer data. For most practical purposes, this linear solution is as accurate as the non-linear ones.

  3. Interaction and charge transfer between dielectric spheres: Exact and approximate analytical solutions.

    PubMed

    Lindén, Fredrik; Cederquist, Henrik; Zettergren, Henning

    2016-11-21

    We present exact analytical solutions for charge transfer reactions between two arbitrarily charged hard dielectric spheres. These solutions, and the corresponding exact ones for sphere-sphere interaction energies, include sums that describe polarization effects to infinite orders in the inverse of the distance between the sphere centers. In addition, we show that these exact solutions may be approximated by much simpler analytical expressions that are useful for many practical applications. This is exemplified through calculations of Langevin type cross sections for forming a compound system of two colliding spheres and through calculations of electron transfer cross sections. We find that it is important to account for dielectric properties and finite sphere sizes in such calculations, which for example may be useful for describing the evolution, growth, and dynamics of nanometer sized dielectric objects such as molecular clusters or dust grains in different environments including astrophysical ones.

  4. A method to approximate a closest loadability limit using multiple load flow solutions

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

    Yorino, Naoto; Harada, Shigemi; Cheng, Haozhong

    A new method is proposed to approximate a closest loadability limit (CLL), or closest saddle node bifurcation point, using a pair of multiple load flow solutions. More strictly, the obtainable points by the method are the stationary points including not only CLL but also farthest and saddle points. An operating solution and a low voltage load flow solution are used to efficiently estimate the node injections at a CLL as well as the left and right eigenvectors corresponding to the zero eigenvalue of the load flow Jacobian. They can be used in monitoring loadability margin, in identification of weak spotsmore » in a power system and in the examination of an optimal control against voltage collapse. Most of the computation time of the proposed method is taken in calculating the load flow solution pair. The remaining computation time is less than that of an ordinary load flow.« less

  5. Approximate Solution to the Angular Speeds of a Nearly-Symmetric Mass-Varying Cylindrical Body

    NASA Astrophysics Data System (ADS)

    Nanjangud, Angadh; Eke, Fidelis

    2017-06-01

    This paper examines the rotational motion of a nearly axisymmetric rocket type system with uniform burn of its propellant. The asymmetry comes from a slight difference in the transverse principal moments of inertia of the system, which then results in a set of nonlinear equations of motion even when no external torque is applied to the system. It is often difficult, or even impossible, to generate analytic solutions for such equations; closed form solutions are even more difficult to obtain. In this paper, a perturbation-based approach is employed to linearize the equations of motion and generate analytic solutions. The solutions for the variables of transverse motion are analytic and a closed-form solution to the spin rate is suggested. The solutions are presented in a compact form that permits rapid computation. The approximate solutions are then applied to the torque-free motion of a typical solid rocket system and the results are found to agree with those obtained from the numerical solution of the full non-linear equations of motion of the mass varying system.

  6. Comparison of exact solution with Eikonal approximation for elastic heavy ion scattering

    NASA Technical Reports Server (NTRS)

    Dubey, Rajendra R.; Khandelwal, Govind S.; Cucinotta, Francis A.; Maung, Khin Maung

    1995-01-01

    A first-order optical potential is used to calculate the total and absorption cross sections for nucleus-nucleus scattering. The differential cross section is calculated by using a partial-wave expansion of the Lippmann-Schwinger equation in momentum space. The results are compared with solutions in the Eikonal approximation for the equivalent potential and with experimental data in the energy range from 25A to 1000A MeV.

  7. Approximate analytical solutions of a pair of coupled anharmonic oscillators

    NASA Astrophysics Data System (ADS)

    Alam, Nasir; Mandal, Swapan; Öhberg, Patrik

    2015-02-01

    The Hamiltonian and the corresponding equations of motion involving the field operators of two quartic anharmonic oscillators indirectly coupled via a linear oscillator are constructed. The approximate analytical solutions of the coupled differential equations involving the non-commuting field operators are solved up to the second order in the anharmonic coupling. In the absence of nonlinearity these solutions are used to calculate the second order variances and hence the squeezing in pure and in mixed modes. The higher order quadrature squeezing and the amplitude squared squeezing of various field modes are also investigated where the squeezing in pure and in mixed modes are found to be suppressed. Moreover, the absence of a nonlinearity prohibits the higher order quadrature and higher ordered amplitude squeezing of the input coherent states. It is established that the mere coupling of two oscillators through a third one is unable to produce any squeezing effects of input coherent light, but the presence of a nonlinear interaction may provide squeezed states and other nonclassical phenomena.

  8. An approximate stationary solution for multi-allele neutral diffusion with low mutation rates.

    PubMed

    Burden, Conrad J; Tang, Yurong

    2016-12-01

    We address the problem of determining the stationary distribution of the multi-allelic, neutral-evolution Wright-Fisher model in the diffusion limit. A full solution to this problem for an arbitrary K×K mutation rate matrix involves solving for the stationary solution of a forward Kolmogorov equation over a (K-1)-dimensional simplex, and remains intractable. In most practical situations mutations rates are slow on the scale of the diffusion limit and the solution is heavily concentrated on the corners and edges of the simplex. In this paper we present a practical approximate solution for slow mutation rates in the form of a set of line densities along the edges of the simplex. The method of solution relies on parameterising the general non-reversible rate matrix as the sum of a reversible part and a set of (K-1)(K-2)/2 independent terms corresponding to fluxes of probability along closed paths around faces of the simplex. The solution is potentially a first step in estimating non-reversible evolutionary rate matrices from observed allele frequency spectra. Copyright © 2016 Elsevier Inc. All rights reserved.

  9. Approximate solution of space and time fractional higher order phase field equation

    NASA Astrophysics Data System (ADS)

    Shamseldeen, S.

    2018-03-01

    This paper is concerned with a class of space and time fractional partial differential equation (STFDE) with Riesz derivative in space and Caputo in time. The proposed STFDE is considered as a generalization of a sixth-order partial phase field equation. We describe the application of the optimal homotopy analysis method (OHAM) to obtain an approximate solution for the suggested fractional initial value problem. An averaged-squared residual error function is defined and used to determine the optimal convergence control parameter. Two numerical examples are studied, considering periodic and non-periodic initial conditions, to justify the efficiency and the accuracy of the adopted iterative approach. The dependence of the solution on the order of the fractional derivative in space and time and model parameters is investigated.

  10. Green-Ampt approximations: A comprehensive analysis

    NASA Astrophysics Data System (ADS)

    Ali, Shakir; Islam, Adlul; Mishra, P. K.; Sikka, Alok K.

    2016-04-01

    Green-Ampt (GA) model and its modifications are widely used for simulating infiltration process. Several explicit approximate solutions to the implicit GA model have been developed with varying degree of accuracy. In this study, performance of nine explicit approximations to the GA model is compared with the implicit GA model using the published data for broad range of soil classes and infiltration time. The explicit GA models considered are Li et al. (1976) (LI), Stone et al. (1994) (ST), Salvucci and Entekhabi (1994) (SE), Parlange et al. (2002) (PA), Barry et al. (2005) (BA), Swamee et al. (2012) (SW), Ali et al. (2013) (AL), Almedeij and Esen (2014) (AE), and Vatankhah (2015) (VA). Six statistical indicators (e.g., percent relative error, maximum absolute percent relative error, average absolute percent relative errors, percent bias, index of agreement, and Nash-Sutcliffe efficiency) and relative computer computation time are used for assessing the model performance. Models are ranked based on the overall performance index (OPI). The BA model is found to be the most accurate followed by the PA and VA models for variety of soil classes and infiltration periods. The AE, SW, SE, and LI model also performed comparatively better. Based on the overall performance index, the explicit models are ranked as BA > PA > VA > LI > AE > SE > SW > ST > AL. Results of this study will be helpful in selection of accurate and simple explicit approximate GA models for solving variety of hydrological problems.

  11. Accurate modelling of unsteady flows in collapsible tubes.

    PubMed

    Marchandise, Emilie; Flaud, Patrice

    2010-01-01

    The context of this paper is the development of a general and efficient numerical haemodynamic tool to help clinicians and researchers in understanding of physiological flow phenomena. We propose an accurate one-dimensional Runge-Kutta discontinuous Galerkin (RK-DG) method coupled with lumped parameter models for the boundary conditions. The suggested model has already been successfully applied to haemodynamics in arteries and is now extended for the flow in collapsible tubes such as veins. The main difference with cardiovascular simulations is that the flow may become supercritical and elastic jumps may appear with the numerical consequence that scheme may not remain monotone if no limiting procedure is introduced. We show that our second-order RK-DG method equipped with an approximate Roe's Riemann solver and a slope-limiting procedure allows us to capture elastic jumps accurately. Moreover, this paper demonstrates that the complex physics associated with such flows is more accurately modelled than with traditional methods such as finite difference methods or finite volumes. We present various benchmark problems that show the flexibility and applicability of the numerical method. Our solutions are compared with analytical solutions when they are available and with solutions obtained using other numerical methods. Finally, to illustrate the clinical interest, we study the emptying process in a calf vein squeezed by contracting skeletal muscle in a normal and pathological subject. We compare our results with experimental simulations and discuss the sensitivity to parameters of our model.

  12. Dipole Approximation to Predict the Resonances of Dimers Composed of Dielectric Resonators for Directional Emission: Dielectric Dimers Dipole Approximation

    DOE PAGES

    Campione, Salvatore; Warne, Larry K.; Basilio, Lorena I.

    2017-09-29

    In this paper we develop a fully-retarded, dipole approximation model to estimate the effective polarizabilities of a dimer made of dielectric resonators. They are computed from the polarizabilities of the two resonators composing the dimer. We analyze the situation of full-cubes as well as split-cubes, which have been shown to exhibit overlapping electric and magnetic resonances. We compare the effective dimer polarizabilities to ones retrieved via full-wave simulations as well as ones computed via a quasi-static, dipole approximation. We observe good agreement between the fully-retarded solution and the full-wave results, whereas the quasi-static approximation is less accurate for the problemmore » at hand. The developed model can be used to predict the electric and magnetic resonances of a dimer under parallel or orthogonal (to the dimer axis) excitation. This is particularly helpful when interested in locating frequencies at which the dimer will emit directional radiation.« less

  13. Application of geometric approximation to the CPMG experiment: Two- and three-site exchange.

    PubMed

    Chao, Fa-An; Byrd, R Andrew

    2017-04-01

    The Carr-Purcell-Meiboom-Gill (CPMG) experiment is one of the most classical and well-known relaxation dispersion experiments in NMR spectroscopy, and it has been successfully applied to characterize biologically relevant conformational dynamics in many cases. Although the data analysis of the CPMG experiment for the 2-site exchange model can be facilitated by analytical solutions, the data analysis in a more complex exchange model generally requires computationally-intensive numerical analysis. Recently, a powerful computational strategy, geometric approximation, has been proposed to provide approximate numerical solutions for the adiabatic relaxation dispersion experiments where analytical solutions are neither available nor feasible. Here, we demonstrate the general potential of geometric approximation by providing a data analysis solution of the CPMG experiment for both the traditional 2-site model and a linear 3-site exchange model. The approximate numerical solution deviates less than 0.5% from the numerical solution on average, and the new approach is computationally 60,000-fold more efficient than the numerical approach. Moreover, we find that accurate dynamic parameters can be determined in most cases, and, for a range of experimental conditions, the relaxation can be assumed to follow mono-exponential decay. The method is general and applicable to any CPMG RD experiment (e.g. N, C', C α , H α , etc.) The approach forms a foundation of building solution surfaces to analyze the CPMG experiment for different models of 3-site exchange. Thus, the geometric approximation is a general strategy to analyze relaxation dispersion data in any system (biological or chemical) if the appropriate library can be built in a physically meaningful domain. Published by Elsevier Inc.

  14. A uniformly valid approximation algorithm for nonlinear ordinary singular perturbation problems with boundary layer solutions.

    PubMed

    Cengizci, Süleyman; Atay, Mehmet Tarık; Eryılmaz, Aytekin

    2016-01-01

    This paper is concerned with two-point boundary value problems for singularly perturbed nonlinear ordinary differential equations. The case when the solution only has one boundary layer is examined. An efficient method so called Successive Complementary Expansion Method (SCEM) is used to obtain uniformly valid approximations to this kind of solutions. Four test problems are considered to check the efficiency and accuracy of the proposed method. The numerical results are found in good agreement with exact and existing solutions in literature. The results confirm that SCEM has a superiority over other existing methods in terms of easy-applicability and effectiveness.

  15. An approximate JKR solution for a general contact, including rough contacts

    NASA Astrophysics Data System (ADS)

    Ciavarella, M.

    2018-05-01

    In the present note, we suggest a simple closed form approximate solution to the adhesive contact problem under the so-called JKR regime. The derivation is based on generalizing the original JKR energetic derivation assuming calculation of the strain energy in adhesiveless contact, and unloading at constant contact area. The underlying assumption is that the contact area distributions are the same as under adhesiveless conditions (for an appropriately increased normal load), so that in general the stress intensity factors will not be exactly equal at all contact edges. The solution is simply that the indentation is δ =δ1 -√{ 2 wA‧ /P″ } where w is surface energy, δ1 is the adhesiveless indentation, A‧ is the first derivative of contact area and P‧‧ the second derivative of the load with respect to δ1. The solution only requires macroscopic quantities, and not very elaborate local distributions, and is exact in many configurations like axisymmetric contacts, but also sinusoidal waves contact and correctly predicts some features of an ideal asperity model used as a test case and not as a real description of a rough contact problem. The solution permits therefore an estimate of the full solution for elastic rough solids with Gaussian multiple scales of roughness, which so far was lacking, using known adhesiveless simple results. The result turns out to depend only on rms amplitude and slopes of the surface, and as in the fractal limit, slopes would grow without limit, tends to the adhesiveless result - although in this limit the JKR model is inappropriate. The solution would also go to adhesiveless result for large rms amplitude of roughness hrms, irrespective of the small scale details, and in agreement with common sense, well known experiments and previous models by the author.

  16. An equation-free probabilistic steady-state approximation: dynamic application to the stochastic simulation of biochemical reaction networks.

    PubMed

    Salis, Howard; Kaznessis, Yiannis N

    2005-12-01

    Stochastic chemical kinetics more accurately describes the dynamics of "small" chemical systems, such as biological cells. Many real systems contain dynamical stiffness, which causes the exact stochastic simulation algorithm or other kinetic Monte Carlo methods to spend the majority of their time executing frequently occurring reaction events. Previous methods have successfully applied a type of probabilistic steady-state approximation by deriving an evolution equation, such as the chemical master equation, for the relaxed fast dynamics and using the solution of that equation to determine the slow dynamics. However, because the solution of the chemical master equation is limited to small, carefully selected, or linear reaction networks, an alternate equation-free method would be highly useful. We present a probabilistic steady-state approximation that separates the time scales of an arbitrary reaction network, detects the convergence of a marginal distribution to a quasi-steady-state, directly samples the underlying distribution, and uses those samples to accurately predict the state of the system, including the effects of the slow dynamics, at future times. The numerical method produces an accurate solution of both the fast and slow reaction dynamics while, for stiff systems, reducing the computational time by orders of magnitude. The developed theory makes no approximations on the shape or form of the underlying steady-state distribution and only assumes that it is ergodic. We demonstrate the accuracy and efficiency of the method using multiple interesting examples, including a highly nonlinear protein-protein interaction network. The developed theory may be applied to any type of kinetic Monte Carlo simulation to more efficiently simulate dynamically stiff systems, including existing exact, approximate, or hybrid stochastic simulation techniques.

  17. Legendre-Tau approximation for functional differential equations. Part 3: Eigenvalue approximations and uniform stability

    NASA Technical Reports Server (NTRS)

    Ito, K.

    1984-01-01

    The stability and convergence properties of the Legendre-tau approximation for hereditary differential systems are analyzed. A charactristic equation is derived for the eigenvalues of the resulting approximate system. As a result of this derivation the uniform exponential stability of the solution semigroup is preserved under approximation. It is the key to obtaining the convergence of approximate solutions of the algebraic Riccati equation in trace norm.

  18. A useful approximation for the flat surface impulse response

    NASA Technical Reports Server (NTRS)

    Brown, Gary S.

    1989-01-01

    The flat surface impulse response (FSIR) is a very useful quantity in computing the mean return power for near-nadir-oriented short-pulse radar altimeters. However, for very small antenna beamwidths and relatively large pointing angles, previous analytical descriptions become very difficult to compute accurately. An asymptotic approximation is developed to overcome these computational problems. Since accuracy is of key importance, a condition is developed under which this solution is within 2 percent of the exact answer. The asymptotic solution is shown to be in functional agreement with a conventional clutter power result and gives a 1.25-dB correction to this formula to account properly for the antenna-pattern variation over the illuminated area.

  19. Numerical solution of 2D-vector tomography problem using the method of approximate inverse

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

    Svetov, Ivan; Maltseva, Svetlana; Polyakova, Anna

    2016-08-10

    We propose a numerical solution of reconstruction problem of a two-dimensional vector field in a unit disk from the known values of the longitudinal and transverse ray transforms. The algorithm is based on the method of approximate inverse. Numerical simulations confirm that the proposed method yields good results of reconstruction of vector fields.

  20. Parametric study of the Orbiter rollout using an approximate solution

    NASA Technical Reports Server (NTRS)

    Garland, B. J.

    1979-01-01

    An approximate solution to the motion of the Orbiter during rollout is used to perform a parametric study of the rollout distance required by the Orbiter. The study considers the maximum expected dispersions in the landing speed and the touchdown point. These dispersions are assumed to be correlated so that a fast landing occurs before the nominal touchdown point. The maximum rollout distance is required by the maximum landing speed with a 10 knot tailwind and the center of mass at the forward limit of its longitudinal travel. The maximum weight that can be stopped within 15,000 feet on a hot day at Kennedy Space Center is 248,800 pounds. The energy absorbed by the brakes would exceed the limit for reuse of the brakes.

  1. Approximation solution of Schrodinger equation for Q-deformed Rosen-Morse using supersymmetry quantum mechanics (SUSY QM)

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

    Alemgadmi, Khaled I. K., E-mail: azozkied@yahoo.com; Suparmi; Cari

    2015-09-30

    The approximate analytical solution of Schrodinger equation for Q-Deformed Rosen-Morse potential was investigated using Supersymmetry Quantum Mechanics (SUSY QM) method. The approximate bound state energy is given in the closed form and the corresponding approximate wave function for arbitrary l-state given for ground state wave function. The first excited state obtained using upper operator and ground state wave function. The special case is given for the ground state in various number of q. The existence of Rosen-Morse potential reduce energy spectra of system. The larger value of q, the smaller energy spectra of system.

  2. On the dynamics of approximating schemes for dissipative nonlinear equations

    NASA Technical Reports Server (NTRS)

    Jones, Donald A.

    1993-01-01

    Since one can rarely write down the analytical solutions to nonlinear dissipative partial differential equations (PDE's), it is important to understand whether, and in what sense, the behavior of approximating schemes to these equations reflects the true dynamics of the original equations. Further, because standard error estimates between approximations of the true solutions coming from spectral methods - finite difference or finite element schemes, for example - and the exact solutions grow exponentially in time, this analysis provides little value in understanding the infinite time behavior of a given approximating scheme. The notion of the global attractor has been useful in quantifying the infinite time behavior of dissipative PDEs, such as the Navier-Stokes equations. Loosely speaking, the global attractor is all that remains of a sufficiently large bounded set in phase space mapped infinitely forward in time under the evolution of the PDE. Though the attractor has been shown to have some nice properties - it is compact, connected, and finite dimensional, for example - it is in general quite complicated. Nevertheless, the global attractor gives a way to understand how the infinite time behavior of approximating schemes such as the ones coming from a finite difference, finite element, or spectral method relates to that of the original PDE. Indeed, one can often show that such approximations also have a global attractor. We therefore only need to understand how the structure of the attractor for the PDE behaves under approximation. This is by no means a trivial task. Several interesting results have been obtained in this direction. However, we will not go into the details. We mention here that approximations generally lose information about the system no matter how accurate they are. There are examples that show certain parts of the attractor may be lost by arbitrary small perturbations of the original equations.

  3. Analytical approximations to the dynamics of an array of coupled DC SQUIDs

    NASA Astrophysics Data System (ADS)

    Berggren, Susan; Palacios, Antonio

    2014-04-01

    Coupled dynamical systems that operate near the onset of a bifurcation can lead, under certain conditions, to strong signal amplification effects. Over the past years we have studied this generic feature on a wide range of systems, including: magnetic and electric fields sensors, gyroscopic devices, and arrays of loops of superconducting quantum interference devices, also known as SQUIDs. In this work, we consider an array of SQUID loops connected in series as a case study to derive asymptotic analytical approximations to the exact solutions through perturbation analysis. Two approaches are considered. First, a straightforward expansion in which the non-linear parameter related to the inductance of the DC SQUID is treated as the small perturbation parameter. Second, a more accurate procedure that considers the SQUID phase dynamics as non-uniform motion on a circle. This second procedure is readily extended to the series array and it could serve as a mathematical framework to find approximate solutions to related complex systems with high-dimensionality. To the best of our knowledge, an approximate analytical solutions to an array of SQUIDs has not been reported yet in the literature.

  4. Testing approximations for non-linear gravitational clustering

    NASA Technical Reports Server (NTRS)

    Coles, Peter; Melott, Adrian L.; Shandarin, Sergei F.

    1993-01-01

    The accuracy of various analytic approximations for following the evolution of cosmological density fluctuations into the nonlinear regime is investigated. The Zel'dovich approximation is found to be consistently the best approximation scheme. It is extremely accurate for power spectra characterized by n = -1 or less; when the approximation is 'enhanced' by truncating highly nonlinear Fourier modes the approximation is excellent even for n = +1. The performance of linear theory is less spectrum-dependent, but this approximation is less accurate than the Zel'dovich one for all cases because of the failure to treat dynamics. The lognormal approximation generally provides a very poor fit to the spatial pattern.

  5. Accurate upwind methods for the Euler equations

    NASA Technical Reports Server (NTRS)

    Huynh, Hung T.

    1993-01-01

    A new class of piecewise linear methods for the numerical solution of the one-dimensional Euler equations of gas dynamics is presented. These methods are uniformly second-order accurate, and can be considered as extensions of Godunov's scheme. With an appropriate definition of monotonicity preservation for the case of linear convection, it can be shown that they preserve monotonicity. Similar to Van Leer's MUSCL scheme, they consist of two key steps: a reconstruction step followed by an upwind step. For the reconstruction step, a monotonicity constraint that preserves uniform second-order accuracy is introduced. Computational efficiency is enhanced by devising a criterion that detects the 'smooth' part of the data where the constraint is redundant. The concept and coding of the constraint are simplified by the use of the median function. A slope steepening technique, which has no effect at smooth regions and can resolve a contact discontinuity in four cells, is described. As for the upwind step, existing and new methods are applied in a manner slightly different from those in the literature. These methods are derived by approximating the Euler equations via linearization and diagonalization. At a 'smooth' interface, Harten, Lax, and Van Leer's one intermediate state model is employed. A modification for this model that can resolve contact discontinuities is presented. Near a discontinuity, either this modified model or a more accurate one, namely, Roe's flux-difference splitting. is used. The current presentation of Roe's method, via the conceptually simple flux-vector splitting, not only establishes a connection between the two splittings, but also leads to an admissibility correction with no conditional statement, and an efficient approximation to Osher's approximate Riemann solver. These reconstruction and upwind steps result in schemes that are uniformly second-order accurate and economical at smooth regions, and yield high resolution at discontinuities.

  6. Benchmark solutions for the galactic ion transport equations: Energy and spatially dependent problems

    NASA Technical Reports Server (NTRS)

    Ganapol, Barry D.; Townsend, Lawrence W.; Wilson, John W.

    1989-01-01

    Nontrivial benchmark solutions are developed for the galactic ion transport (GIT) equations in the straight-ahead approximation. These equations are used to predict potential radiation hazards in the upper atmosphere and in space. Two levels of difficulty are considered: (1) energy independent, and (2) spatially independent. The analysis emphasizes analytical methods never before applied to the GIT equations. Most of the representations derived have been numerically implemented and compared to more approximate calculations. Accurate ion fluxes are obtained (3 to 5 digits) for nontrivial sources. For monoenergetic beams, both accurate doses and fluxes are found. The benchmarks presented are useful in assessing the accuracy of transport algorithms designed to accommodate more complex radiation protection problems. In addition, these solutions can provide fast and accurate assessments of relatively simple shield configurations.

  7. Polynomial approximation of functions of matrices and its application to the solution of a general system of linear equations

    NASA Technical Reports Server (NTRS)

    Tal-Ezer, Hillel

    1987-01-01

    During the process of solving a mathematical model numerically, there is often a need to operate on a vector v by an operator which can be expressed as f(A) while A is NxN matrix (ex: exp(A), sin(A), A sup -1). Except for very simple matrices, it is impractical to construct the matrix f(A) explicitly. Usually an approximation to it is used. In the present research, an algorithm is developed which uses a polynomial approximation to f(A). It is reduced to a problem of approximating f(z) by a polynomial in z while z belongs to the domain D in the complex plane which includes all the eigenvalues of A. This problem of approximation is approached by interpolating the function f(z) in a certain set of points which is known to have some maximal properties. The approximation thus achieved is almost best. Implementing the algorithm to some practical problem is described. Since a solution to a linear system Ax = b is x= A sup -1 b, an iterative solution to it can be regarded as a polynomial approximation to f(A) = A sup -1. Implementing the algorithm in this case is also described.

  8. Systems-based decomposition schemes for the approximate solution of multi-term fractional differential equations

    NASA Astrophysics Data System (ADS)

    Ford, Neville J.; Connolly, Joseph A.

    2009-07-01

    We give a comparison of the efficiency of three alternative decomposition schemes for the approximate solution of multi-term fractional differential equations using the Caputo form of the fractional derivative. The schemes we compare are based on conversion of the original problem into a system of equations. We review alternative approaches and consider how the most appropriate numerical scheme may be chosen to solve a particular equation.

  9. Approximate solutions for radial travel time and capture zone in unconfined aquifers.

    PubMed

    Zhou, Yangxiao; Haitjema, Henk

    2012-01-01

    Radial time-of-travel (TOT) capture zones have been evaluated for unconfined aquifers with and without recharge. The solutions of travel time for unconfined aquifers are rather complex and have been replaced with much simpler approximate solutions without significant loss of accuracy in most practical cases. The current "volumetric method" for calculating the radius of a TOT capture zone assumes no recharge and a constant aquifer thickness. It was found that for unconfined aquifers without recharge, the volumetric method leads to a smaller and less protective wellhead protection zone when ignoring drawdowns. However, if the saturated thickness near the well is used in the volumetric method a larger more protective TOT capture zone is obtained. The same is true when the volumetric method is used in the presence of recharge. However, for that case it leads to unreasonableness over the prediction of a TOT capture zone of 5 years or more. © 2011, The Author(s). Ground Water © 2011, National Ground Water Association.

  10. Extracting Time-Accurate Acceleration Vectors From Nontrivial Accelerometer Arrangements.

    PubMed

    Franck, Jennifer A; Blume, Janet; Crisco, Joseph J; Franck, Christian

    2015-09-01

    Sports-related concussions are of significant concern in many impact sports, and their detection relies on accurate measurements of the head kinematics during impact. Among the most prevalent recording technologies are videography, and more recently, the use of single-axis accelerometers mounted in a helmet, such as the HIT system. Successful extraction of the linear and angular impact accelerations depends on an accurate analysis methodology governed by the equations of motion. Current algorithms are able to estimate the magnitude of acceleration and hit location, but make assumptions about the hit orientation and are often limited in the position and/or orientation of the accelerometers. The newly formulated algorithm presented in this manuscript accurately extracts the full linear and rotational acceleration vectors from a broad arrangement of six single-axis accelerometers directly from the governing set of kinematic equations. The new formulation linearizes the nonlinear centripetal acceleration term with a finite-difference approximation and provides a fast and accurate solution for all six components of acceleration over long time periods (>250 ms). The approximation of the nonlinear centripetal acceleration term provides an accurate computation of the rotational velocity as a function of time and allows for reconstruction of a multiple-impact signal. Furthermore, the algorithm determines the impact location and orientation and can distinguish between glancing, high rotational velocity impacts, or direct impacts through the center of mass. Results are shown for ten simulated impact locations on a headform geometry computed with three different accelerometer configurations in varying degrees of signal noise. Since the algorithm does not require simplifications of the actual impacted geometry, the impact vector, or a specific arrangement of accelerometer orientations, it can be easily applied to many impact investigations in which accurate kinematics need

  11. On the Accuracy of Double Scattering Approximation for Atmospheric Polarization Computations

    NASA Technical Reports Server (NTRS)

    Korkin, Sergey V.; Lyapustin, Alexei I.; Marshak, Alexander L.

    2011-01-01

    Interpretation of multi-angle spectro-polarimetric data in remote sensing of atmospheric aerosols require fast and accurate methods of solving the vector radiative transfer equation (VRTE). The single and double scattering approximations could provide an analytical framework for the inversion algorithms and are relatively fast, however accuracy assessments of these approximations for the aerosol atmospheres in the atmospheric window channels have been missing. This paper provides such analysis for a vertically homogeneous aerosol atmosphere with weak and strong asymmetry of scattering. In both cases, the double scattering approximation gives a high accuracy result (relative error approximately 0.2%) only for the low optical path - 10(sup -2) As the error rapidly grows with optical thickness, a full VRTE solution is required for the practical remote sensing analysis. It is shown that the scattering anisotropy is not important at low optical thicknesses neither for reflected nor for transmitted polarization components of radiation.

  12. Accurate and Efficient Parallel Implementation of an Effective Linear-Scaling Direct Random Phase Approximation Method.

    PubMed

    Graf, Daniel; Beuerle, Matthias; Schurkus, Henry F; Luenser, Arne; Savasci, Gökcen; Ochsenfeld, Christian

    2018-05-08

    An efficient algorithm for calculating the random phase approximation (RPA) correlation energy is presented that is as accurate as the canonical molecular orbital resolution-of-the-identity RPA (RI-RPA) with the important advantage of an effective linear-scaling behavior (instead of quartic) for large systems due to a formulation in the local atomic orbital space. The high accuracy is achieved by utilizing optimized minimax integration schemes and the local Coulomb metric attenuated by the complementary error function for the RI approximation. The memory bottleneck of former atomic orbital (AO)-RI-RPA implementations ( Schurkus, H. F.; Ochsenfeld, C. J. Chem. Phys. 2016 , 144 , 031101 and Luenser, A.; Schurkus, H. F.; Ochsenfeld, C. J. Chem. Theory Comput. 2017 , 13 , 1647 - 1655 ) is addressed by precontraction of the large 3-center integral matrix with the Cholesky factors of the ground state density reducing the memory requirements of that matrix by a factor of [Formula: see text]. Furthermore, we present a parallel implementation of our method, which not only leads to faster RPA correlation energy calculations but also to a scalable decrease in memory requirements, opening the door for investigations of large molecules even on small- to medium-sized computing clusters. Although it is known that AO methods are highly efficient for extended systems, where sparsity allows for reaching the linear-scaling regime, we show that our work also extends the applicability when considering highly delocalized systems for which no linear scaling can be achieved. As an example, the interlayer distance of two covalent organic framework pore fragments (comprising 384 atoms in total) is analyzed.

  13. Development of highly accurate approximate scheme for computing the charge transfer integral

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

    Pershin, Anton; Szalay, Péter G.

    The charge transfer integral is a key parameter required by various theoretical models to describe charge transport properties, e.g., in organic semiconductors. The accuracy of this important property depends on several factors, which include the level of electronic structure theory and internal simplifications of the applied formalism. The goal of this paper is to identify the performance of various approximate approaches of the latter category, while using the high level equation-of-motion coupled cluster theory for the electronic structure. The calculations have been performed on the ethylene dimer as one of the simplest model systems. By studying different spatial perturbations, itmore » was shown that while both energy split in dimer and fragment charge difference methods are equivalent with the exact formulation for symmetrical displacements, they are less efficient when describing transfer integral along the asymmetric alteration coordinate. Since the “exact” scheme was found computationally expensive, we examine the possibility to obtain the asymmetric fluctuation of the transfer integral by a Taylor expansion along the coordinate space. By exploring the efficiency of this novel approach, we show that the Taylor expansion scheme represents an attractive alternative to the “exact” calculations due to a substantial reduction of computational costs, when a considerably large region of the potential energy surface is of interest. Moreover, we show that the Taylor expansion scheme, irrespective of the dimer symmetry, is very accurate for the entire range of geometry fluctuations that cover the space the molecule accesses at room temperature.« less

  14. An approximation function for frequency constrained structural optimization

    NASA Technical Reports Server (NTRS)

    Canfield, R. A.

    1989-01-01

    The purpose is to examine a function for approximating natural frequency constraints during structural optimization. The nonlinearity of frequencies has posed a barrier to constructing approximations for frequency constraints of high enough quality to facilitate efficient solutions. A new function to represent frequency constraints, called the Rayleigh Quotient Approximation (RQA), is presented. Its ability to represent the actual frequency constraint results in stable convergence with effectively no move limits. The objective of the optimization problem is to minimize structural weight subject to some minimum (or maximum) allowable frequency and perhaps subject to other constraints such as stress, displacement, and gage size, as well. A reason for constraining natural frequencies during design might be to avoid potential resonant frequencies due to machinery or actuators on the structure. Another reason might be to satisy requirements of an aircraft or spacecraft's control law. Whatever the structure supports may be sensitive to a frequency band that must be avoided. Any of these situations or others may require the designer to insure the satisfaction of frequency constraints. A further motivation for considering accurate approximations of natural frequencies is that they are fundamental to dynamic response constraints.

  15. Born approximation in linear-time invariant system

    NASA Astrophysics Data System (ADS)

    Gumjudpai, Burin

    2017-09-01

    An alternative way of finding the LTI’s solution with the Born approximation, is investigated. We use Born approximation in the LTI and in the transformed LTI in form of Helmholtz equation. General solution are considered as infinite series or Feynman graph. Slow-roll approximation are explored. Transforming the LTI system into Helmholtz equation, approximated general solution can be found for any given forms of force with its initial value.

  16. An Approximate Solution to the Equation of Motion for Large-Angle Oscillations of the Simple Pendulum with Initial Velocity

    ERIC Educational Resources Information Center

    Johannessen, Kim

    2010-01-01

    An analytic approximation of the solution to the differential equation describing the oscillations of a simple pendulum at large angles and with initial velocity is discussed. In the derivation, a sinusoidal approximation has been applied, and an analytic formula for the large-angle period of the simple pendulum is obtained, which also includes…

  17. Using Stochastic Approximation Techniques to Efficiently Construct Confidence Intervals for Heritability.

    PubMed

    Schweiger, Regev; Fisher, Eyal; Rahmani, Elior; Shenhav, Liat; Rosset, Saharon; Halperin, Eran

    2018-06-22

    Estimation of heritability is an important task in genetics. The use of linear mixed models (LMMs) to determine narrow-sense single-nucleotide polymorphism (SNP)-heritability and related quantities has received much recent attention, due of its ability to account for variants with small effect sizes. Typically, heritability estimation under LMMs uses the restricted maximum likelihood (REML) approach. The common way to report the uncertainty in REML estimation uses standard errors (SEs), which rely on asymptotic properties. However, these assumptions are often violated because of the bounded parameter space, statistical dependencies, and limited sample size, leading to biased estimates and inflated or deflated confidence intervals (CIs). In addition, for larger data sets (e.g., tens of thousands of individuals), the construction of SEs itself may require considerable time, as it requires expensive matrix inversions and multiplications. Here, we present FIESTA (Fast confidence IntErvals using STochastic Approximation), a method for constructing accurate CIs. FIESTA is based on parametric bootstrap sampling, and, therefore, avoids unjustified assumptions on the distribution of the heritability estimator. FIESTA uses stochastic approximation techniques, which accelerate the construction of CIs by several orders of magnitude, compared with previous approaches as well as to the analytical approximation used by SEs. FIESTA builds accurate CIs rapidly, for example, requiring only several seconds for data sets of tens of thousands of individuals, making FIESTA a very fast solution to the problem of building accurate CIs for heritability for all data set sizes.

  18. An analytic, approximate method for modeling steady, three-dimensional flow to partially penetrating wells

    NASA Astrophysics Data System (ADS)

    Bakker, Mark

    2001-05-01

    An analytic, approximate solution is derived for the modeling of three-dimensional flow to partially penetrating wells. The solution is written in terms of a correction on the solution for a fully penetrating well and is obtained by dividing the aquifer up, locally, in a number of aquifer layers. The resulting system of differential equations is solved by application of the theory for multiaquifer flow. The presented approach has three major benefits. First, the solution may be applied to any groundwater model that can simulate flow to a fully penetrating well; the solution may be superimposed onto the solution for the fully penetrating well to simulate the local three-dimensional drawdown and flow field. Second, the approach is applicable to isotropic, anisotropic, and stratified aquifers and to both confined and unconfined flow. Third, the solution extends over a small area around the well only; outside this area the three-dimensional effect of the partially penetrating well is negligible, and no correction to the fully penetrating well is needed. A number of comparisons are made to existing three-dimensional, analytic solutions, including radial confined and unconfined flow and a well in a uniform flow field. It is shown that a subdivision in three layers is accurate for many practical cases; very accurate solutions are obtained with more layers.

  19. Approximate solution to the Callan-Giddings-Harvey-Strominger field equations for two-dimensional evaporating black holes

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

    Ori, Amos

    2010-11-15

    Callan, Giddings, Harvey, and Strominger (CGHS) previously introduced a two-dimensional semiclassical model of gravity coupled to a dilaton and to matter fields. Their model yields a system of field equations which may describe the formation of a black hole in gravitational collapse as well as its subsequent evaporation. Here we present an approximate analytical solution to the semiclassical CGHS field equations. This solution is constructed using the recently introduced formalism of flux-conserving hyperbolic systems. We also explore the asymptotic behavior at the horizon of the evaporating black hole.

  20. Electrolyte diodes with weak acids and bases. I. Theory and an approximate analytical solution.

    PubMed

    Iván, Kristóf; Simon, Péter L; Wittmann, Mária; Noszticzius, Zoltán

    2005-10-22

    Until now acid-base diodes and transistors applied strong mineral acids and bases exclusively. In this work properties of electrolyte diodes with weak electrolytes are studied and compared with those of diodes with strong ones to show the advantages of weak acids and bases in these applications. The theoretical model is a one dimensional piece of gel containing fixed ionizable groups and connecting reservoirs of an acid and a base. The electric current flowing through the gel is measured as a function of the applied voltage. The steady-state current-voltage characteristic (CVC) of such a gel looks like that of a diode under these conditions. Results of our theoretical, numerical, and experimental investigations are reported in two parts. In this first, theoretical part governing equations necessary to calculate the steady-state CVC of a reverse-biased electrolyte diode are presented together with an approximate analytical solution of this reaction-diffusion-ionic migration problem. The applied approximations are quasielectroneutrality and quasiequilibrium. It is shown that the gel can be divided into an alkaline and an acidic zone separated by a middle weakly acidic region. As a further approximation it is assumed that the ionization of the fixed acidic groups is complete in the alkaline zone and that it is completely suppressed in the acidic one. The general solution given here describes the CVC and the potential and ionic concentration profiles of diodes applying either strong or weak electrolytes. It is proven that previous formulas valid for a strong acid-strong base diode can be regarded as a special case of the more general formulas presented here.

  1. Combination of the pair density approximation and the Takahashi–Imada approximation for path integral Monte Carlo simulations

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

    Zillich, Robert E., E-mail: robert.zillich@jku.at

    2015-11-15

    We construct an accurate imaginary time propagator for path integral Monte Carlo simulations for heterogeneous systems consisting of a mixture of atoms and molecules. We combine the pair density approximation, which is highly accurate but feasible only for the isotropic interactions between atoms, with the Takahashi–Imada approximation for general interactions. We present finite temperature simulations results for energy and structure of molecules–helium clusters X{sup 4}He{sub 20} (X=HCCH and LiH) which show a marked improvement over the Trotter approximation which has a 2nd-order time step bias. We show that the 4th-order corrections of the Takahashi–Imada approximation can also be applied perturbativelymore » to a 2nd-order simulation.« less

  2. Finite-dimensional linear approximations of solutions to general irregular nonlinear operator equations and equations with quadratic operators

    NASA Astrophysics Data System (ADS)

    Kokurin, M. Yu.

    2010-11-01

    A general scheme for improving approximate solutions to irregular nonlinear operator equations in Hilbert spaces is proposed and analyzed in the presence of errors. A modification of this scheme designed for equations with quadratic operators is also examined. The technique of universal linear approximations of irregular equations is combined with the projection onto finite-dimensional subspaces of a special form. It is shown that, for finite-dimensional quadratic problems, the proposed scheme provides information about the global geometric properties of the intersections of quadrics.

  3. One-dimensional model and solutions for creeping gas flows in the approximation of uniform pressure

    NASA Astrophysics Data System (ADS)

    Vedernikov, A.; Balapanov, D.

    2016-11-01

    A model, along with analytical and numerical solutions, is presented to describe a wide variety of one-dimensional slow flows of compressible heat-conductive fluids. The model is based on the approximation of uniform pressure valid for the flows, in which the sound propagation time is much shorter than the duration of any meaningful density variation in the system. The energy balance is described by the heat equation that is solved independently. This approach enables the explicit solution for the fluid velocity to be obtained. Interfacial and volumetric heat and mass sources as well as boundary motion are considered as possible sources of density variation in the fluid. A set of particular tasks is analyzed for different motion sources in planar, axial, and central symmetries in the quasistationary limit of heat conduction (i.e., for large Fourier number). The analytical solutions are in excellent agreement with corresponding numerical solutions of the whole system of the Navier-Stokes equations. This work deals with the ideal gas. The approach is also valid for other equations of state.

  4. Bohm-criterion approximation versus optimal matched solution for a cylindrical probe in radial-motion theory

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

    Din, Alif

    2016-08-15

    The theory of positive-ion collection by a probe immersed in a low-pressure plasma was reviewed and extended by Allen et al. [Proc. Phys. Soc. 70, 297 (1957)]. The numerical computations for cylindrical and spherical probes in a sheath region were presented by F. F. Chen [J. Nucl. Energy C 7, 41 (1965)]. Here, in this paper, the sheath and presheath solutions for a cylindrical probe are matched through a numerical matching procedure to yield “matched” potential profile or “M solution.” The solution based on the Bohm criterion approach “B solution” is discussed for this particular problem. The comparison of cylindricalmore » probe characteristics obtained from the correct potential profile (M solution) and the approximated Bohm-criterion approach are different. This raises questions about the correctness of cylindrical probe theories relying only on the Bohm-criterion approach. Also the comparison between theoretical and experimental ion current characteristics shows that in an argon plasma the ions motion towards the probe is almost radial.« less

  5. Slender-Body Theory Based On Approximate Solution of the Transonic Flow Equation

    NASA Technical Reports Server (NTRS)

    Spreiter, John R.; Alksne, Alberta Y.

    1959-01-01

    Approximate solution of the nonlinear equations of the small disturbance theory of transonic flow are found for the pressure distribution on pointed slender bodies of revolution for flows with free-stream, Mach number 1, and for flows that are either purely subsonic or purely supersonic. These results are obtained by application of a method based on local linearization that was introduced recently in the analysis of similar problems in two-dimensional flows. The theory is developed for bodies of arbitrary shape, and specific results are given for cone-cylinders and for parabolic-arc bodies at zero angle of attack. All results are compared either with existing theoretical results or with experimental data.

  6. Difference equation state approximations for nonlinear hereditary control problems

    NASA Technical Reports Server (NTRS)

    Rosen, I. G.

    1982-01-01

    Discrete approximation schemes for the solution of nonlinear hereditary control problems are constructed. The methods involve approximation by a sequence of optimal control problems in which the original infinite dimensional state equation has been approximated by a finite dimensional discrete difference equation. Convergence of the state approximations is argued using linear semigroup theory and is then used to demonstrate that solutions to the approximating optimal control problems in some sense approximate solutions to the original control problem. Two schemes, one based upon piecewise constant approximation, and the other involving spline functions are discussed. Numerical results are presented, analyzed and used to compare the schemes to other available approximation methods for the solution of hereditary control problems.

  7. Wavelets and distributed approximating functionals

    NASA Astrophysics Data System (ADS)

    Wei, G. W.; Kouri, D. J.; Hoffman, D. K.

    1998-07-01

    A general procedure is proposed for constructing father and mother wavelets that have excellent time-frequency localization and can be used to generate entire wavelet families for use as wavelet transforms. One interesting feature of our father wavelets (scaling functions) is that they belong to a class of generalized delta sequences, which we refer to as distributed approximating functionals (DAFs). We indicate this by the notation wavelet-DAFs. Correspondingly, the mother wavelets generated from these wavelet-DAFs are appropriately called DAF-wavelets. Wavelet-DAFs can be regarded as providing a pointwise (localized) spectral method, which furnishes a bridge between the traditional global methods and local methods for solving partial differential equations. They are shown to provide extremely accurate numerical solutions for a number of nonlinear partial differential equations, including the Korteweg-de Vries (KdV) equation, for which a previous method has encountered difficulties (J. Comput. Phys. 132 (1997) 233).

  8. Difference equation state approximations for nonlinear hereditary control problems

    NASA Technical Reports Server (NTRS)

    Rosen, I. G.

    1984-01-01

    Discrete approximation schemes for the solution of nonlinear hereditary control problems are constructed. The methods involve approximation by a sequence of optimal control problems in which the original infinite dimensional state equation has been approximated by a finite dimensional discrete difference equation. Convergence of the state approximations is argued using linear semigroup theory and is then used to demonstrate that solutions to the approximating optimal control problems in some sense approximate solutions to the original control problem. Two schemes, one based upon piecewise constant approximation, and the other involving spline functions are discussed. Numerical results are presented, analyzed and used to compare the schemes to other available approximation methods for the solution of hereditary control problems. Previously announced in STAR as N83-33589

  9. Discrete ordinates solutions of nongray radiative transfer with diffusely reflecting walls

    NASA Technical Reports Server (NTRS)

    Menart, J. A.; Lee, Haeok S.; Kim, Tae-Kuk

    1993-01-01

    Nongray gas radiation in a plane parallel slab bounded by gray, diffusely reflecting walls is studied using the discrete ordinates method. The spectral equation of transfer is averaged over a narrow wavenumber interval preserving the spectral correlation effect. The governing equations are derived by considering the history of multiple reflections between two reflecting wails. A closure approximation is applied so that only a finite number of reflections have to be explicitly included. The closure solutions express the physics of the problem to a very high degree and show relatively little error. Numerical solutions are obtained by applying a statistical narrow-band model for gas properties and a discrete ordinates code. The net radiative wail heat fluxes and the radiative source distributions are obtained for different temperature profiles. A zeroth-degree formulation, where no wall reflection is handled explicitly, is sufficient to predict the radiative transfer accurately for most cases considered, when compared with increasingly accurate solutions based on explicitly tracing a larger number of wail reflections without any closure approximation applied.

  10. Uniformly high-order accurate non-oscillatory schemes, 1

    NASA Technical Reports Server (NTRS)

    Harten, A.; Osher, S.

    1985-01-01

    The construction and the analysis of nonoscillatory shock capturing methods for the approximation of hyperbolic conservation laws was begun. These schemes share many desirable properties with total variation diminishing schemes (TVD), but TVD schemes have at most first order accuracy, in the sense of truncation error, at extreme of the solution. A uniformly second order approximation was constucted, which is nonoscillatory in the sense that the number of extrema of the discrete solution is not increasing in time. This is achieved via a nonoscillatory piecewise linear reconstruction of the solution from its cell averages, time evolution through an approximate solution of the resulting initial value problem, and averaging of this approximate solution over each cell.

  11. A test of the adhesion approximation for gravitational clustering

    NASA Technical Reports Server (NTRS)

    Melott, Adrian L.; Shandarin, Sergei; Weinberg, David H.

    1993-01-01

    We quantitatively compare a particle implementation of the adhesion approximation to fully non-linear, numerical 'N-body' simulations. Our primary tool, cross-correlation of N-body simulations with the adhesion approximation, indicates good agreement, better than that found by the same test performed with the Zel-dovich approximation (hereafter ZA). However, the cross-correlation is not as good as that of the truncated Zel-dovich approximation (TZA), obtained by applying the Zel'dovich approximation after smoothing the initial density field with a Gaussian filter. We confirm that the adhesion approximation produces an excessively filamentary distribution. Relative to the N-body results, we also find that: (a) the power spectrum obtained from the adhesion approximation is more accurate than that from ZA or TZA, (b) the error in the phase angle of Fourier components is worse than that from TZA, and (c) the mass distribution function is more accurate than that from ZA or TZA. It appears that adhesion performs well statistically, but that TZA is more accurate dynamically, in the sense of moving mass to the right place.

  12. A test of the adhesion approximation for gravitational clustering

    NASA Technical Reports Server (NTRS)

    Melott, Adrian L.; Shandarin, Sergei F.; Weinberg, David H.

    1994-01-01

    We quantitatively compare a particle implementation of the adhesion approximation to fully nonlinear, numerical 'N-body' simulations. Our primary tool, cross-correlation of N-body simulations with the adhesion approximation, indicates good agreement, better than that found by the same test performed with the Zel'dovich approximation (hereafter ZA). However, the cross-correlation is not as good as that of the truncated Zel'dovich approximation (TZA), obtained by applying the Zel'dovich approximation after smoothing the initial density field with a Gaussian filter. We confirm that the adhesion approximation produces an excessively filamentary distribution. Relative to the N-body results, we also find that: (a) the power spectrum obtained from the adhesion approximation is more accurate that that from ZA to TZA, (b) the error in the phase angle of Fourier components is worse that that from TZA, and (c) the mass distribution function is more accurate than that from ZA or TZA. It appears that adhesion performs well statistically, but that TZA is more accurate dynamically, in the sense of moving mass to the right place.

  13. Interpreting the Coulomb-field approximation for generalized-Born electrostatics using boundary-integral equation theory.

    PubMed

    Bardhan, Jaydeep P

    2008-10-14

    The importance of molecular electrostatic interactions in aqueous solution has motivated extensive research into physical models and numerical methods for their estimation. The computational costs associated with simulations that include many explicit water molecules have driven the development of implicit-solvent models, with generalized-Born (GB) models among the most popular of these. In this paper, we analyze a boundary-integral equation interpretation for the Coulomb-field approximation (CFA), which plays a central role in most GB models. This interpretation offers new insights into the nature of the CFA, which traditionally has been assessed using only a single point charge in the solute. The boundary-integral interpretation of the CFA allows the use of multiple point charges, or even continuous charge distributions, leading naturally to methods that eliminate the interpolation inaccuracies associated with the Still equation. This approach, which we call boundary-integral-based electrostatic estimation by the CFA (BIBEE/CFA), is most accurate when the molecular charge distribution generates a smooth normal displacement field at the solute-solvent boundary, and CFA-based GB methods perform similarly. Conversely, both methods are least accurate for charge distributions that give rise to rapidly varying or highly localized normal displacement fields. Supporting this analysis are comparisons of the reaction-potential matrices calculated using GB methods and boundary-element-method (BEM) simulations. An approximation similar to BIBEE/CFA exhibits complementary behavior, with superior accuracy for charge distributions that generate rapidly varying normal fields and poorer accuracy for distributions that produce smooth fields. This approximation, BIBEE by preconditioning (BIBEE/P), essentially generates initial guesses for preconditioned Krylov-subspace iterative BEMs. Thus, iterative refinement of the BIBEE/P results recovers the BEM solution; excellent agreement

  14. Approximate inference on planar graphs using loop calculus and belief progagation

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

    Chertkov, Michael; Gomez, Vicenc; Kappen, Hilbert

    We introduce novel results for approximate inference on planar graphical models using the loop calculus framework. The loop calculus (Chertkov and Chernyak, 2006b) allows to express the exact partition function Z of a graphical model as a finite sum of terms that can be evaluated once the belief propagation (BP) solution is known. In general, full summation over all correction terms is intractable. We develop an algorithm for the approach presented in Chertkov et al. (2008) which represents an efficient truncation scheme on planar graphs and a new representation of the series in terms of Pfaffians of matrices. We analyzemore » in detail both the loop series and the Pfaffian series for models with binary variables and pairwise interactions, and show that the first term of the Pfaffian series can provide very accurate approximations. The algorithm outperforms previous truncation schemes of the loop series and is competitive with other state-of-the-art methods for approximate inference.« less

  15. Numerical solution of nonlinear partial differential equations of mixed type. [finite difference approximation

    NASA Technical Reports Server (NTRS)

    Jameson, A.

    1976-01-01

    A review is presented of some recently developed numerical methods for the solution of nonlinear equations of mixed type. The methods considered use finite difference approximations to the differential equation. Central difference formulas are employed in the subsonic zone and upwind difference formulas are used in the supersonic zone. The relaxation method for the small disturbance equation is discussed and a description is given of difference schemes for the potential flow equation in quasi-linear form. Attention is also given to difference schemes for the potential flow equation in conservation form, the analysis of relaxation schemes by the time dependent analogy, the accelerated iterative method, and three-dimensional calculations.

  16. Solution of the mean spherical approximation for polydisperse multi-Yukawa hard-sphere fluid mixture using orthogonal polynomial expansions

    NASA Astrophysics Data System (ADS)

    Kalyuzhnyi, Yurij V.; Cummings, Peter T.

    2006-03-01

    The Blum-Høye [J. Stat. Phys. 19 317 (1978)] solution of the mean spherical approximation for a multicomponent multi-Yukawa hard-sphere fluid is extended to a polydisperse multi-Yukawa hard-sphere fluid. Our extension is based on the application of the orthogonal polynomial expansion method of Lado [Phys. Rev. E 54, 4411 (1996)]. Closed form analytical expressions for the structural and thermodynamic properties of the model are presented. They are given in terms of the parameters that follow directly from the solution. By way of illustration the method of solution is applied to describe the thermodynamic properties of the one- and two-Yukawa versions of the model.

  17. Exact Solution of Gas Dynamics Equations Through Reduced Differential Transform and Sumudu Transform Linked with Pades Approximants

    NASA Astrophysics Data System (ADS)

    Rao, T. R. Ramesh

    2018-04-01

    In this paper, we study the analytical method based on reduced differential transform method coupled with sumudu transform through Pades approximants. The proposed method may be considered as alternative approach for finding exact solution of Gas dynamics equation in an effective manner. This method does not require any discretization, linearization and perturbation.

  18. Laplacian versus topography in the solution of the linear gravimetric boundary value problem by means of successive approximations

    NASA Astrophysics Data System (ADS)

    Holota, Petr; Nesvadba, Otakar

    2017-04-01

    The aim of this paper is to discuss the solution of the linearized gravimetric boundary value problem by means of the method of successive approximations. We start with the relation between the geometry of the solution domain and the structure of Laplace's operator. Similarly as in other branches of engineering and mathematical physics a transformation of coordinates is used that offers a possibility to solve an alternative between the boundary complexity and the complexity of the coefficients of the partial differential equation governing the solution. Laplace's operator has a relatively simple structure in terms of ellipsoidal coordinates which are frequently used in geodesy. However, the physical surface of the Earth substantially differs from an oblate ellipsoid of revolution, even if it is optimally fitted. Therefore, an alternative is discussed. A system of general curvilinear coordinates such that the physical surface of the Earth is imbedded in the family of coordinate surfaces is used. Clearly, the structure of Laplace's operator is more complicated in this case. It was deduced by means of tensor calculus and in a sense it represents the topography of the physical surface of the Earth. Nevertheless, the construction of the respective Green's function is more simple, if the solution domain is transformed. This enables the use of the classical Green's function method together with the method of successive approximations for the solution of the linear gravimetric boundary value problem expressed in terms of new coordinates. The structure of iteration steps is analyzed and where useful also modified by means of the integration by parts. Comparison with other methods is discussed.

  19. Technical Note: Approximate solution of transient drawdown for constant-flux pumping at a partially penetrating well in a radial two-zone confined aquifer

    NASA Astrophysics Data System (ADS)

    Huang, C.-S.; Yang, S.-Y.; Yeh, H.-D.

    2015-03-01

    An aquifer consisting of a skin zone and a formation zone is considered as a two-zone aquifer. Existing solutions for the problem of constant-flux pumping (CFP) in a two-zone confined aquifer involve laborious calculation. This study develops a new approximate solution for the problem based on a mathematical model including two steady-state flow equations with different hydraulic parameters for the skin and formation zones. A partially penetrating well may be treated as the Neumann condition with a known flux along the screened part and zero flux along the unscreened part. The aquifer domain is finite with an outer circle boundary treated as the Dirichlet condition. The steady-state drawdown solution of the model is derived by the finite Fourier cosine transform. Then, an approximate transient solution is developed by replacing the radius of the boundary in the steady-state solution with an analytical expression for a dimensionless time-dependent radius of influence. The approximate solution is capable of predicting good temporal drawdown distributions over the whole pumping period except at the early stage. A quantitative criterion for the validity of neglecting the vertical flow component due to a partially penetrating well is also provided. Conventional models considering radial flow without the vertical component for the CFP have good accuracy if satisfying the criterion.

  20. Technical Note: Approximate solution of transient drawdown for constant-flux pumping at a partially penetrating well in a radial two-zone confined aquifer

    NASA Astrophysics Data System (ADS)

    Huang, C.-S.; Yang, S.-Y.; Yeh, H.-D.

    2015-06-01

    An aquifer consisting of a skin zone and a formation zone is considered as a two-zone aquifer. Existing solutions for the problem of constant-flux pumping in a two-zone confined aquifer involve laborious calculation. This study develops a new approximate solution for the problem based on a mathematical model describing steady-state radial and vertical flows in a two-zone aquifer. Hydraulic parameters in these two zones can be different but are assumed homogeneous in each zone. A partially penetrating well may be treated as the Neumann condition with a known flux along the screened part and zero flux along the unscreened part. The aquifer domain is finite with an outer circle boundary treated as the Dirichlet condition. The steady-state drawdown solution of the model is derived by the finite Fourier cosine transform. Then, an approximate transient solution is developed by replacing the radius of the aquifer domain in the steady-state solution with an analytical expression for a dimensionless time-dependent radius of influence. The approximate solution is capable of predicting good temporal drawdown distributions over the whole pumping period except at the early stage. A quantitative criterion for the validity of neglecting the vertical flow due to a partially penetrating well is also provided. Conventional models considering radial flow without the vertical component for the constant-flux pumping have good accuracy if satisfying the criterion.

  1. Calcium ions in aqueous solutions: Accurate force field description aided by ab initio molecular dynamics and neutron scattering

    NASA Astrophysics Data System (ADS)

    Martinek, Tomas; Duboué-Dijon, Elise; Timr, Štěpán; Mason, Philip E.; Baxová, Katarina; Fischer, Henry E.; Schmidt, Burkhard; Pluhařová, Eva; Jungwirth, Pavel

    2018-06-01

    We present a combination of force field and ab initio molecular dynamics simulations together with neutron scattering experiments with isotopic substitution that aim at characterizing ion hydration and pairing in aqueous calcium chloride and formate/acetate solutions. Benchmarking against neutron scattering data on concentrated solutions together with ion pairing free energy profiles from ab initio molecular dynamics allows us to develop an accurate calcium force field which accounts in a mean-field way for electronic polarization effects via charge rescaling. This refined calcium parameterization is directly usable for standard molecular dynamics simulations of processes involving this key biological signaling ion.

  2. Asymptotic solution of the diffusion equation in slender impermeable tubes of revolution. I. The leading-term approximation

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

    Traytak, Sergey D., E-mail: sergtray@mail.ru; Le STUDIUM; Semenov Institute of Chemical Physics RAS, 4 Kosygina St., 117977 Moscow

    The anisotropic 3D equation describing the pointlike particles diffusion in slender impermeable tubes of revolution with cross section smoothly depending on the longitudinal coordinate is the object of our study. We use singular perturbations approach to find the rigorous asymptotic expression for the local particles concentration as an expansion in the ratio of the characteristic transversal and longitudinal diffusion relaxation times. The corresponding leading-term approximation is a generalization of well-known Fick-Jacobs approximation. This result allowed us to delineate the conditions on temporal and spatial scales under which the Fick-Jacobs approximation is valid. A striking analogy between solution of our problemmore » and the method of inner-outer expansions for low Knudsen numbers gas kinetic theory is established. With the aid of this analogy we clarify the physical and mathematical meaning of the obtained results.« less

  3. Analytic Approximate Solutions to the Boundary Layer Flow Equation over a Stretching Wall with Partial Slip at the Boundary.

    PubMed

    Ene, Remus-Daniel; Marinca, Vasile; Marinca, Bogdan

    2016-01-01

    Analytic approximate solutions using Optimal Homotopy Perturbation Method (OHPM) are given for steady boundary layer flow over a nonlinearly stretching wall in presence of partial slip at the boundary. The governing equations are reduced to nonlinear ordinary differential equation by means of similarity transformations. Some examples are considered and the effects of different parameters are shown. OHPM is a very efficient procedure, ensuring a very rapid convergence of the solutions after only two iterations.

  4. Petermann I and II spot size: Accurate semi analytical description involving Nelder-Mead method of nonlinear unconstrained optimization and three parameter fundamental modal field

    NASA Astrophysics Data System (ADS)

    Roy Choudhury, Raja; Roy Choudhury, Arundhati; Kanti Ghose, Mrinal

    2013-01-01

    A semi-analytical model with three optimizing parameters and a novel non-Gaussian function as the fundamental modal field solution has been proposed to arrive at an accurate solution to predict various propagation parameters of graded-index fibers with less computational burden than numerical methods. In our semi analytical formulation the optimization of core parameter U which is usually uncertain, noisy or even discontinuous, is being calculated by Nelder-Mead method of nonlinear unconstrained minimizations as it is an efficient and compact direct search method and does not need any derivative information. Three optimizing parameters are included in the formulation of fundamental modal field of an optical fiber to make it more flexible and accurate than other available approximations. Employing variational technique, Petermann I and II spot sizes have been evaluated for triangular and trapezoidal-index fibers with the proposed fundamental modal field. It has been demonstrated that, the results of the proposed solution identically match with the numerical results over a wide range of normalized frequencies. This approximation can also be used in the study of doped and nonlinear fiber amplifier.

  5. An approximate viscous shock layer technique for calculating chemically reacting hypersonic flows about blunt-nosed bodies

    NASA Technical Reports Server (NTRS)

    Cheatwood, F. Mcneil; Dejarnette, Fred R.

    1991-01-01

    An approximate axisymmetric method was developed which can reliably calculate fully viscous hypersonic flows over blunt nosed bodies. By substituting Maslen's second order pressure expression for the normal momentum equation, a simplified form of the viscous shock layer (VSL) equations is obtained. This approach can solve both the subsonic and supersonic regions of the shock layer without a starting solution for the shock shape. The approach is applicable to perfect gas, equilibrium, and nonequilibrium flowfields. Since the method is fully viscous, the problems associated with a boundary layer solution with an inviscid layer solution are avoided. This procedure is significantly faster than the parabolized Navier-Stokes (PNS) or VSL solvers and would be useful in a preliminary design environment. Problems associated with a previously developed approximate VSL technique are addressed before extending the method to nonequilibrium calculations. Perfect gas (laminar and turbulent), equilibrium, and nonequilibrium solutions were generated for airflows over several analytic body shapes. Surface heat transfer, skin friction, and pressure predictions are comparable to VSL results. In addition, computed heating rates are in good agreement with experimental data. The present technique generates its own shock shape as part of its solution, and therefore could be used to provide more accurate initial shock shapes for higher order procedures which require starting solutions.

  6. Electrostatics of proteins in dielectric solvent continua. I. An accurate and efficient reaction field description

    NASA Astrophysics Data System (ADS)

    Bauer, Sebastian; Mathias, Gerald; Tavan, Paul

    2014-03-01

    We present a reaction field (RF) method which accurately solves the Poisson equation for proteins embedded in dielectric solvent continua at a computational effort comparable to that of an electrostatics calculation with polarizable molecular mechanics (MM) force fields. The method combines an approach originally suggested by Egwolf and Tavan [J. Chem. Phys. 118, 2039 (2003)] with concepts generalizing the Born solution [Z. Phys. 1, 45 (1920)] for a solvated ion. First, we derive an exact representation according to which the sources of the RF potential and energy are inducible atomic anti-polarization densities and atomic shielding charge distributions. Modeling these atomic densities by Gaussians leads to an approximate representation. Here, the strengths of the Gaussian shielding charge distributions are directly given in terms of the static partial charges as defined, e.g., by standard MM force fields for the various atom types, whereas the strengths of the Gaussian anti-polarization densities are calculated by a self-consistency iteration. The atomic volumes are also described by Gaussians. To account for covalently overlapping atoms, their effective volumes are calculated by another self-consistency procedure, which guarantees that the dielectric function ɛ(r) is close to one everywhere inside the protein. The Gaussian widths σi of the atoms i are parameters of the RF approximation. The remarkable accuracy of the method is demonstrated by comparison with Kirkwood's analytical solution for a spherical protein [J. Chem. Phys. 2, 351 (1934)] and with computationally expensive grid-based numerical solutions for simple model systems in dielectric continua including a di-peptide (Ac-Ala-NHMe) as modeled by a standard MM force field. The latter example shows how weakly the RF conformational free energy landscape depends on the parameters σi. A summarizing discussion highlights the achievements of the new theory and of its approximate solution particularly by

  7. Electrostatics of proteins in dielectric solvent continua. I. An accurate and efficient reaction field description.

    PubMed

    Bauer, Sebastian; Mathias, Gerald; Tavan, Paul

    2014-03-14

    We present a reaction field (RF) method which accurately solves the Poisson equation for proteins embedded in dielectric solvent continua at a computational effort comparable to that of an electrostatics calculation with polarizable molecular mechanics (MM) force fields. The method combines an approach originally suggested by Egwolf and Tavan [J. Chem. Phys. 118, 2039 (2003)] with concepts generalizing the Born solution [Z. Phys. 1, 45 (1920)] for a solvated ion. First, we derive an exact representation according to which the sources of the RF potential and energy are inducible atomic anti-polarization densities and atomic shielding charge distributions. Modeling these atomic densities by Gaussians leads to an approximate representation. Here, the strengths of the Gaussian shielding charge distributions are directly given in terms of the static partial charges as defined, e.g., by standard MM force fields for the various atom types, whereas the strengths of the Gaussian anti-polarization densities are calculated by a self-consistency iteration. The atomic volumes are also described by Gaussians. To account for covalently overlapping atoms, their effective volumes are calculated by another self-consistency procedure, which guarantees that the dielectric function ε(r) is close to one everywhere inside the protein. The Gaussian widths σ(i) of the atoms i are parameters of the RF approximation. The remarkable accuracy of the method is demonstrated by comparison with Kirkwood's analytical solution for a spherical protein [J. Chem. Phys. 2, 351 (1934)] and with computationally expensive grid-based numerical solutions for simple model systems in dielectric continua including a di-peptide (Ac-Ala-NHMe) as modeled by a standard MM force field. The latter example shows how weakly the RF conformational free energy landscape depends on the parameters σ(i). A summarizing discussion highlights the achievements of the new theory and of its approximate solution particularly by

  8. A solution for measuring accurate reaction time to visual stimuli realized with a programmable microcontroller.

    PubMed

    Ohyanagi, Toshio; Sengoku, Yasuhito

    2010-02-01

    This article presents a new solution for measuring accurate reaction time (SMART) to visual stimuli. The SMART is a USB device realized with a Cypress Programmable System-on-Chip (PSoC) mixed-signal array programmable microcontroller. A brief overview of the hardware and firmware of the PSoC is provided, together with the results of three experiments. In Experiment 1, we investigated the timing accuracy of the SMART in measuring reaction time (RT) under different conditions of operating systems (OSs; Windows XP or Vista) and monitor displays (a CRT or an LCD). The results indicated that the timing error in measuring RT by the SMART was less than 2 msec, on average, under all combinations of OS and display and that the SMART was tolerant to jitter and noise. In Experiment 2, we tested the SMART with 8 participants. The results indicated that there was no significant difference among RTs obtained with the SMART under the different conditions of OS and display. In Experiment 3, we used Microsoft (MS) PowerPoint to present visual stimuli on the display. We found no significant difference in RTs obtained using MS DirectX technology versus using the PowerPoint file with the SMART. We are certain that the SMART is a simple and practical solution for measuring RTs accurately. Although there are some restrictions in using the SMART with RT paradigms, the SMART is capable of providing both researchers and health professionals working in clinical settings with new ways of using RT paradigms in their work.

  9. On the implementation of an accurate and efficient solver for convection-diffusion equations

    NASA Astrophysics Data System (ADS)

    Wu, Chin-Tien

    In this dissertation, we examine several different aspects of computing the numerical solution of the convection-diffusion equation. The solution of this equation often exhibits sharp gradients due to Dirichlet outflow boundaries or discontinuities in boundary conditions. Because of the singular-perturbed nature of the equation, numerical solutions often have severe oscillations when grid sizes are not small enough to resolve sharp gradients. To overcome such difficulties, the streamline diffusion discretization method can be used to obtain an accurate approximate solution in regions where the solution is smooth. To increase accuracy of the solution in the regions containing layers, adaptive mesh refinement and mesh movement based on a posteriori error estimations can be employed. An error-adapted mesh refinement strategy based on a posteriori error estimations is also proposed to resolve layers. For solving the sparse linear systems that arise from discretization, goemetric multigrid (MG) and algebraic multigrid (AMG) are compared. In addition, both methods are also used as preconditioners for Krylov subspace methods. We derive some convergence results for MG with line Gauss-Seidel smoothers and bilinear interpolation. Finally, while considering adaptive mesh refinement as an integral part of the solution process, it is natural to set a stopping tolerance for the iterative linear solvers on each mesh stage so that the difference between the approximate solution obtained from iterative methods and the finite element solution is bounded by an a posteriori error bound. Here, we present two stopping criteria. The first is based on a residual-type a posteriori error estimator developed by Verfurth. The second is based on an a posteriori error estimator, using local solutions, developed by Kay and Silvester. Our numerical results show the refined mesh obtained from the iterative solution which satisfies the second criteria is similar to the refined mesh obtained from

  10. Analytic Approximate Solutions to the Boundary Layer Flow Equation over a Stretching Wall with Partial Slip at the Boundary

    PubMed Central

    Ene, Remus-Daniel; Marinca, Vasile; Marinca, Bogdan

    2016-01-01

    Analytic approximate solutions using Optimal Homotopy Perturbation Method (OHPM) are given for steady boundary layer flow over a nonlinearly stretching wall in presence of partial slip at the boundary. The governing equations are reduced to nonlinear ordinary differential equation by means of similarity transformations. Some examples are considered and the effects of different parameters are shown. OHPM is a very efficient procedure, ensuring a very rapid convergence of the solutions after only two iterations. PMID:27031232

  11. Approximate Analysis for Interlaminar Stresses in Composite Structures with Thickness Discontinuities

    NASA Technical Reports Server (NTRS)

    Rose, Cheryl A.; Starnes, James H., Jr.

    1996-01-01

    An efficient, approximate analysis for calculating complete three-dimensional stress fields near regions of geometric discontinuities in laminated composite structures is presented. An approximate three-dimensional local analysis is used to determine the detailed local response due to far-field stresses obtained from a global two-dimensional analysis. The stress results from the global analysis are used as traction boundary conditions for the local analysis. A generalized plane deformation assumption is made in the local analysis to reduce the solution domain to two dimensions. This assumption allows out-of-plane deformation to occur. The local analysis is based on the principle of minimum complementary energy and uses statically admissible stress functions that have an assumed through-the-thickness distribution. Examples are presented to illustrate the accuracy and computational efficiency of the local analysis. Comparisons of the results of the present local analysis with the corresponding results obtained from a finite element analysis and from an elasticity solution are presented. These results indicate that the present local analysis predicts the stress field accurately. Computer execution-times are also presented. The demonstrated accuracy and computational efficiency of the analysis make it well suited for parametric and design studies.

  12. APPROXIMATION OF SOLUTIONS OF THE EQUATION \\overline\\partial^jf=0, j\\geq1, IN DOMAINS WITH QUASICONFORMAL BOUNDARY

    NASA Astrophysics Data System (ADS)

    Andrievskiĭ, V. V.; Belyĭ, V. I.; Maĭmeskul, V. V.

    1991-02-01

    This article establishes direct and inverse theorems of approximation theory (of the same type as theorems of Dzyadyk) that describe the quantitative connection between the smoothness properties of solutions of the equation \\overline\\partial^jf=0, j\\geq1, and the rate of their approximation by "module" polynomials of the form \\displaystyle P_N(z)=\\sum_{n=0}^{j-1}\\sum_{m=0}^{N-n}a_{m,n}z^m\\overline{z}^n,\\qquad N\\geq j-1.In particular, a constructive characterization is obtained for generalized Hölder classes of such functions on domains with quasiconformal boundary.Bibliography: 19 titles.

  13. Covariance approximation for fast and accurate computation of channelized Hotelling observer statistics

    NASA Astrophysics Data System (ADS)

    Bonetto, P.; Qi, Jinyi; Leahy, R. M.

    2000-08-01

    Describes a method for computing linear observer statistics for maximum a posteriori (MAP) reconstructions of PET images. The method is based on a theoretical approximation for the mean and covariance of MAP reconstructions. In particular, the authors derive here a closed form for the channelized Hotelling observer (CHO) statistic applied to 2D MAP images. The theoretical analysis models both the Poission statistics of PET data and the inhomogeneity of tracer uptake. The authors show reasonably good correspondence between these theoretical results and Monte Carlo studies. The accuracy and low computational cost of the approximation allow the authors to analyze the observer performance over a wide range of operating conditions and parameter settings for the MAP reconstruction algorithm.

  14. A Hermite WENO reconstruction for fourth order temporal accurate schemes based on the GRP solver for hyperbolic conservation laws

    NASA Astrophysics Data System (ADS)

    Du, Zhifang; Li, Jiequan

    2018-02-01

    This paper develops a new fifth order accurate Hermite WENO (HWENO) reconstruction method for hyperbolic conservation schemes in the framework of the two-stage fourth order accurate temporal discretization in Li and Du (2016) [13]. Instead of computing the first moment of the solution additionally in the conventional HWENO or DG approach, we can directly take the interface values, which are already available in the numerical flux construction using the generalized Riemann problem (GRP) solver, to approximate the first moment. The resulting scheme is fourth order temporal accurate by only invoking the HWENO reconstruction twice so that it becomes more compact. Numerical experiments show that such compactness makes significant impact on the resolution of nonlinear waves.

  15. Accurate quantum chemical calculations

    NASA Technical Reports Server (NTRS)

    Bauschlicher, Charles W., Jr.; Langhoff, Stephen R.; Taylor, Peter R.

    1989-01-01

    An important goal of quantum chemical calculations is to provide an understanding of chemical bonding and molecular electronic structure. A second goal, the prediction of energy differences to chemical accuracy, has been much harder to attain. First, the computational resources required to achieve such accuracy are very large, and second, it is not straightforward to demonstrate that an apparently accurate result, in terms of agreement with experiment, does not result from a cancellation of errors. Recent advances in electronic structure methodology, coupled with the power of vector supercomputers, have made it possible to solve a number of electronic structure problems exactly using the full configuration interaction (FCI) method within a subspace of the complete Hilbert space. These exact results can be used to benchmark approximate techniques that are applicable to a wider range of chemical and physical problems. The methodology of many-electron quantum chemistry is reviewed. Methods are considered in detail for performing FCI calculations. The application of FCI methods to several three-electron problems in molecular physics are discussed. A number of benchmark applications of FCI wave functions are described. Atomic basis sets and the development of improved methods for handling very large basis sets are discussed: these are then applied to a number of chemical and spectroscopic problems; to transition metals; and to problems involving potential energy surfaces. Although the experiences described give considerable grounds for optimism about the general ability to perform accurate calculations, there are several problems that have proved less tractable, at least with current computer resources, and these and possible solutions are discussed.

  16. Approximate solution of the multiple watchman routes problem with restricted visibility range.

    PubMed

    Faigl, Jan

    2010-10-01

    In this paper, a new self-organizing map (SOM) based adaptation procedure is proposed to address the multiple watchman route problem with the restricted visibility range in the polygonal domain W. A watchman route is represented by a ring of connected neuron weights that evolves in W, while obstacles are considered by approximation of the shortest path. The adaptation procedure considers a coverage of W by the ring in order to attract nodes toward uncovered parts of W. The proposed procedure is experimentally verified in a set of environments and several visibility ranges. Performance of the procedure is compared with the decoupled approach based on solutions of the art gallery problem and the consecutive traveling salesman problem. The experimental results show the suitability of the proposed procedure based on relatively simple supporting geometrical structures, enabling application of the SOM principles to watchman route problems in W.

  17. Implicit time accurate simulation of unsteady flow

    NASA Astrophysics Data System (ADS)

    van Buuren, René; Kuerten, Hans; Geurts, Bernard J.

    2001-03-01

    Implicit time integration was studied in the context of unsteady shock-boundary layer interaction flow. With an explicit second-order Runge-Kutta scheme, a reference solution to compare with the implicit second-order Crank-Nicolson scheme was determined. The time step in the explicit scheme is restricted by both temporal accuracy as well as stability requirements, whereas in the A-stable implicit scheme, the time step has to obey temporal resolution requirements and numerical convergence conditions. The non-linear discrete equations for each time step are solved iteratively by adding a pseudo-time derivative. The quasi-Newton approach is adopted and the linear systems that arise are approximately solved with a symmetric block Gauss-Seidel solver. As a guiding principle for properly setting numerical time integration parameters that yield an efficient time accurate capturing of the solution, the global error caused by the temporal integration is compared with the error resulting from the spatial discretization. Focus is on the sensitivity of properties of the solution in relation to the time step. Numerical simulations show that the time step needed for acceptable accuracy can be considerably larger than the explicit stability time step; typical ratios range from 20 to 80. At large time steps, convergence problems that are closely related to a highly complex structure of the basins of attraction of the iterative method may occur. Copyright

  18. Approximate analytic expression for the Skyrmions crystal

    NASA Astrophysics Data System (ADS)

    Grandi, Nicolás; Sturla, Mauricio

    2018-01-01

    We find approximate solutions for the two-dimensional nonlinear Σ-model with Dzyalioshinkii-Moriya term, representing magnetic Skyrmions. They are built in an analytic form, by pasting different approximate solutions found in different regions of space. We verify that our construction reproduces the phenomenology known from numerical solutions and Monte Carlo simulations, giving rise to a Skyrmion lattice at an intermediate range of magnetic field, flanked by spiral and spin-polarized phases for low and high magnetic fields, respectively.

  19. A cubic spline approximation for problems in fluid mechanics

    NASA Technical Reports Server (NTRS)

    Rubin, S. G.; Graves, R. A., Jr.

    1975-01-01

    A cubic spline approximation is presented which is suited for many fluid-mechanics problems. This procedure provides a high degree of accuracy, even with a nonuniform mesh, and leads to an accurate treatment of derivative boundary conditions. The truncation errors and stability limitations of several implicit and explicit integration schemes are presented. For two-dimensional flows, a spline-alternating-direction-implicit method is evaluated. The spline procedure is assessed, and results are presented for the one-dimensional nonlinear Burgers' equation, as well as the two-dimensional diffusion equation and the vorticity-stream function system describing the viscous flow in a driven cavity. Comparisons are made with analytic solutions for the first two problems and with finite-difference calculations for the cavity flow.

  20. Approximations and Solution Estimates in Optimization

    DTIC Science & Technology

    2016-04-06

    comprehensive descriptions of epi-convergence and its connections to variational analysis broadly. Our motivation for going beyond normed linear spaces , which...proper, every closed ball in this metric space is compact and the existence of solutions of such optimal fitting problems is more easily established...lsc-fcns(X), dl(fν , f) → 0 implies that fν epi-converges to f. We recall that a metric space is proper if every closed ball in that space is compact

  1. Exact PDF equations and closure approximations for advective-reactive transport

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

    Venturi, D.; Tartakovsky, Daniel M.; Tartakovsky, Alexandre M.

    2013-06-01

    Mathematical models of advection–reaction phenomena rely on advective flow velocity and (bio) chemical reaction rates that are notoriously random. By using functional integral methods, we derive exact evolution equations for the probability density function (PDF) of the state variables of the advection–reaction system in the presence of random transport velocity and random reaction rates with rather arbitrary distributions. These PDF equations are solved analytically for transport with deterministic flow velocity and a linear reaction rate represented mathematically by a heterog eneous and strongly-correlated random field. Our analytical solution is then used to investigate the accuracy and robustness of the recentlymore » proposed large-eddy diffusivity (LED) closure approximation [1]. We find that the solution to the LED-based PDF equation, which is exact for uncorrelated reaction rates, is accurate even in the presence of strong correlations and it provides an upper bound of predictive uncertainty.« less

  2. An approximate analytic solution of a set of nonlinear model alpha-omega-dynamo equations for marginally unstable systems

    NASA Technical Reports Server (NTRS)

    Hinata, S.

    1989-01-01

    An approximate analytic solution of a set of nonlinear model alpha-omega-dynamo equations is obtained. The reaction of the Lorentz force on the velocity shear which stretches and, hence, amplifies the magnetic field is incorporated into the model. To single out the effect of the Lorentz force on the omega-effect, the effect of the Lorentz force on the alpha-effect is neglected in this study. The solution represents a nonlinear oscillation with the amplitude and period determined by the dynamo number N. The amplitude is proportional to N - 1, while the period is almost exactly the same as the dissipation time of the unstable mode (proportional to N).

  3. Weber's gravitational force as static weak field approximation

    NASA Astrophysics Data System (ADS)

    Tiandho, Yuant

    2016-02-01

    Weber's gravitational force (WGF) is one of gravitational model that can accommodate a non-static system because it depends not only on the distance but also on the velocity and the acceleration. Unlike Newton's law of gravitation, WGF can predict the anomalous of Mercury and gravitational bending of light near massive object very well. Then, some researchers use WGF as an alternative model of gravitation and propose a new mechanics theory namely the relational mechanics theory. However, currently we have known that the theory of general relativity which proposed by Einstein can explain gravity with very accurate. Through the static weak field approximation for the non-relativistic object, we also have known that the theory of general relativity will reduce to Newton's law of gravity. In this work, we expand the static weak field approximation that compatible with relativistic object and we obtain a force equation which correspond to WGF. Therefore, WGF is more precise than Newton's gravitational law. The static-weak gravitational field that we used is a solution of the Einstein's equation in the vacuum that satisfy the linear field approximation. The expression of WGF with ξ = 1 and satisfy the requirement of energy conservation are obtained after resolving the geodesic equation. By this result, we can conclude that WGF can be derived from the general relativity.

  4. Non-Schwarzschild black-hole metric in four dimensional higher derivative gravity: Analytical approximation

    NASA Astrophysics Data System (ADS)

    Kokkotas, K. D.; Konoplya, R. A.; Zhidenko, A.

    2017-09-01

    Higher derivative extensions of Einstein gravity are important within the string theory approach to gravity and as alternative and effective theories of gravity. H. Lü, A. Perkins, C. Pope, and K. Stelle [Phys. Rev. Lett. 114, 171601 (2015), 10.1103/PhysRevLett.114.171601] found a numerical solution describing a spherically symmetric non-Schwarzschild asymptotically flat black hole in Einstein gravity with added higher derivative terms. Using the general and quickly convergent parametrization in terms of the continued fractions, we represent this numerical solution in the analytical form, which is accurate not only near the event horizon or far from the black hole, but in the whole space. Thereby, the obtained analytical form of the metric allows one to study easily all the further properties of the black hole, such as thermodynamics, Hawking radiation, particle motion, accretion, perturbations, stability, quasinormal spectrum, etc. Thus, the found analytical approximate representation can serve in the same way as an exact solution.

  5. Highly accurate analytic formulae for projectile motion subjected to quadratic drag

    NASA Astrophysics Data System (ADS)

    Turkyilmazoglu, Mustafa

    2016-05-01

    The classical phenomenon of motion of a projectile fired (thrown) into the horizon through resistive air charging a quadratic drag onto the object is revisited in this paper. No exact solution is known that describes the full physical event under such an exerted resistance force. Finding elegant analytical approximations for the most interesting engineering features of dynamical behavior of the projectile is the principal target. Within this purpose, some analytical explicit expressions are derived that accurately predict the maximum height, its arrival time as well as the flight range of the projectile at the highest ascent. The most significant property of the proposed formulas is that they are not restricted to the initial speed and firing angle of the object, nor to the drag coefficient of the medium. In combination with the available approximations in the literature, it is possible to gain information about the flight and complete the picture of a trajectory with high precision, without having to numerically simulate the full governing equations of motion.

  6. SOLUTIONS APPROXIMATING SOLUTE TRANSPORT IN A LEAKY AQUIFER RECEIVING WASTEWATER INJECTION

    EPA Science Inventory

    A mathematical model amenable to analytical solution techniques is developed for the investigation of contaminant transport from an injection well into a leaky aquifer system, which comprises a pumped and an unpumped aquifer connected to each other by an aquitard. A steady state ...

  7. A More Accurate and Efficient Technique Developed for Using Computational Methods to Obtain Helical Traveling-Wave Tube Interaction Impedance

    NASA Technical Reports Server (NTRS)

    Kory, Carol L.

    1999-01-01

    The phenomenal growth of commercial communications has created a great demand for traveling-wave tube (TWT) amplifiers. Although the helix slow-wave circuit remains the mainstay of the TWT industry because of its exceptionally wide bandwidth, until recently it has been impossible to accurately analyze a helical TWT using its exact dimensions because of the complexity of its geometrical structure. For the first time, an accurate three-dimensional helical model was developed that allows accurate prediction of TWT cold-test characteristics including operating frequency, interaction impedance, and attenuation. This computational model, which was developed at the NASA Lewis Research Center, allows TWT designers to obtain a more accurate value of interaction impedance than is possible using experimental methods. Obtaining helical slow-wave circuit interaction impedance is an important part of the design process for a TWT because it is related to the gain and efficiency of the tube. This impedance cannot be measured directly; thus, conventional methods involve perturbing a helical circuit with a cylindrical dielectric rod placed on the central axis of the circuit and obtaining the difference in resonant frequency between the perturbed and unperturbed circuits. A mathematical relationship has been derived between this frequency difference and the interaction impedance (ref. 1). However, because of the complex configuration of the helical circuit, deriving this relationship involves several approximations. In addition, this experimental procedure is time-consuming and expensive, but until recently it was widely accepted as the most accurate means of determining interaction impedance. The advent of an accurate three-dimensional helical circuit model (ref. 2) made it possible for Lewis researchers to fully investigate standard approximations made in deriving the relationship between measured perturbation data and interaction impedance. The most prominent approximations made

  8. Logical gaps in the approximate solutions of the social learning game and an exact solution.

    PubMed

    Dai, Wenjie; Wang, Xin; Di, Zengru; Wu, Jinshan

    2014-01-01

    After the social learning models were proposed, finding solutions to the games becomes a well-defined mathematical question. However, almost all papers on the games and their applications are based on solutions built either upon an ad-hoc argument or a twisted Bayesian analysis of the games. Here, we present logical gaps in those solutions and offer an exact solution of our own. We also introduce a minor extension to the original game so that not only logical differences but also differences in action outcomes among those solutions become visible.

  9. Approximate Solutions for Certain Optimal Stopping Problems

    DTIC Science & Technology

    1978-01-05

    one-armed bandit problem) has arisen in a number of statistical applications (Chernoff and Ray (1965);, Chernoff (±9&]), Mallik (1971)): Let X(t... Mallik (1971) and Chernoff (1972). These previous approximations were determined without the benefit of the "correction for continuity" given in (5.1...Vol. 1, 3rd edition, John Wiley and Sons, Inc., New York» 7. Mallik , A.K» (1971), "Sequential estimation of the common mean of two normal

  10. Flexible Approximation Model Approach for Bi-Level Integrated System Synthesis

    NASA Technical Reports Server (NTRS)

    Sobieszczanski-Sobieski, Jaroslaw; Kim, Hongman; Ragon, Scott; Soremekun, Grant; Malone, Brett

    2004-01-01

    Bi-Level Integrated System Synthesis (BLISS) is an approach that allows design problems to be naturally decomposed into a set of subsystem optimizations and a single system optimization. In the BLISS approach, approximate mathematical models are used to transfer information from the subsystem optimizations to the system optimization. Accurate approximation models are therefore critical to the success of the BLISS procedure. In this paper, new capabilities that are being developed to generate accurate approximation models for BLISS procedure will be described. The benefits of using flexible approximation models such as Kriging will be demonstrated in terms of convergence characteristics and computational cost. An approach of dealing with cases where subsystem optimization cannot find a feasible design will be investigated by using the new flexible approximation models for the violated local constraints.

  11. Padé Approximant and Minimax Rational Approximation in Standard Cosmology

    NASA Astrophysics Data System (ADS)

    Zaninetti, Lorenzo

    2016-02-01

    The luminosity distance in the standard cosmology as given by $\\Lambda$CDM and consequently the distance modulus for supernovae can be defined by the Pad\\'e approximant. A comparison with a known analytical solution shows that the Pad\\'e approximant for the luminosity distance has an error of $4\\%$ at redshift $= 10$. A similar procedure for the Taylor expansion of the luminosity distance gives an error of $4\\%$ at redshift $=0.7 $; this means that for the luminosity distance, the Pad\\'e approximation is superior to the Taylor series. The availability of an analytical expression for the distance modulus allows applying the Levenberg--Marquardt method to derive the fundamental parameters from the available compilations for supernovae. A new luminosity function for galaxies derived from the truncated gamma probability density function models the observed luminosity function for galaxies when the observed range in absolute magnitude is modeled by the Pad\\'e approximant. A comparison of $\\Lambda$CDM with other cosmologies is done adopting a statistical point of view.

  12. Simulations of free-solution electrophoresis of polyelectrolytes with a finite Debye length using the Debye-Hückel approximation.

    PubMed

    Hickey, Owen A; Shendruk, Tyler N; Harden, James L; Slater, Gary W

    2012-08-31

    We introduce a mesoscale simulation method based on multiparticle collision dynamics (MPCD) for the electrohydrodynamics of polyelectrolytes with finite Debye lengths. By applying the Debye-Hückel approximation to assign an effective charge to MPCD particles near charged monomers, our simulations are able to reproduce the rapid rise in the electrophoretic mobility with respect to the degree of polymerization for the shortest polymer lengths followed by a small decrease for longer polymers due to charge condensation. Moreover, these simulations demonstrate the importance of a finite Debye length in accurately determining the mobility of uniformly charged polyelectrolytes and net neutral polyampholytes.

  13. Osmotic potential calculations of inorganic and organic aqueous solutions over wide solute concentration levels and temperatures.

    PubMed

    Cochrane, T T; Cochrane, T A

    2016-01-01

    To demonstrate that the authors' new "aqueous solution vs pure water" equation to calculate osmotic potential may be used to calculate the osmotic potentials of inorganic and organic aqueous solutions over wide ranges of solute concentrations and temperatures. Currently, the osmotic potentials of solutions used for medical purposes are calculated from equations based on the thermodynamics of the gas laws which are only accurate at low temperature and solute concentration levels. Some solutions used in medicine may need their osmotic potentials calculated more accurately to take into account solute concentrations and temperatures. The authors experimented with their new equation for calculating the osmotic potentials of inorganic and organic aqueous solutions up to and beyond body temperatures by adjusting three of its factors; (a) the volume property of pure water, (b) the number of "free" water molecules per unit volume of solution, "Nf," and (c) the "t" factor expressing the cooperative structural relaxation time of pure water at given temperatures. Adequate information on the volume property of pure water at different temperatures is available in the literature. However, as little information on the relative densities of inorganic and organic solutions, respectively, at varying temperatures needed to calculate Nf was available, provisional equations were formulated to approximate values. Those values together with tentative t values for different temperatures chosen from values calculated by different workers were substituted into the authors' equation to demonstrate how osmotic potentials could be estimated over temperatures up to and beyond bodily temperatures. The provisional equations formulated to calculate Nf, the number of free water molecules per unit volume of inorganic and organic solute solutions, respectively, over wide concentration ranges compared well with the calculations of Nf using recorded relative density data at 20 °C. They were

  14. Vibration-translation energy transfer in anharmonic diatomic molecules. 1: A critical evaluation of the semiclassical approximation

    NASA Technical Reports Server (NTRS)

    Mckenzie, R. L.

    1974-01-01

    The semiclassical approximation is applied to anharmonic diatomic oscillators in excited initial states. Multistate numerical solutions giving the vibrational transition probabilities for collinear collisions with an inert atom are compared with equivalent, exact quantum-mechanical calculations. Several symmetrization methods are shown to correlate accurately the predictions of both theories for all initial states, transitions, and molecular types tested, but only if coupling of the oscillator motion and the classical trajectory of the incident particle is considered. In anharmonic heteronuclear molecules, the customary semiclassical method of computing the classical trajectory independently leads to transition probabilities with anomalous low-energy resonances. Proper accounting of the effects of oscillator compression and recoil on the incident particle trajectory removes the anomalies and restores the applicability of the semiclassical approximation.

  15. Applicability of the Rayleigh-Gans approximation for scattering by snowflakes at microwave frequencies in vertical incidence

    NASA Astrophysics Data System (ADS)

    Tyynelä, J.; Leinonen, J.; Westbrook, C. D.; Moisseev, D.; Nousiainen, T.

    2013-02-01

    The applicability of the Rayleigh-Gans approximation (RGA) for scattering by snowflakes is studied in the microwave region of the electromagnetic spectrum. Both the shapes of the single ice crystals, or monomers, and their amounts in the modeled snowflakes are varied. For reference, the discrete-dipole approximation (DDA) is used to produce numerically accurate solutions to the single-scattering properties, such as the backscattering and extinction cross-sections, single-scattering albedo, and the asymmetry parameter. We find that the single-scattering albedo is the most accurate with only about 10% relative bias at maximum. The asymmetry parameter has about 0.12 absolute bias at maximum. The backscattering and extinction cross-sections show about - 65% relative biases at maximum, corresponding to about - 4.6 dB difference. Overall, the RGA agrees well with the DDA computations for all the cases studied and is more accurate for the integrated quantities, such as the single-scattering albedo and the asymmetry parameter than the cross-sections for the same snowflakes. The accuracy of the RGA seems to improve, when the number of monomers is increased in an aggregate, and decrease, when the frequency increases. It is also more accurate for less dense monomer shapes, such as stellar dendrites. The DDA and RGA results are well correlated; the sample correlation coefficients of those are close to unity throughout the study. Therefore, the accuracy of the RGA could be improved by applying appropriate correction factors.

  16. An approximation theory for the identification of linear thermoelastic systems

    NASA Technical Reports Server (NTRS)

    Rosen, I. G.; Su, Chien-Hua Frank

    1990-01-01

    An abstract approximation framework and convergence theory for the identification of thermoelastic systems is developed. Starting from an abstract operator formulation consisting of a coupled second order hyperbolic equation of elasticity and first order parabolic equation for heat conduction, well-posedness is established using linear semigroup theory in Hilbert space, and a class of parameter estimation problems is then defined involving mild solutions. The approximation framework is based upon generic Galerkin approximation of the mild solutions, and convergence of solutions of the resulting sequence of approximating finite dimensional parameter identification problems to a solution of the original infinite dimensional inverse problem is established using approximation results for operator semigroups. An example involving the basic equations of one dimensional linear thermoelasticity and a linear spline based scheme are discussed. Numerical results indicate how the approach might be used in a study of damping mechanisms in flexible structures.

  17. High Order Approximations for Compressible Fluid Dynamics on Unstructured and Cartesian Meshes

    NASA Technical Reports Server (NTRS)

    Barth, Timothy (Editor); Deconinck, Herman (Editor)

    1999-01-01

    The development of high-order accurate numerical discretization techniques for irregular domains and meshes is often cited as one of the remaining challenges facing the field of computational fluid dynamics. In structural mechanics, the advantages of high-order finite element approximation are widely recognized. This is especially true when high-order element approximation is combined with element refinement (h-p refinement). In computational fluid dynamics, high-order discretization methods are infrequently used in the computation of compressible fluid flow. The hyperbolic nature of the governing equations and the presence of solution discontinuities makes high-order accuracy difficult to achieve. Consequently, second-order accurate methods are still predominately used in industrial applications even though evidence suggests that high-order methods may offer a way to significantly improve the resolution and accuracy for these calculations. To address this important topic, a special course was jointly organized by the Applied Vehicle Technology Panel of NATO's Research and Technology Organization (RTO), the von Karman Institute for Fluid Dynamics, and the Numerical Aerospace Simulation Division at the NASA Ames Research Center. The NATO RTO sponsored course entitled "Higher Order Discretization Methods in Computational Fluid Dynamics" was held September 14-18, 1998 at the von Karman Institute for Fluid Dynamics in Belgium and September 21-25, 1998 at the NASA Ames Research Center in the United States. During this special course, lecturers from Europe and the United States gave a series of comprehensive lectures on advanced topics related to the high-order numerical discretization of partial differential equations with primary emphasis given to computational fluid dynamics (CFD). Additional consideration was given to topics in computational physics such as the high-order discretization of the Hamilton-Jacobi, Helmholtz, and elasticity equations. This volume consists

  18. Accurate and Efficient Approximation to the Optimized Effective Potential for Exchange

    NASA Astrophysics Data System (ADS)

    Ryabinkin, Ilya G.; Kananenka, Alexei A.; Staroverov, Viktor N.

    2013-07-01

    We devise an efficient practical method for computing the Kohn-Sham exchange-correlation potential corresponding to a Hartree-Fock electron density. This potential is almost indistinguishable from the exact-exchange optimized effective potential (OEP) and, when used as an approximation to the OEP, is vastly better than all existing models. Using our method one can obtain unambiguous, nearly exact OEPs for any reasonable finite one-electron basis set at the same low cost as the Krieger-Li-Iafrate and Becke-Johnson potentials. For all practical purposes, this solves the long-standing problem of black-box construction of OEPs in exact-exchange calculations.

  19. An Approximate Approach to Automatic Kernel Selection.

    PubMed

    Ding, Lizhong; Liao, Shizhong

    2016-02-02

    Kernel selection is a fundamental problem of kernel-based learning algorithms. In this paper, we propose an approximate approach to automatic kernel selection for regression from the perspective of kernel matrix approximation. We first introduce multilevel circulant matrices into automatic kernel selection, and develop two approximate kernel selection algorithms by exploiting the computational virtues of multilevel circulant matrices. The complexity of the proposed algorithms is quasi-linear in the number of data points. Then, we prove an approximation error bound to measure the effect of the approximation in kernel matrices by multilevel circulant matrices on the hypothesis and further show that the approximate hypothesis produced with multilevel circulant matrices converges to the accurate hypothesis produced with kernel matrices. Experimental evaluations on benchmark datasets demonstrate the effectiveness of approximate kernel selection.

  20. Low-frequency sound propagation modeling over a locally-reacting boundary using the parabolic approximation

    NASA Technical Reports Server (NTRS)

    Robertson, J. S.; Siegman, W. L.; Jacobson, M. J.

    1989-01-01

    There is substantial interest in the analytical and numerical modeling of low-frequency, long-range atmospheric acoustic propagation. Ray-based models, because of frequency limitations, do not always give an adequate prediction of quantities such as sound pressure or intensity levels. However, the parabolic approximation method, widely used in ocean acoustics, and often more accurate than ray models for lower frequencies of interest, can be applied to acoustic propagation in the atmosphere. Modifications of an existing implicit finite-difference implementation for computing solutions to the parabolic approximation are discussed. A locally-reacting boundary is used together with a one-parameter impedance model. Intensity calculations are performed for a number of flow resistivity values in both quiescent and windy atmospheres. Variations in the value of this parameter are shown to have substantial effects on the spatial variation of the acoustic signal.

  1. Accurate solution of the Poisson equation with discontinuities

    NASA Astrophysics Data System (ADS)

    Nave, Jean-Christophe; Marques, Alexandre; Rosales, Rodolfo

    2017-11-01

    Solving the Poisson equation in the presence of discontinuities is of great importance in many applications of science and engineering. In many cases, the discontinuities are caused by interfaces between different media, such as in multiphase flows. These interfaces are themselves solutions to differential equations, and can assume complex configurations. For this reason, it is convenient to embed the interface into a regular triangulation or Cartesian grid and solve the Poisson equation in this regular domain. We present an extension of the Correction Function Method (CFM), which was developed to solve the Poisson equation in the context of embedded interfaces. The distinctive feature of the CFM is that it uses partial differential equations to construct smooth extensions of the solution in the vicinity of interfaces. A consequence of this approach is that it can achieve high order of accuracy while maintaining compact discretizations. The extension we present removes the restrictions of the original CFM, and yields a method that can solve the Poisson equation when discontinuities are present in the solution, the coefficients of the equation (material properties), and the source term. We show results computed to fourth order of accuracy in two and three dimensions. This work was partially funded by DARPA, NSF, and NSERC.

  2. Exact and approximate solutions for the decades-old Michaelis-Menten equation: Progress-curve analysis through integrated rate equations.

    PubMed

    Goličnik, Marko

    2011-01-01

    The Michaelis-Menten rate equation can be found in most general biochemistry textbooks, where the time derivative of the substrate is a hyperbolic function of two kinetic parameters (the limiting rate V, and the Michaelis constant K(M) ) and the amount of substrate. However, fundamental concepts of enzyme kinetics can be difficult to understand fully, or can even be misunderstood, by students when based only on the differential form of the Michaelis-Menten equation, and the variety of methods available to calculate the kinetic constants from rate versus substrate concentration "textbook data." Consequently, enzyme kinetics can be confusing if an analytical solution of the Michaelis-Menten equation is not available. Therefore, the still rarely known exact solution to the Michaelis-Menten equation is presented here through the explicit closed-form equation in terms of the Lambert W(x) function. Unfortunately, as the W(x) is not available in standard curve-fitting computer programs, the practical use of this direct solution is limited for most life-science students. Thus, the purpose of this article is to provide analytical approximations to the equation for modeling Michaelis-Menten kinetics. The elementary and explicit nature of these approximations can provide students with direct and simple estimations of kinetic parameters from raw experimental time-course data. The Michaelis-Menten kinetics studied in the latter context can provide an ideal alternative to the 100-year-old problems of data transformation, graphical visualization, and data analysis of enzyme-catalyzed reactions. Hence, the content of the course presented here could gradually become an important component of the modern biochemistry curriculum in the 21st century. Copyright © 2011 Wiley Periodicals, Inc.

  3. Numerical approximations for fractional diffusion equations via a Chebyshev spectral-tau method

    NASA Astrophysics Data System (ADS)

    Doha, Eid H.; Bhrawy, Ali H.; Ezz-Eldien, Samer S.

    2013-10-01

    In this paper, a class of fractional diffusion equations with variable coefficients is considered. An accurate and efficient spectral tau technique for solving the fractional diffusion equations numerically is proposed. This method is based upon Chebyshev tau approximation together with Chebyshev operational matrix of Caputo fractional differentiation. Such approach has the advantage of reducing the problem to the solution of a system of algebraic equations, which may then be solved by any standard numerical technique. We apply this general method to solve four specific examples. In each of the examples considered, the numerical results show that the proposed method is of high accuracy and is efficient for solving the time-dependent fractional diffusion equations.

  4. Replace-approximation method for ambiguous solutions in factor analysis of ultrasonic hepatic perfusion

    NASA Astrophysics Data System (ADS)

    Zhang, Ji; Ding, Mingyue; Yuchi, Ming; Hou, Wenguang; Ye, Huashan; Qiu, Wu

    2010-03-01

    Factor analysis is an efficient technique to the analysis of dynamic structures in medical image sequences and recently has been used in contrast-enhanced ultrasound (CEUS) of hepatic perfusion. Time-intensity curves (TICs) extracted by factor analysis can provide much more diagnostic information for radiologists and improve the diagnostic rate of focal liver lesions (FLLs). However, one of the major drawbacks of factor analysis of dynamic structures (FADS) is nonuniqueness of the result when only the non-negativity criterion is used. In this paper, we propose a new method of replace-approximation based on apex-seeking for ambiguous FADS solutions. Due to a partial overlap of different structures, factor curves are assumed to be approximately replaced by the curves existing in medical image sequences. Therefore, how to find optimal curves is the key point of the technique. No matter how many structures are assumed, our method always starts to seek apexes from one-dimensional space where the original high-dimensional data is mapped. By finding two stable apexes from one dimensional space, the method can ascertain the third one. The process can be continued until all structures are found. This technique were tested on two phantoms of blood perfusion and compared to the two variants of apex-seeking method. The results showed that the technique outperformed two variants in comparison of region of interest measurements from phantom data. It can be applied to the estimation of TICs derived from CEUS images and separation of different physiological regions in hepatic perfusion.

  5. Accurate thermoelastic tensor and acoustic velocities of NaCl

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

    Marcondes, Michel L., E-mail: michel@if.usp.br; Chemical Engineering and Material Science, University of Minnesota, Minneapolis, 55455; Shukla, Gaurav, E-mail: shukla@physics.umn.edu

    Despite the importance of thermoelastic properties of minerals in geology and geophysics, their measurement at high pressures and temperatures are still challenging. Thus, ab initio calculations are an essential tool for predicting these properties at extreme conditions. Owing to the approximate description of the exchange-correlation energy, approximations used in calculations of vibrational effects, and numerical/methodological approximations, these methods produce systematic deviations. Hybrid schemes combining experimental data and theoretical results have emerged as a way to reconcile available information and offer more reliable predictions at experimentally inaccessible thermodynamics conditions. Here we introduce a method to improve the calculated thermoelastic tensor bymore » using highly accurate thermal equation of state (EoS). The corrective scheme is general, applicable to crystalline solids with any symmetry, and can produce accurate results at conditions where experimental data may not exist. We apply it to rock-salt-type NaCl, a material whose structural properties have been challenging to describe accurately by standard ab initio methods and whose acoustic/seismic properties are important for the gas and oil industry.« less

  6. Approximate bound-state solutions of the Dirac equation for the generalized yukawa potential plus the generalized tensor interaction

    NASA Astrophysics Data System (ADS)

    Ikot, Akpan N.; Maghsoodi, Elham; Hassanabadi, Hassan; Obu, Joseph A.

    2014-05-01

    In this paper, we obtain the approximate analytical bound-state solutions of the Dirac particle with the generalized Yukawa potential within the framework of spin and pseudospin symmetries for the arbitrary к state with a generalized tensor interaction. The generalized parametric Nikiforov-Uvarov method is used to obtain the energy eigenvalues and the corresponding wave functions in closed form. We also report some numerical results and present figures to show the effect of the tensor interaction.

  7. Numerical solution of sixth-order boundary-value problems using Legendre wavelet collocation method

    NASA Astrophysics Data System (ADS)

    Sohaib, Muhammad; Haq, Sirajul; Mukhtar, Safyan; Khan, Imad

    2018-03-01

    An efficient method is proposed to approximate sixth order boundary value problems. The proposed method is based on Legendre wavelet in which Legendre polynomial is used. The mechanism of the method is to use collocation points that converts the differential equation into a system of algebraic equations. For validation two test problems are discussed. The results obtained from proposed method are quite accurate, also close to exact solution, and other different methods. The proposed method is computationally more effective and leads to more accurate results as compared to other methods from literature.

  8. Fast viscosity solutions for shape from shading under a more realistic imaging model

    NASA Astrophysics Data System (ADS)

    Wang, Guohui; Han, Jiuqiang; Jia, Honghai; Zhang, Xinman

    2009-11-01

    Shape from shading (SFS) has been a classical and important problem in the domain of computer vision. The goal of SFS is to reconstruct the 3-D shape of an object from its 2-D intensity image. To this end, an image irradiance equation describing the relation between the shape of a surface and its corresponding brightness variations is used. Then it is derived as an explicit partial differential equation (PDE). Using the nonlinear programming principle, we propose a detailed solution to Prados and Faugeras's implicit scheme for approximating the viscosity solution of the resulting PDE. Furthermore, by combining implicit and semi-implicit schemes, a new approximation scheme is presented. In order to accelerate the convergence speed, we adopt the Gauss-Seidel idea and alternating sweeping strategy to the approximation schemes. Experimental results on both synthetic and real images are performed to demonstrate that the proposed methods are fast and accurate.

  9. Computing approximate solutions of the protein structure determination problem using global constraints on discrete crystal lattices.

    PubMed

    Dal Palù, Alessandro; Dovier, Agostino; Pontelli, Enrico

    2010-01-01

    Crystal lattices are discrete models of the three-dimensional space that have been effectively employed to facilitate the task of determining proteins' natural conformation. This paper investigates alternative global constraints that can be introduced in a constraint solver over discrete crystal lattices. The objective is to enhance the efficiency of lattice solvers in dealing with the construction of approximate solutions of the protein structure determination problem. Some of them (e.g., self-avoiding-walk) have been explicitly or implicitly already used in previous approaches, while others (e.g., the density constraint) are new. The intrinsic complexities of all of them are studied and preliminary experimental results are discussed.

  10. Spacing of bending-induced fractures at saturation: Numerical models and approximate analytical solution

    NASA Astrophysics Data System (ADS)

    Schöpfer, Martin; Lehner, Florian; Grasemann, Bernhard; Kaserer, Klemens; Hinsch, Ralph

    2017-04-01

    analytical solution is derived for the critical fracture spacing, i.e. the spacing below which the maximum tensile stress cannot reach the layer strength. The model results are consistent with an approximate analytical solution, and illustrate that the spacing of bending-induced fractures is proportional to layer thickness and a square root function of the ratio of layer tensile strength to confining pressure. Although highly idealised, models and analysis presented in this study offer an explanation for fracture saturation during folding and point towards certain key factors that may control fracture spacing in natural systems.

  11. On the Rigid-Lid Approximation for Two Shallow Layers of Immiscible Fluids with Small Density Contrast

    NASA Astrophysics Data System (ADS)

    Duchêne, Vincent

    2014-08-01

    The rigid-lid approximation is a commonly used simplification in the study of density-stratified fluids in oceanography. Roughly speaking, one assumes that the displacements of the surface are negligible compared with interface displacements. In this paper, we offer a rigorous justification of this approximation in the case of two shallow layers of immiscible fluids with constant and quasi-equal mass density. More precisely, we control the difference between the solutions of the Cauchy problem predicted by the shallow-water (Saint-Venant) system in the rigid-lid and free-surface configuration. We show that in the limit of a small density contrast, the flow may be accurately described as the superposition of a baroclinic (or slow) mode, which is well predicted by the rigid-lid approximation, and a barotropic (or fast) mode, whose initial smallness persists for large time. We also describe explicitly the first-order behavior of the deformation of the surface and discuss the case of a nonsmall initial barotropic mode.

  12. Osmotic potential calculations of inorganic and organic aqueous solutions over wide solute concentration levels and temperatures

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

    Cochrane, T. T., E-mail: agteca@hotmail.com; Cochrane, T. A., E-mail: tom.cochrane@canterbury.ac.nz

    Purpose: To demonstrate that the authors’ new “aqueous solution vs pure water” equation to calculate osmotic potential may be used to calculate the osmotic potentials of inorganic and organic aqueous solutions over wide ranges of solute concentrations and temperatures. Currently, the osmotic potentials of solutions used for medical purposes are calculated from equations based on the thermodynamics of the gas laws which are only accurate at low temperature and solute concentration levels. Some solutions used in medicine may need their osmotic potentials calculated more accurately to take into account solute concentrations and temperatures. Methods: The authors experimented with their newmore » equation for calculating the osmotic potentials of inorganic and organic aqueous solutions up to and beyond body temperatures by adjusting three of its factors; (a) the volume property of pure water, (b) the number of “free” water molecules per unit volume of solution, “N{sub f},” and (c) the “t” factor expressing the cooperative structural relaxation time of pure water at given temperatures. Adequate information on the volume property of pure water at different temperatures is available in the literature. However, as little information on the relative densities of inorganic and organic solutions, respectively, at varying temperatures needed to calculate N{sub f} was available, provisional equations were formulated to approximate values. Those values together with tentative t values for different temperatures chosen from values calculated by different workers were substituted into the authors’ equation to demonstrate how osmotic potentials could be estimated over temperatures up to and beyond bodily temperatures. Results: The provisional equations formulated to calculate N{sub f}, the number of free water molecules per unit volume of inorganic and organic solute solutions, respectively, over wide concentration ranges compared well with the calculations of

  13. Accurate and efficient calculation of excitation energies with the active-space particle-particle random phase approximation

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

    Zhang, Du; Yang, Weitao

    An efficient method for calculating excitation energies based on the particle-particle random phase approximation (ppRPA) is presented. Neglecting the contributions from the high-lying virtual states and the low-lying core states leads to the significantly smaller active-space ppRPA matrix while keeping the error to within 0.05 eV from the corresponding full ppRPA excitation energies. The resulting computational cost is significantly reduced and becomes less than the construction of the non-local Fock exchange potential matrix in the self-consistent-field (SCF) procedure. With only a modest number of active orbitals, the original ppRPA singlet-triplet (ST) gaps as well as the low-lying single and doublemore » excitation energies can be accurately reproduced at much reduced computational costs, up to 100 times faster than the iterative Davidson diagonalization of the original full ppRPA matrix. For high-lying Rydberg excitations where the Davidson algorithm fails, the computational savings of active-space ppRPA with respect to the direct diagonalization is even more dramatic. The virtues of the underlying full ppRPA combined with the significantly lower computational cost of the active-space approach will significantly expand the applicability of the ppRPA method to calculate excitation energies at a cost of O(K^{4}), with a prefactor much smaller than a single SCF Hartree-Fock (HF)/hybrid functional calculation, thus opening up new possibilities for the quantum mechanical study of excited state electronic structure of large systems.« less

  14. Accurate and efficient calculation of excitation energies with the active-space particle-particle random phase approximation

    DOE PAGES

    Zhang, Du; Yang, Weitao

    2016-10-13

    An efficient method for calculating excitation energies based on the particle-particle random phase approximation (ppRPA) is presented. Neglecting the contributions from the high-lying virtual states and the low-lying core states leads to the significantly smaller active-space ppRPA matrix while keeping the error to within 0.05 eV from the corresponding full ppRPA excitation energies. The resulting computational cost is significantly reduced and becomes less than the construction of the non-local Fock exchange potential matrix in the self-consistent-field (SCF) procedure. With only a modest number of active orbitals, the original ppRPA singlet-triplet (ST) gaps as well as the low-lying single and doublemore » excitation energies can be accurately reproduced at much reduced computational costs, up to 100 times faster than the iterative Davidson diagonalization of the original full ppRPA matrix. For high-lying Rydberg excitations where the Davidson algorithm fails, the computational savings of active-space ppRPA with respect to the direct diagonalization is even more dramatic. The virtues of the underlying full ppRPA combined with the significantly lower computational cost of the active-space approach will significantly expand the applicability of the ppRPA method to calculate excitation energies at a cost of O(K^{4}), with a prefactor much smaller than a single SCF Hartree-Fock (HF)/hybrid functional calculation, thus opening up new possibilities for the quantum mechanical study of excited state electronic structure of large systems.« less

  15. Analytic Approximations to the Free Boundary and Multi-dimensional Problems in Financial Derivatives Pricing

    NASA Astrophysics Data System (ADS)

    Lau, Chun Sing

    This thesis studies two types of problems in financial derivatives pricing. The first type is the free boundary problem, which can be formulated as a partial differential equation (PDE) subject to a set of free boundary condition. Although the functional form of the free boundary condition is given explicitly, the location of the free boundary is unknown and can only be determined implicitly by imposing continuity conditions on the solution. Two specific problems are studied in details, namely the valuation of fixed-rate mortgages and CEV American options. The second type is the multi-dimensional problem, which involves multiple correlated stochastic variables and their governing PDE. One typical problem we focus on is the valuation of basket-spread options, whose underlying asset prices are driven by correlated geometric Brownian motions (GBMs). Analytic approximate solutions are derived for each of these three problems. For each of the two free boundary problems, we propose a parametric moving boundary to approximate the unknown free boundary, so that the original problem transforms into a moving boundary problem which can be solved analytically. The governing parameter of the moving boundary is determined by imposing the first derivative continuity condition on the solution. The analytic form of the solution allows the price and the hedging parameters to be computed very efficiently. When compared against the benchmark finite-difference method, the computational time is significantly reduced without compromising the accuracy. The multi-stage scheme further allows the approximate results to systematically converge to the benchmark results as one recasts the moving boundary into a piecewise smooth continuous function. For the multi-dimensional problem, we generalize the Kirk (1995) approximate two-asset spread option formula to the case of multi-asset basket-spread option. Since the final formula is in closed form, all the hedging parameters can also be derived in

  16. Construction and accuracy of partial differential equation approximations to the chemical master equation.

    PubMed

    Grima, Ramon

    2011-11-01

    The mesoscopic description of chemical kinetics, the chemical master equation, can be exactly solved in only a few simple cases. The analytical intractability stems from the discrete character of the equation, and hence considerable effort has been invested in the development of Fokker-Planck equations, second-order partial differential equation approximations to the master equation. We here consider two different types of higher-order partial differential approximations, one derived from the system-size expansion and the other from the Kramers-Moyal expansion, and derive the accuracy of their predictions for chemical reactive networks composed of arbitrary numbers of unimolecular and bimolecular reactions. In particular, we show that the partial differential equation approximation of order Q from the Kramers-Moyal expansion leads to estimates of the mean number of molecules accurate to order Ω(-(2Q-3)/2), of the variance of the fluctuations in the number of molecules accurate to order Ω(-(2Q-5)/2), and of skewness accurate to order Ω(-(Q-2)). We also show that for large Q, the accuracy in the estimates can be matched only by a partial differential equation approximation from the system-size expansion of approximate order 2Q. Hence, we conclude that partial differential approximations based on the Kramers-Moyal expansion generally lead to considerably more accurate estimates in the mean, variance, and skewness than approximations of the same order derived from the system-size expansion.

  17. Structural optimization with approximate sensitivities

    NASA Technical Reports Server (NTRS)

    Patnaik, S. N.; Hopkins, D. A.; Coroneos, R.

    1994-01-01

    Computational efficiency in structural optimization can be enhanced if the intensive computations associated with the calculation of the sensitivities, that is, gradients of the behavior constraints, are reduced. Approximation to gradients of the behavior constraints that can be generated with small amount of numerical calculations is proposed. Structural optimization with these approximate sensitivities produced correct optimum solution. Approximate gradients performed well for different nonlinear programming methods, such as the sequence of unconstrained minimization technique, method of feasible directions, sequence of quadratic programming, and sequence of linear programming. Structural optimization with approximate gradients can reduce by one third the CPU time that would otherwise be required to solve the problem with explicit closed-form gradients. The proposed gradient approximation shows potential to reduce intensive computation that has been associated with traditional structural optimization.

  18. Second order accurate finite difference approximations for the transonic small disturbance equation and the full potential equation

    NASA Technical Reports Server (NTRS)

    Mostrel, M. M.

    1988-01-01

    New shock-capturing finite difference approximations for solving two scalar conservation law nonlinear partial differential equations describing inviscid, isentropic, compressible flows of aerodynamics at transonic speeds are presented. A global linear stability theorem is applied to these schemes in order to derive a necessary and sufficient condition for the finite element method. A technique is proposed to render the described approximations total variation-stable by applying the flux limiters to the nonlinear terms of the difference equation dimension by dimension. An entropy theorem applying to the approximations is proved, and an implicit, forward Euler-type time discretization of the approximation is presented. Results of some numerical experiments using the approximations are reported.

  19. Insight solutions are correct more often than analytic solutions

    PubMed Central

    Salvi, Carola; Bricolo, Emanuela; Kounios, John; Bowden, Edward; Beeman, Mark

    2016-01-01

    How accurate are insights compared to analytical solutions? In four experiments, we investigated how participants’ solving strategies influenced their solution accuracies across different types of problems, including one that was linguistic, one that was visual and two that were mixed visual-linguistic. In each experiment, participants’ self-judged insight solutions were, on average, more accurate than their analytic ones. We hypothesised that insight solutions have superior accuracy because they emerge into consciousness in an all-or-nothing fashion when the unconscious solving process is complete, whereas analytic solutions can be guesses based on conscious, prematurely terminated, processing. This hypothesis is supported by the finding that participants’ analytic solutions included relatively more incorrect responses (i.e., errors of commission) than timeouts (i.e., errors of omission) compared to their insight responses. PMID:27667960

  20. Approximation methods in gravitational-radiation theory

    NASA Technical Reports Server (NTRS)

    Will, C. M.

    1986-01-01

    The observation of gravitational-radiation damping in the binary pulsar PSR 1913 + 16 and the ongoing experimental search for gravitational waves of extraterrestrial origin have made the theory of gravitational radiation an active branch of classical general relativity. In calculations of gravitational radiation, approximation methods play a crucial role. Recent developments are summarized in two areas in which approximations are important: (a) the quadrupole approxiamtion, which determines the energy flux and the radiation reaction forces in weak-field, slow-motion, source-within-the-near-zone systems such as the binary pulsar; and (b) the normal modes of oscillation of black holes, where the Wentzel-Kramers-Brillouin approximation gives accurate estimates of the complex frequencies of the modes.

  1. A parallel offline CFD and closed-form approximation strategy for computationally efficient analysis of complex fluid flows

    NASA Astrophysics Data System (ADS)

    Allphin, Devin

    Computational fluid dynamics (CFD) solution approximations for complex fluid flow problems have become a common and powerful engineering analysis technique. These tools, though qualitatively useful, remain limited in practice by their underlying inverse relationship between simulation accuracy and overall computational expense. While a great volume of research has focused on remedying these issues inherent to CFD, one traditionally overlooked area of resource reduction for engineering analysis concerns the basic definition and determination of functional relationships for the studied fluid flow variables. This artificial relationship-building technique, called meta-modeling or surrogate/offline approximation, uses design of experiments (DOE) theory to efficiently approximate non-physical coupling between the variables of interest in a fluid flow analysis problem. By mathematically approximating these variables, DOE methods can effectively reduce the required quantity of CFD simulations, freeing computational resources for other analytical focuses. An idealized interpretation of a fluid flow problem can also be employed to create suitably accurate approximations of fluid flow variables for the purposes of engineering analysis. When used in parallel with a meta-modeling approximation, a closed-form approximation can provide useful feedback concerning proper construction, suitability, or even necessity of an offline approximation tool. It also provides a short-circuit pathway for further reducing the overall computational demands of a fluid flow analysis, again freeing resources for otherwise unsuitable resource expenditures. To validate these inferences, a design optimization problem was presented requiring the inexpensive estimation of aerodynamic forces applied to a valve operating on a simulated piston-cylinder heat engine. The determination of these forces was to be found using parallel surrogate and exact approximation methods, thus evidencing the comparative

  2. Approximate solution of coupled cluster equations: application to the coupled cluster doubles method and non-covalent interacting systems.

    PubMed

    Smiga, Szymon; Fabiano, Eduardo

    2017-11-15

    We have developed a simplified coupled cluster (SCC) methodology, using the basic idea of scaled MP2 methods. The scheme has been applied to the coupled cluster double equations and implemented in three different non-iterative variants. This new method (especially the SCCD[3] variant, which utilizes a spin-resolved formalism) has been found to be very efficient and to yield an accurate approximation of the reference CCD results for both total and interaction energies of different atoms and molecules. Furthermore, we demonstrate that the equations determining the scaling coefficients for the SCCD[3] approach can generate non-empirical SCS-MP2 scaling coefficients which are in good agreement with previous theoretical investigations.

  3. Risk approximation in decision making: approximative numeric abilities predict advantageous decisions under objective risk.

    PubMed

    Mueller, Silke M; Schiebener, Johannes; Delazer, Margarete; Brand, Matthias

    2018-01-22

    Many decision situations in everyday life involve mathematical considerations. In decisions under objective risk, i.e., when explicit numeric information is available, executive functions and abilities to handle exact numbers and ratios are predictors of objectively advantageous choices. Although still debated, exact numeric abilities, e.g., normative calculation skills, are assumed to be related to approximate number processing skills. The current study investigates the effects of approximative numeric abilities on decision making under objective risk. Participants (N = 153) performed a paradigm measuring number-comparison, quantity-estimation, risk-estimation, and decision-making skills on the basis of rapid dot comparisons. Additionally, a risky decision-making task with exact numeric information was administered, as well as tasks measuring executive functions and exact numeric abilities, e.g., mental calculation and ratio processing skills, were conducted. Approximative numeric abilities significantly predicted advantageous decision making, even beyond the effects of executive functions and exact numeric skills. Especially being able to make accurate risk estimations seemed to contribute to superior choices. We recommend approximation skills and approximate number processing to be subject of future investigations on decision making under risk.

  4. Accurate approximation method for prediction of class I MHC affinities for peptides of length 8, 10 and 11 using prediction tools trained on 9mers.

    PubMed

    Lundegaard, Claus; Lund, Ole; Nielsen, Morten

    2008-06-01

    Several accurate prediction systems have been developed for prediction of class I major histocompatibility complex (MHC):peptide binding. Most of these are trained on binding affinity data of primarily 9mer peptides. Here, we show how prediction methods trained on 9mer data can be used for accurate binding affinity prediction of peptides of length 8, 10 and 11. The method gives the opportunity to predict peptides with a different length than nine for MHC alleles where no such peptides have been measured. As validation, the performance of this approach is compared to predictors trained on peptides of the peptide length in question. In this validation, the approximation method has an accuracy that is comparable to or better than methods trained on a peptide length identical to the predicted peptides. The algorithm has been implemented in the web-accessible servers NetMHC-3.0: http://www.cbs.dtu.dk/services/NetMHC-3.0, and NetMHCpan-1.1: http://www.cbs.dtu.dk/services/NetMHCpan-1.1

  5. Numerical solution to generalized Burgers'-Fisher equation using Exp-function method hybridized with heuristic computation.

    PubMed

    Malik, Suheel Abdullah; Qureshi, Ijaz Mansoor; Amir, Muhammad; Malik, Aqdas Naveed; Haq, Ihsanul

    2015-01-01

    In this paper, a new heuristic scheme for the approximate solution of the generalized Burgers'-Fisher equation is proposed. The scheme is based on the hybridization of Exp-function method with nature inspired algorithm. The given nonlinear partial differential equation (NPDE) through substitution is converted into a nonlinear ordinary differential equation (NODE). The travelling wave solution is approximated by the Exp-function method with unknown parameters. The unknown parameters are estimated by transforming the NODE into an equivalent global error minimization problem by using a fitness function. The popular genetic algorithm (GA) is used to solve the minimization problem, and to achieve the unknown parameters. The proposed scheme is successfully implemented to solve the generalized Burgers'-Fisher equation. The comparison of numerical results with the exact solutions, and the solutions obtained using some traditional methods, including adomian decomposition method (ADM), homotopy perturbation method (HPM), and optimal homotopy asymptotic method (OHAM), show that the suggested scheme is fairly accurate and viable for solving such problems.

  6. Numerical Solution to Generalized Burgers'-Fisher Equation Using Exp-Function Method Hybridized with Heuristic Computation

    PubMed Central

    Malik, Suheel Abdullah; Qureshi, Ijaz Mansoor; Amir, Muhammad; Malik, Aqdas Naveed; Haq, Ihsanul

    2015-01-01

    In this paper, a new heuristic scheme for the approximate solution of the generalized Burgers'-Fisher equation is proposed. The scheme is based on the hybridization of Exp-function method with nature inspired algorithm. The given nonlinear partial differential equation (NPDE) through substitution is converted into a nonlinear ordinary differential equation (NODE). The travelling wave solution is approximated by the Exp-function method with unknown parameters. The unknown parameters are estimated by transforming the NODE into an equivalent global error minimization problem by using a fitness function. The popular genetic algorithm (GA) is used to solve the minimization problem, and to achieve the unknown parameters. The proposed scheme is successfully implemented to solve the generalized Burgers'-Fisher equation. The comparison of numerical results with the exact solutions, and the solutions obtained using some traditional methods, including adomian decomposition method (ADM), homotopy perturbation method (HPM), and optimal homotopy asymptotic method (OHAM), show that the suggested scheme is fairly accurate and viable for solving such problems. PMID:25811858

  7. New realisation of Preisach model using adaptive polynomial approximation

    NASA Astrophysics Data System (ADS)

    Liu, Van-Tsai; Lin, Chun-Liang; Wing, Home-Young

    2012-09-01

    Modelling system with hysteresis has received considerable attention recently due to the increasing accurate requirement in engineering applications. The classical Preisach model (CPM) is the most popular model to demonstrate hysteresis which can be represented by infinite but countable first-order reversal curves (FORCs). The usage of look-up tables is one way to approach the CPM in actual practice. The data in those tables correspond with the samples of a finite number of FORCs. This approach, however, faces two major problems: firstly, it requires a large amount of memory space to obtain an accurate prediction of hysteresis; secondly, it is difficult to derive efficient ways to modify the data table to reflect the timing effect of elements with hysteresis. To overcome, this article proposes the idea of using a set of polynomials to emulate the CPM instead of table look-up. The polynomial approximation requires less memory space for data storage. Furthermore, the polynomial coefficients can be obtained accurately by using the least-square approximation or adaptive identification algorithm, such as the possibility of accurate tracking of hysteresis model parameters.

  8. Recognition of computerized facial approximations by familiar assessors.

    PubMed

    Richard, Adam H; Monson, Keith L

    2017-11-01

    Studies testing the effectiveness of facial approximations typically involve groups of participants who are unfamiliar with the approximated individual(s). This limitation requires the use of photograph arrays including a picture of the subject for comparison to the facial approximation. While this practice is often necessary due to the difficulty in obtaining a group of assessors who are familiar with the approximated subject, it may not accurately simulate the thought process of the target audience (friends and family members) in comparing a mental image of the approximated subject to the facial approximation. As part of a larger process to evaluate the effectiveness and best implementation of the ReFace facial approximation software program, the rare opportunity arose to conduct a recognition study using assessors who were personally acquainted with the subjects of the approximations. ReFace facial approximations were generated based on preexisting medical scans, and co-workers of the scan donors were tested on whether they could accurately pick out the approximation of their colleague from arrays of facial approximations. Results from the study demonstrated an overall poor recognition performance (i.e., where a single choice within a pool is not enforced) for individuals who were familiar with the approximated subjects. Out of 220 recognition tests only 10.5% resulted in the assessor selecting the correct approximation (or correctly choosing not to make a selection when the array consisted only of foils), an outcome that was not significantly different from the 9% random chance rate. When allowed to select multiple approximations the assessors felt resembled the target individual, the overall sensitivity for ReFace approximations was 16.0% and the overall specificity was 81.8%. These results differ markedly from the results of a previous study using assessors who were unfamiliar with the approximated subjects. Some possible explanations for this disparity in

  9. Contextual classification of multispectral image data: Approximate algorithm

    NASA Technical Reports Server (NTRS)

    Tilton, J. C. (Principal Investigator)

    1980-01-01

    An approximation to a classification algorithm incorporating spatial context information in a general, statistical manner is presented which is computationally less intensive. Classifications that are nearly as accurate are produced.

  10. Automatically high accurate and efficient photomask defects management solution for advanced lithography manufacture

    NASA Astrophysics Data System (ADS)

    Zhu, Jun; Chen, Lijun; Ma, Lantao; Li, Dejian; Jiang, Wei; Pan, Lihong; Shen, Huiting; Jia, Hongmin; Hsiang, Chingyun; Cheng, Guojie; Ling, Li; Chen, Shijie; Wang, Jun; Liao, Wenkui; Zhang, Gary

    2014-04-01

    Defect review is a time consuming job. Human error makes result inconsistent. The defects located on don't care area would not hurt the yield and no need to review them such as defects on dark area. However, critical area defects can impact yield dramatically and need more attention to review them such as defects on clear area. With decrease in integrated circuit dimensions, mask defects are always thousands detected during inspection even more. Traditional manual or simple classification approaches are unable to meet efficient and accuracy requirement. This paper focuses on automatic defect management and classification solution using image output of Lasertec inspection equipment and Anchor pattern centric image process technology. The number of mask defect found during an inspection is always in the range of thousands or even more. This system can handle large number defects with quick and accurate defect classification result. Our experiment includes Die to Die and Single Die modes. The classification accuracy can reach 87.4% and 93.3%. No critical or printable defects are missing in our test cases. The missing classification defects are 0.25% and 0.24% in Die to Die mode and Single Die mode. This kind of missing rate is encouraging and acceptable to apply on production line. The result can be output and reloaded back to inspection machine to have further review. This step helps users to validate some unsure defects with clear and magnification images when captured images can't provide enough information to make judgment. This system effectively reduces expensive inline defect review time. As a fully inline automated defect management solution, the system could be compatible with current inspection approach and integrated with optical simulation even scoring function and guide wafer level defect inspection.

  11. Correction of the near threshold behavior of electron collisional excitation cross-sections in the plane-wave Born approximation

    NASA Astrophysics Data System (ADS)

    Kilcrease, D. P.; Brookes, S.

    2013-12-01

    The modeling of NLTE plasmas requires the solution of population rate equations to determine the populations of the various atomic levels relevant to a particular problem. The equations require many cross sections for excitation, de-excitation, ionization and recombination. A simple and computational fast way to calculate electron collisional excitation cross-sections for ions is by using the plane-wave Born approximation. This is essentially a high-energy approximation and the cross section suffers from the unphysical problem of going to zero near threshold. Various remedies for this problem have been employed with varying degrees of success. We present a correction procedure for the Born cross-sections that employs the Elwert-Sommerfeld factor to correct for the use of plane waves instead of Coulomb waves in an attempt to produce a cross-section similar to that from using the more time consuming Coulomb Born approximation. We compare this new approximation with other, often employed correction procedures. We also look at some further modifications to our Born Elwert procedure and its combination with Y.K. Kim's correction of the Coulomb Born approximation for singly charged ions that more accurately approximate convergent close coupling calculations.

  12. Approximation theory for LQG (Linear-Quadratic-Gaussian) optimal control of flexible structures

    NASA Technical Reports Server (NTRS)

    Gibson, J. S.; Adamian, A.

    1988-01-01

    An approximation theory is presented for the LQG (Linear-Quadratic-Gaussian) optimal control problem for flexible structures whose distributed models have bounded input and output operators. The main purpose of the theory is to guide the design of finite dimensional compensators that approximate closely the optimal compensator. The optimal LQG problem separates into an optimal linear-quadratic regulator problem and an optimal state estimation problem. The solution of the former problem lies in the solution to an infinite dimensional Riccati operator equation. The approximation scheme approximates the infinite dimensional LQG problem with a sequence of finite dimensional LQG problems defined for a sequence of finite dimensional, usually finite element or modal, approximations of the distributed model of the structure. Two Riccati matrix equations determine the solution to each approximating problem. The finite dimensional equations for numerical approximation are developed, including formulas for converting matrix control and estimator gains to their functional representation to allow comparison of gains based on different orders of approximation. Convergence of the approximating control and estimator gains and of the corresponding finite dimensional compensators is studied. Also, convergence and stability of the closed-loop systems produced with the finite dimensional compensators are discussed. The convergence theory is based on the convergence of the solutions of the finite dimensional Riccati equations to the solutions of the infinite dimensional Riccati equations. A numerical example with a flexible beam, a rotating rigid body, and a lumped mass is given.

  13. Analytical approximation of the InGaZnO thin-film transistors surface potential

    NASA Astrophysics Data System (ADS)

    Colalongo, Luigi

    2016-10-01

    Surface-potential-based mathematical models are among the most accurate and physically based compact models of thin-film transistors, and in turn of indium gallium zinc oxide TFTs, available today. However, the need of iterative computations of the surface potential limits their computational efficiency and diffusion in CAD applications. The existing closed-form approximations of the surface potential are based on regional approximations and empirical smoothing functions that could result not accurate enough in particular to model transconductances and transcapacitances. In this work we present an extremely accurate (in the range of nV) and computationally efficient non-iterative approximation of the surface potential that can serve as a basis for advanced surface-potential-based indium gallium zinc oxide TFTs models.

  14. Minimax rational approximation of the Fermi-Dirac distribution.

    PubMed

    Moussa, Jonathan E

    2016-10-28

    Accurate rational approximations of the Fermi-Dirac distribution are a useful component in many numerical algorithms for electronic structure calculations. The best known approximations use O(log(βΔ)log(ϵ -1 )) poles to achieve an error tolerance ϵ at temperature β -1 over an energy interval Δ. We apply minimax approximation to reduce the number of poles by a factor of four and replace Δ with Δ occ , the occupied energy interval. This is particularly beneficial when Δ ≫ Δ occ , such as in electronic structure calculations that use a large basis set.

  15. Minimax rational approximation of the Fermi-Dirac distribution

    NASA Astrophysics Data System (ADS)

    Moussa, Jonathan E.

    2016-10-01

    Accurate rational approximations of the Fermi-Dirac distribution are a useful component in many numerical algorithms for electronic structure calculations. The best known approximations use O(log(βΔ)log(ɛ-1)) poles to achieve an error tolerance ɛ at temperature β-1 over an energy interval Δ. We apply minimax approximation to reduce the number of poles by a factor of four and replace Δ with Δocc, the occupied energy interval. This is particularly beneficial when Δ ≫ Δocc, such as in electronic structure calculations that use a large basis set.

  16. Analytical approximation schemes for solving exact renormalization group equations in the local potential approximation

    NASA Astrophysics Data System (ADS)

    Bervillier, C.; Boisseau, B.; Giacomini, H.

    2008-02-01

    The relation between the Wilson-Polchinski and the Litim optimized ERGEs in the local potential approximation is studied with high accuracy using two different analytical approaches based on a field expansion: a recently proposed genuine analytical approximation scheme to two-point boundary value problems of ordinary differential equations, and a new one based on approximating the solution by generalized hypergeometric functions. A comparison with the numerical results obtained with the shooting method is made. A similar accuracy is reached in each case. Both two methods appear to be more efficient than the usual field expansions frequently used in the current studies of ERGEs (in particular for the Wilson-Polchinski case in the study of which they fail).

  17. Approximation methods for combined thermal/structural design

    NASA Technical Reports Server (NTRS)

    Haftka, R. T.; Shore, C. P.

    1979-01-01

    Two approximation concepts for combined thermal/structural design are evaluated. The first concept is an approximate thermal analysis based on the first derivatives of structural temperatures with respect to design variables. Two commonly used first-order Taylor series expansions are examined. The direct and reciprocal expansions are special members of a general family of approximations, and for some conditions other members of that family of approximations are more accurate. Several examples are used to compare the accuracy of the different expansions. The second approximation concept is the use of critical time points for combined thermal and stress analyses of structures with transient loading conditions. Significant time savings are realized by identifying critical time points and performing the stress analysis for those points only. The design of an insulated panel which is exposed to transient heating conditions is discussed.

  18. Exact and approximate Fourier rebinning algorithms for the solution of the data truncation problem in 3-D PET.

    PubMed

    Bouallègue, Fayçal Ben; Crouzet, Jean-François; Comtat, Claude; Fourcade, Marjolaine; Mohammadi, Bijan; Mariano-Goulart, Denis

    2007-07-01

    This paper presents an extended 3-D exact rebinning formula in the Fourier space that leads to an iterative reprojection algorithm (iterative FOREPROJ), which enables the estimation of unmeasured oblique projection data on the basis of the whole set of measured data. In first approximation, this analytical formula also leads to an extended Fourier rebinning equation that is the basis for an approximate reprojection algorithm (extended FORE). These algorithms were evaluated on numerically simulated 3-D positron emission tomography (PET) data for the solution of the truncation problem, i.e., the estimation of the missing portions in the oblique projection data, before the application of algorithms that require complete projection data such as some rebinning methods (FOREX) or 3-D reconstruction algorithms (3DRP or direct Fourier methods). By taking advantage of all the 3-D data statistics, the iterative FOREPROJ reprojection provides a reliable alternative to the classical FOREPROJ method, which only exploits the low-statistics nonoblique data. It significantly improves the quality of the external reconstructed slices without loss of spatial resolution. As for the approximate extended FORE algorithm, it clearly exhibits limitations due to axial interpolations, but will require clinical studies with more realistic measured data in order to decide on its pertinence.

  19. Engineering the Charge Transport of Ag Nanocrystals for Highly Accurate, Wearable Temperature Sensors through All-Solution Processes.

    PubMed

    Joh, Hyungmok; Lee, Seung-Wook; Seong, Mingi; Lee, Woo Seok; Oh, Soong Ju

    2017-06-01

    All-nanocrystal (NC)-based and all-solution-processed wearable resistance temperature detectors (RTDs) are introduced. The charge transport mechanisms of Ag NC thin films are engineered through various ligand treatments to design high performance RTDs. Highly conductive Ag NC thin films exhibiting metallic transport behavior with high positive temperature coefficients of resistance (TCRs) are achieved through tetrabutylammonium bromide treatment. Ag NC thin films showing hopping transport with high negative TCRs are created through organic ligand treatment. All-solution-based, one-step photolithography techniques that integrate two distinct opposite-sign TCR Ag NC thin films into an ultrathin single device are developed to decouple the mechanical effects such as human motion. The unconventional materials design and strategy enables highly accurate, sensitive, wearable and motion-free RTDs, demonstrated by experiments on moving or curved objects such as human skin, and simulation results based on charge transport analysis. This strategy provides a low cost and simple method to design wearable multifunctional sensors with high sensitivity which could be utilized in various fields such as biointegrated sensors or electronic skin. © 2017 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

  20. Embedding impedance approximations in the analysis of SIS mixers

    NASA Technical Reports Server (NTRS)

    Kerr, A. R.; Pan, S.-K.; Withington, S.

    1992-01-01

    Future millimeter-wave radio astronomy instruments will use arrays of many SIS receivers, either as focal plane arrays on individual radio telescopes, or as individual receivers on the many antennas of radio interferometers. Such applications will require broadband integrated mixers without mechanical tuners. To produce such mixers, it will be necessary to improve present mixer design techniques, most of which use the three-frequency approximation to Tucker's quantum mixer theory. This paper examines the adequacy of three approximations to Tucker's theory: (1) the usual three-frequency approximation which assumes a sinusoidal LO voltage at the junction, and a short-circuit at all frequencies above the upper sideband; (2) a five-frequency approximation which allows two LO voltage harmonics and five small-signal sidebands; and (3) a quasi five-frequency approximation in which five small-signal sidebands are allowed, but the LO voltage is assumed sinusoidal. These are compared with a full harmonic-Newton solution of Tucker's equations, including eight LO harmonics and their corresponding sidebands, for realistic SIS mixer circuits. It is shown that the accuracy of the three approximations depends strongly on the value of omega R(sub N)C for the SIS junctions used. For large omega R(sub N)C, all three approximations approach the eight-harmonic solution. For omega R(sub N)C values in the range 0.5 to 10, the range of most practical interest, the quasi five-frequency approximation is a considerable improvement over the three-frequency approximation, and should be suitable for much design work. For the realistic SIS mixers considered here, the five-frequency approximation gives results very close to those of the eight-harmonic solution. Use of these approximations, where appropriate, considerably reduces the computational effort needed to analyze an SIS mixer, and allows the design and optimization of mixers using a personal computer.

  1. Extension of the KLI approximation toward the exact optimized effective potential.

    PubMed

    Iafrate, G J; Krieger, J B

    2013-03-07

    The integral equation for the optimized effective potential (OEP) is utilized in a compact form from which an accurate OEP solution for the spin-unrestricted exchange-correlation potential, Vxcσ, is obtained for any assumed orbital-dependent exchange-correlation energy functional. The method extends beyond the Krieger-Li-Iafrate (KLI) approximation toward the exact OEP result. The compact nature of the OEP equation arises by replacing the integrals involving the Green's function terms in the traditional OEP equation by an equivalent first-order perturbation theory wavefunction often referred to as the "orbital shift" function. Significant progress is then obtained by solving the equation for the first order perturbation theory wavefunction by use of Dalgarno functions which are determined from well known methods of partial differential equations. The use of Dalgarno functions circumvents the need to explicitly address the Green's functions and the associated problems with "sum over states" numerics; as well, the Dalgarno functions provide ease in dealing with inherent singularities arising from the origin and the zeros of the occupied orbital wavefunctions. The Dalgarno approach for finding a solution to the OEP equation is described herein, and a detailed illustrative example is presented for the special case of a spherically symmetric exchange-correlation potential. For the case of spherical symmetry, the relevant Dalgarno function is derived by direct integration of the appropriate radial equation while utilizing a user friendly method which explicitly treats the singular behavior at the origin and at the nodal singularities arising from the zeros of the occupied states. The derived Dalgarno function is shown to be an explicit integral functional of the exact OEP Vxcσ, thus allowing for the reduction of the OEP equation to a self-consistent integral equation for the exact exchange-correlation potential; the exact solution to this integral equation can be

  2. Extension of the KLI approximation toward the exact optimized effective potential

    NASA Astrophysics Data System (ADS)

    Iafrate, G. J.; Krieger, J. B.

    2013-03-01

    The integral equation for the optimized effective potential (OEP) is utilized in a compact form from which an accurate OEP solution for the spin-unrestricted exchange-correlation potential, Vxcσ, is obtained for any assumed orbital-dependent exchange-correlation energy functional. The method extends beyond the Krieger-Li-Iafrate (KLI) approximation toward the exact OEP result. The compact nature of the OEP equation arises by replacing the integrals involving the Green's function terms in the traditional OEP equation by an equivalent first-order perturbation theory wavefunction often referred to as the "orbital shift" function. Significant progress is then obtained by solving the equation for the first order perturbation theory wavefunction by use of Dalgarno functions which are determined from well known methods of partial differential equations. The use of Dalgarno functions circumvents the need to explicitly address the Green's functions and the associated problems with "sum over states" numerics; as well, the Dalgarno functions provide ease in dealing with inherent singularities arising from the origin and the zeros of the occupied orbital wavefunctions. The Dalgarno approach for finding a solution to the OEP equation is described herein, and a detailed illustrative example is presented for the special case of a spherically symmetric exchange-correlation potential. For the case of spherical symmetry, the relevant Dalgarno function is derived by direct integration of the appropriate radial equation while utilizing a user friendly method which explicitly treats the singular behavior at the origin and at the nodal singularities arising from the zeros of the occupied states. The derived Dalgarno function is shown to be an explicit integral functional of the exact OEP Vxcσ, thus allowing for the reduction of the OEP equation to a self-consistent integral equation for the exact exchange-correlation potential; the exact solution to this integral equation can be

  3. Spatial adaption procedures on unstructured meshes for accurate unsteady aerodynamic flow computation

    NASA Technical Reports Server (NTRS)

    Rausch, Russ D.; Batina, John T.; Yang, Henry T. Y.

    1991-01-01

    Spatial adaption procedures for the accurate and efficient solution of steady and unsteady inviscid flow problems are described. The adaption procedures were developed and implemented within a two-dimensional unstructured-grid upwind-type Euler code. These procedures involve mesh enrichment and mesh coarsening to either add points in a high gradient region or the flow or remove points where they are not needed, respectively, to produce solutions of high spatial accuracy at minimal computational costs. A detailed description is given of the enrichment and coarsening procedures and comparisons with alternative results and experimental data are presented to provide an assessment of the accuracy and efficiency of the capability. Steady and unsteady transonic results, obtained using spatial adaption for the NACA 0012 airfoil, are shown to be of high spatial accuracy, primarily in that the shock waves are very sharply captured. The results were obtained with a computational savings of a factor of approximately fifty-three for a steady case and as much as twenty-five for the unsteady cases.

  4. Spatial adaption procedures on unstructured meshes for accurate unsteady aerodynamic flow computation

    NASA Technical Reports Server (NTRS)

    Rausch, Russ D.; Yang, Henry T. Y.; Batina, John T.

    1991-01-01

    Spatial adaption procedures for the accurate and efficient solution of steady and unsteady inviscid flow problems are described. The adaption procedures were developed and implemented within a two-dimensional unstructured-grid upwind-type Euler code. These procedures involve mesh enrichment and mesh coarsening to either add points in high gradient regions of the flow or remove points where they are not needed, respectively, to produce solutions of high spatial accuracy at minimal computational cost. The paper gives a detailed description of the enrichment and coarsening procedures and presents comparisons with alternative results and experimental data to provide an assessment of the accuracy and efficiency of the capability. Steady and unsteady transonic results, obtained using spatial adaption for the NACA 0012 airfoil, are shown to be of high spatial accuracy, primarily in that the shock waves are very sharply captured. The results were obtained with a computational savings of a factor of approximately fifty-three for a steady case and as much as twenty-five for the unsteady cases.

  5. An Approximate Solution and Master Curves for Buckling of Symmetrically Laminated Composite Cylinders

    NASA Technical Reports Server (NTRS)

    Nemeth, Michael P.

    2013-01-01

    Nondimensional linear-bifurcation buckling equations for balanced, symmetrically laminated cylinders with negligible shell-wall anisotropies and subjected to uniform axial compression loads are presented. These equations are solved exactly for the practical case of simply supported ends. Nondimensional quantities are used to characterize the buckling behavior that consist of a stiffness-weighted length-to-radius parameter, a stiffness-weighted shell-thinness parameter, a shell-wall nonhomogeneity parameter, two orthotropy parameters, and a nondimensional buckling load. Ranges for the nondimensional parameters are established that encompass a wide range of laminated-wall constructions and numerous generic plots of nondimensional buckling load versus a stiffness-weighted length-to-radius ratio are presented for various combinations of the other parameters. These plots are expected to include many practical cases of interest to designers. Additionally, these plots show how the parameter values affect the distribution and size of the festoons forming each response curve and how they affect the attenuation of each response curve to the corresponding solution for an infinitely long cylinder. To aid in preliminary design studies, approximate formulas for the nondimensional buckling load are derived, and validated against the corresponding exact solution, that give the attenuated buckling response of an infinitely long cylinder in terms of the nondimensional parameters presented herein. A relatively small number of "master curves" are identified that give a nondimensional measure of the buckling load of an infinitely long cylinder as a function of the orthotropy and wall inhomogeneity parameters. These curves reduce greatly the complexity of the design-variable space as compared to representations that use dimensional quantities as design variables. As a result of their inherent simplicity, these master curves are anticipated to be useful in the ongoing development of

  6. Engine With Regression and Neural Network Approximators Designed

    NASA Technical Reports Server (NTRS)

    Patnaik, Surya N.; Hopkins, Dale A.

    2001-01-01

    At the NASA Glenn Research Center, the NASA engine performance program (NEPP, ref. 1) and the design optimization testbed COMETBOARDS (ref. 2) with regression and neural network analysis-approximators have been coupled to obtain a preliminary engine design methodology. The solution to a high-bypass-ratio subsonic waverotor-topped turbofan engine, which is shown in the preceding figure, was obtained by the simulation depicted in the following figure. This engine is made of 16 components mounted on two shafts with 21 flow stations. The engine is designed for a flight envelope with 47 operating points. The design optimization utilized both neural network and regression approximations, along with the cascade strategy (ref. 3). The cascade used three algorithms in sequence: the method of feasible directions, the sequence of unconstrained minimizations technique, and sequential quadratic programming. The normalized optimum thrusts obtained by the three methods are shown in the following figure: the cascade algorithm with regression approximation is represented by a triangle, a circle is shown for the neural network solution, and a solid line indicates original NEPP results. The solutions obtained from both approximate methods lie within one standard deviation of the benchmark solution for each operating point. The simulation improved the maximum thrust by 5 percent. The performance of the linear regression and neural network methods as alternate engine analyzers was found to be satisfactory for the analysis and operation optimization of air-breathing propulsion engines (ref. 4).

  7. A comparison of the reduced and approximate systems for the time dependent computation of the polar wind and multiconstituent stellar winds

    NASA Technical Reports Server (NTRS)

    Browning, G. L.; Holzer, T. E.

    1992-01-01

    The paper derives the 'reduced' system of equations commonly used to describe the time evolution of the polar wind and multiconstituent stellar winds from the equations for a multispecies plasma with known temperature profiles by assuming that the electron thermal speed approaches infinity. The reduced system is proved to have unbounded growth near the sonic point of the protons for many of the standard parameter cases. For the same parameter cases, the unmodified system exhibits growth in some of the Fourier modes, but this growth is bounded. An alternate system (the 'approximate' system) in which the electron thermal speed is slowed down is introduced. The approximate system retains the mathematical behavior of the unmodified system and can be shown to accurately describe the smooth solutions of the unmodified system. Other advantages of the approximate system over the reduced system are discussed.

  8. Fast iterative solution of the Bethe-Salpeter eigenvalue problem using low-rank and QTT tensor approximation

    NASA Astrophysics Data System (ADS)

    Benner, Peter; Dolgov, Sergey; Khoromskaia, Venera; Khoromskij, Boris N.

    2017-04-01

    In this paper, we propose and study two approaches to approximate the solution of the Bethe-Salpeter equation (BSE) by using structured iterative eigenvalue solvers. Both approaches are based on the reduced basis method and low-rank factorizations of the generating matrices. We also propose to represent the static screen interaction part in the BSE matrix by a small active sub-block, with a size balancing the storage for rank-structured representations of other matrix blocks. We demonstrate by various numerical tests that the combination of the diagonal plus low-rank plus reduced-block approximation exhibits higher precision with low numerical cost, providing as well a distinct two-sided error estimate for the smallest eigenvalues of the Bethe-Salpeter operator. The complexity is reduced to O (Nb2) in the size of the atomic orbitals basis set, Nb, instead of the practically intractable O (Nb6) scaling for the direct diagonalization. In the second approach, we apply the quantized-TT (QTT) tensor representation to both, the long eigenvectors and the column vectors in the rank-structured BSE matrix blocks, and combine this with the ALS-type iteration in block QTT format. The QTT-rank of the matrix entities possesses almost the same magnitude as the number of occupied orbitals in the molecular systems, No approximation is estimated by O (log ⁡ (No)No2). We confirm numerically a considerable decrease in computational time for the presented iterative approaches applied to various compact and chain-type molecules, while supporting sufficient accuracy.

  9. Effectiveness of “Thin-Layer” and “Effective Medium” Approximations in Numerical Simulation of Dielectric Spectra of Biological Cell Suspensions

    NASA Astrophysics Data System (ADS)

    Asami, Koji

    2010-12-01

    There are a few concerns in dielectric modeling of biological cells by the finite-element method (FEM) to simulate their dielectric spectra. Cells possess thin plasma membranes and membrane-bound intracellular organelles, requiring extra fine meshes and considerable computational tasks in the simulation. To solve the problems, the “thin-layer” approximation (TLA) and the “effective medium” approximation (EMA) were adopted. TLA deals with the membrane as an interface of the specific membrane impedance, and therefore it is not necessary to divide the membrane region. EMA regards the composite cytoplasm as an effective homogeneous phase whose dielectric properties are calculated separately. It was proved that TLA and EMA were both useful for greatly reducing computational tasks while accurately coinciding with analytical solutions.

  10. Approximate solution of the p-median minimization problem

    NASA Astrophysics Data System (ADS)

    Il'ev, V. P.; Il'eva, S. D.; Navrotskaya, A. A.

    2016-09-01

    A version of the facility location problem (the well-known p-median minimization problem) and its generalization—the problem of minimizing a supermodular set function—is studied. These problems are NP-hard, and they are approximately solved by a gradient algorithm that is a discrete analog of the steepest descent algorithm. A priori bounds on the worst-case behavior of the gradient algorithm for the problems under consideration are obtained. As a consequence, a bound on the performance guarantee of the gradient algorithm for the p-median minimization problem in terms of the production and transportation cost matrix is obtained.

  11. Minimax rational approximation of the Fermi-Dirac distribution

    DOE PAGES

    Moussa, Jonathan E.

    2016-10-27

    Accurate rational approximations of the Fermi-Dirac distribution are a useful component in many numerical algorithms for electronic structure calculations. The best known approximations use O(log(βΔ)log(ϵ –1)) poles to achieve an error tolerance ϵ at temperature β –1 over an energy interval Δ. We apply minimax approximation to reduce the number of poles by a factor of four and replace Δ with Δ occ, the occupied energy interval. Furthermore, this is particularly beneficial when Δ >> Δ occ, such as in electronic structure calculations that use a large basis set.

  12. Phase-field approach to implicit solvation of biomolecules with Coulomb-field approximation

    NASA Astrophysics Data System (ADS)

    Zhao, Yanxiang; Kwan, Yuen-Yick; Che, Jianwei; Li, Bo; McCammon, J. Andrew

    2013-07-01

    A phase-field variational implicit-solvent approach is developed for the solvation of charged molecules. The starting point of such an approach is the representation of a solute-solvent interface by a phase field that takes one value in the solute region and another in the solvent region, with a smooth transition from one to the other on a small transition layer. The minimization of an effective free-energy functional of all possible phase fields determines the equilibrium conformations and free energies of an underlying molecular system. All the surface energy, the solute-solvent van der Waals interaction, and the electrostatic interaction are coupled together self-consistently through a phase field. The surface energy results from the minimization of a double-well potential and the gradient of a field. The electrostatic interaction is described by the Coulomb-field approximation. Accurate and efficient methods are designed and implemented to numerically relax an underlying charged molecular system. Applications to single ions, a two-plate system, and a two-domain protein reveal that the new theory and methods can capture capillary evaporation in hydrophobic confinement and corresponding multiple equilibrium states as found in molecular dynamics simulations. Comparisons of the phase-field and the original sharp-interface variational approaches are discussed.

  13. Analytic approximations to the modon dispersion relation. [in oceanography

    NASA Technical Reports Server (NTRS)

    Boyd, J. P.

    1981-01-01

    Three explicit analytic approximations are given to the modon dispersion relation developed by Flierl et al. (1980) to describe Gulf Stream rings and related phenomena in the oceans and atmosphere. The solutions are in the form of k(q), and are developed in the form of a power series in q for small q, an inverse power series in 1/q for large q, and a two-point Pade approximant. The low order Pade approximant is shown to yield a solution for the dispersion relation with a maximum relative error for the lowest branch of the function equal to one in 700 in the q interval zero to infinity.

  14. Second-order accurate nonoscillatory schemes for scalar conservation laws

    NASA Technical Reports Server (NTRS)

    Huynh, Hung T.

    1989-01-01

    Explicit finite difference schemes for the computation of weak solutions of nonlinear scalar conservation laws is presented and analyzed. These schemes are uniformly second-order accurate and nonoscillatory in the sense that the number of extrema of the discrete solution is not increasing in time.

  15. Recent advances in approximation concepts for optimum structural design

    NASA Technical Reports Server (NTRS)

    Barthelemy, Jean-Francois M.; Haftka, Raphael T.

    1991-01-01

    The basic approximation concepts used in structural optimization are reviewed. Some of the most recent developments in that area since the introduction of the concept in the mid-seventies are discussed. The paper distinguishes between local, medium-range, and global approximations; it covers functions approximations and problem approximations. It shows that, although the lack of comparative data established on reference test cases prevents an accurate assessment, there have been significant improvements. The largest number of developments have been in the areas of local function approximations and use of intermediate variable and response quantities. It also appears that some new methodologies are emerging which could greatly benefit from the introduction of new computer architecture.

  16. State space truncation with quantified errors for accurate solutions to discrete Chemical Master Equation

    PubMed Central

    Cao, Youfang; Terebus, Anna; Liang, Jie

    2016-01-01

    truncation and error analysis methods developed here can be used to ensure accurate direct solutions to the dCME for a large number of stochastic networks. PMID:27105653

  17. State Space Truncation with Quantified Errors for Accurate Solutions to Discrete Chemical Master Equation

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

    Cao, Youfang; Terebus, Anna; Liang, Jie

    state space truncation and error analysis methods developed here can be used to ensure accurate direct solutions to the dCME for a large number of stochastic networks.« less

  18. State Space Truncation with Quantified Errors for Accurate Solutions to Discrete Chemical Master Equation

    DOE PAGES

    Cao, Youfang; Terebus, Anna; Liang, Jie

    2016-04-22

    state space truncation and error analysis methods developed here can be used to ensure accurate direct solutions to the dCME for a large number of stochastic networks.« less

  19. Sparse approximation problem: how rapid simulated annealing succeeds and fails

    NASA Astrophysics Data System (ADS)

    Obuchi, Tomoyuki; Kabashima, Yoshiyuki

    2016-03-01

    Information processing techniques based on sparseness have been actively studied in several disciplines. Among them, a mathematical framework to approximately express a given dataset by a combination of a small number of basis vectors of an overcomplete basis is termed the sparse approximation. In this paper, we apply simulated annealing, a metaheuristic algorithm for general optimization problems, to sparse approximation in the situation where the given data have a planted sparse representation and noise is present. The result in the noiseless case shows that our simulated annealing works well in a reasonable parameter region: the planted solution is found fairly rapidly. This is true even in the case where a common relaxation of the sparse approximation problem, the G-relaxation, is ineffective. On the other hand, when the dimensionality of the data is close to the number of non-zero components, another metastable state emerges, and our algorithm fails to find the planted solution. This phenomenon is associated with a first-order phase transition. In the case of very strong noise, it is no longer meaningful to search for the planted solution. In this situation, our algorithm determines a solution with close-to-minimum distortion fairly quickly.

  20. Correction of the near threshold behavior of electron collisional excitation cross-sections in the plane-wave Born approximation

    DOE PAGES

    Kilcrease, D. P.; Brookes, S.

    2013-08-19

    The modeling of NLTE plasmas requires the solution of population rate equations to determine the populations of the various atomic levels relevant to a particular problem. The equations require many cross sections for excitation, de-excitation, ionization and recombination. Additionally, a simple and computational fast way to calculate electron collisional excitation cross-sections for ions is by using the plane-wave Born approximation. This is essentially a high-energy approximation and the cross section suffers from the unphysical problem of going to zero near threshold. Various remedies for this problem have been employed with varying degrees of success. We present a correction procedure formore » the Born cross-sections that employs the Elwert–Sommerfeld factor to correct for the use of plane waves instead of Coulomb waves in an attempt to produce a cross-section similar to that from using the more time consuming Coulomb Born approximation. We compare this new approximation with other, often employed correction procedures. Furthermore, we also look at some further modifications to our Born Elwert procedure and its combination with Y.K. Kim's correction of the Coulomb Born approximation for singly charged ions that more accurately approximate convergent close coupling calculations.« less

  1. Interference effects in phased beam tracing using exact half-space solutions.

    PubMed

    Boucher, Matthew A; Pluymers, Bert; Desmet, Wim

    2016-12-01

    Geometrical acoustics provides a correct solution to the wave equation for rectangular rooms with rigid boundaries and is an accurate approximation at high frequencies with nearly hard walls. When interference effects are important, phased geometrical acoustics is employed in order to account for phase shifts due to propagation and reflection. Error increases, however, with more absorption, complex impedance values, grazing incidence, smaller volumes and lower frequencies. Replacing the plane wave reflection coefficient with a spherical one reduces the error but results in slower convergence. Frequency-dependent stopping criteria are then applied to avoid calculating higher order reflections for frequencies that have already converged. Exact half-space solutions are used to derive two additional spherical wave reflection coefficients: (i) the Sommerfeld integral, consisting of a plane wave decomposition of a point source and (ii) a line of image sources located at complex coordinates. Phased beam tracing using exact half-space solutions agrees well with the finite element method for rectangular rooms with absorbing boundaries, at low frequencies and for rooms with different aspect ratios. Results are accurate even for long source-to-receiver distances. Finally, the crossover frequency between the plane and spherical wave reflection coefficients is discussed.

  2. A Bayesian Hierarchical Model for Glacial Dynamics Based on the Shallow Ice Approximation and its Evaluation Using Analytical Solutions

    NASA Astrophysics Data System (ADS)

    Gopalan, Giri; Hrafnkelsson, Birgir; Aðalgeirsdóttir, Guðfinna; Jarosch, Alexander H.; Pálsson, Finnur

    2018-03-01

    Bayesian hierarchical modeling can assist the study of glacial dynamics and ice flow properties. This approach will allow glaciologists to make fully probabilistic predictions for the thickness of a glacier at unobserved spatio-temporal coordinates, and it will also allow for the derivation of posterior probability distributions for key physical parameters such as ice viscosity and basal sliding. The goal of this paper is to develop a proof of concept for a Bayesian hierarchical model constructed, which uses exact analytical solutions for the shallow ice approximation (SIA) introduced by Bueler et al. (2005). A suite of test simulations utilizing these exact solutions suggests that this approach is able to adequately model numerical errors and produce useful physical parameter posterior distributions and predictions. A byproduct of the development of the Bayesian hierarchical model is the derivation of a novel finite difference method for solving the SIA partial differential equation (PDE). An additional novelty of this work is the correction of numerical errors induced through a numerical solution using a statistical model. This error correcting process models numerical errors that accumulate forward in time and spatial variation of numerical errors between the dome, interior, and margin of a glacier.

  3. Application of the Parabolic Approximation to Predict Acoustical Propagation in the Ocean.

    ERIC Educational Resources Information Center

    McDaniel, Suzanne T.

    1979-01-01

    A simplified derivation of the parabolic approximation to the acoustical wave equation is presented. Exact solutions to this approximate equation are compared with solutions to the wave equation to demonstrate the applicability of this method to the study of underwater sound propagation. (Author/BB)

  4. Nonexistence of exact solutions agreeing with the Gaussian beam on the beam axis or in the focal plane

    NASA Astrophysics Data System (ADS)

    Lekner, John; Andrejic, Petar

    2018-01-01

    Solutions of the Helmholtz equation which describe electromagnetic beams (and also acoustic or particle beams) are discussed. We show that an exact solution which reproduces the Gaussian beam waveform on the beam axis does not exist. This is surprising, since the Gaussian beam is a solution of the paraxial equation, and thus supposedly accurate on and near the beam axis. Likewise, a solution of the Helmholtz equation which exactly reproduces the Gaussian beam in the focal plane does not exist. We show that the last statement also holds for Bessel-Gauss beams. However, solutions of the Helmholtz equation (one of which is discussed in detail) can approximate the Gaussian waveform within the central focal region.

  5. An efficient and accurate solution methodology for bilevel multi-objective programming problems using a hybrid evolutionary-local-search algorithm.

    PubMed

    Deb, Kalyanmoy; Sinha, Ankur

    2010-01-01

    Bilevel optimization problems involve two optimization tasks (upper and lower level), in which every feasible upper level solution must correspond to an optimal solution to a lower level optimization problem. These problems commonly appear in many practical problem solving tasks including optimal control, process optimization, game-playing strategy developments, transportation problems, and others. However, they are commonly converted into a single level optimization problem by using an approximate solution procedure to replace the lower level optimization task. Although there exist a number of theoretical, numerical, and evolutionary optimization studies involving single-objective bilevel programming problems, not many studies look at the context of multiple conflicting objectives in each level of a bilevel programming problem. In this paper, we address certain intricate issues related to solving multi-objective bilevel programming problems, present challenging test problems, and propose a viable and hybrid evolutionary-cum-local-search based algorithm as a solution methodology. The hybrid approach performs better than a number of existing methodologies and scales well up to 40-variable difficult test problems used in this study. The population sizing and termination criteria are made self-adaptive, so that no additional parameters need to be supplied by the user. The study indicates a clear niche of evolutionary algorithms in solving such difficult problems of practical importance compared to their usual solution by a computationally expensive nested procedure. The study opens up many issues related to multi-objective bilevel programming and hopefully this study will motivate EMO and other researchers to pay more attention to this important and difficult problem solving activity.

  6. Estimating ice particle scattering properties using a modified Rayleigh-Gans approximation

    NASA Astrophysics Data System (ADS)

    Lu, Yinghui; Clothiaux, Eugene E.; Aydin, Kültegin; Verlinde, Johannes

    2014-09-01

    A modification to the Rayleigh-Gans approximation is made that includes self-interactions between different parts of an ice crystal, which both improves the accuracy of the Rayleigh-Gans approximation and extends its applicability to polarization-dependent parameters. This modified Rayleigh-Gans approximation is both efficient and reasonably accurate for particles with at least one dimension much smaller than the wavelength (e.g., dendrites at millimeter or longer wavelengths) or particles with sparse structures (e.g., low-density aggregates). Relative to the Generalized Multiparticle Mie method, backscattering reflectivities at horizontal transmit and receive polarization (HH) (ZHH) computed with this modified Rayleigh-Gans approach are about 3 dB more accurate than with the traditional Rayleigh-Gans approximation. For realistic particle size distributions and pristine ice crystals the modified Rayleigh-Gans approach agrees with the Generalized Multiparticle Mie method to within 0.5 dB for ZHH whereas for the polarimetric radar observables differential reflectivity (ZDR) and specific differential phase (KDP) agreement is generally within 0.7 dB and 13%, respectively. Compared to the A-DDA code, the modified Rayleigh-Gans approximation is several to tens of times faster if scattering properties for different incident angles and particle orientations are calculated. These accuracies and computational efficiencies are sufficient to make this modified Rayleigh-Gans approach a viable alternative to the Rayleigh-Gans approximation in some applications such as millimeter to centimeter wavelength radars and to other methods that assume simpler, less accurate shapes for ice crystals. This method should not be used on materials with dielectric properties much different from ice and on compact particles much larger than the wavelength.

  7. NONLINEAR MULTIGRID SOLVER EXPLOITING AMGe COARSE SPACES WITH APPROXIMATION PROPERTIES

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

    Christensen, Max La Cour; Villa, Umberto E.; Engsig-Karup, Allan P.

    The paper introduces a nonlinear multigrid solver for mixed nite element discretizations based on the Full Approximation Scheme (FAS) and element-based Algebraic Multigrid (AMGe). The main motivation to use FAS for unstruc- tured problems is the guaranteed approximation property of the AMGe coarse spaces that were developed recently at Lawrence Livermore National Laboratory. These give the ability to derive stable and accurate coarse nonlinear discretization problems. The previous attempts (including ones with the original AMGe method, [5, 11]), were less successful due to lack of such good approximation properties of the coarse spaces. With coarse spaces with approximation properties, ourmore » FAS approach on un- structured meshes should be as powerful/successful as FAS on geometrically re ned meshes. For comparison, Newton's method and Picard iterations with an inner state-of-the-art linear solver is compared to FAS on a nonlinear saddle point problem with applications to porous media ow. It is demonstrated that FAS is faster than Newton's method and Picard iterations for the experiments considered here. Due to the guaranteed approximation properties of our AMGe, the coarse spaces are very accurate, providing a solver with the potential for mesh-independent convergence on general unstructured meshes.« less

  8. Accurate hybrid stochastic simulation of a system of coupled chemical or biochemical reactions.

    PubMed

    Salis, Howard; Kaznessis, Yiannis

    2005-02-01

    The dynamical solution of a well-mixed, nonlinear stochastic chemical kinetic system, described by the Master equation, may be exactly computed using the stochastic simulation algorithm. However, because the computational cost scales with the number of reaction occurrences, systems with one or more "fast" reactions become costly to simulate. This paper describes a hybrid stochastic method that partitions the system into subsets of fast and slow reactions, approximates the fast reactions as a continuous Markov process, using a chemical Langevin equation, and accurately describes the slow dynamics using the integral form of the "Next Reaction" variant of the stochastic simulation algorithm. The key innovation of this method is its mechanism of efficiently monitoring the occurrences of slow, discrete events while simultaneously simulating the dynamics of a continuous, stochastic or deterministic process. In addition, by introducing an approximation in which multiple slow reactions may occur within a time step of the numerical integration of the chemical Langevin equation, the hybrid stochastic method performs much faster with only a marginal decrease in accuracy. Multiple examples, including a biological pulse generator and a large-scale system benchmark, are simulated using the exact and proposed hybrid methods as well as, for comparison, a previous hybrid stochastic method. Probability distributions of the solutions are compared and the weak errors of the first two moments are computed. In general, these hybrid methods may be applied to the simulation of the dynamics of a system described by stochastic differential, ordinary differential, and Master equations.

  9. A Discrete Approximation Framework for Hereditary Systems.

    DTIC Science & Technology

    1980-05-01

    schemes which are included in the general framework and which may be implemented directly on high-speed computing machines are developed. A numerical...an appropriately chosen Hilbert space. We then proceed to develop general approximation schemes for the solutions to the homogeneous AEE which in turn...rich classes of these schemes . In addition, two particular families of approximation schemes included in the general framework are developed and

  10. An accurate solution of the gas lubricated, flat sector thrust bearing

    NASA Technical Reports Server (NTRS)

    Etsion, I.; Fleming, D. P.

    1976-01-01

    A flat sector shaped pad geometry for gas lubricated thrust bearings is analyzed considering both pitch and roll angles of the pad and the true film thickness distribution. Maximum load capacity is achieved when the pad is tilted so as to create a uniform minimum film thickness along the pad trailing edge. Performance characteristics for various geometries and operating conditions of gas thrust bearings are presented in the form of design curves. A comparison is made with the rectangular slider approximation. It is found that this approximation is unsafe for practical design, since it always overestimates load capacity.

  11. Nonlinear Schroedinger Approximations for Partial Differential Equations with Quadratic and Quasilinear Terms

    NASA Astrophysics Data System (ADS)

    Cummings, Patrick

    We consider the approximation of solutions of two complicated, physical systems via the nonlinear Schrodinger equation (NLS). In particular, we discuss the evolution of wave packets and long waves in two physical models. Due to the complicated nature of the equations governing many physical systems and the in-depth knowledge we have for solutions of the nonlinear Schrodinger equation, it is advantageous to use approximation results of this kind to model these physical systems. The approximations are simple enough that we can use them to understand the qualitative and quantitative behavior of the solutions, and by justifying them we can show that the behavior of the approximation captures the behavior of solutions to the original equation, at least for long, but finite time. We first consider a model of the water wave equations which can be approximated by wave packets using the NLS equation. We discuss a new proof that both simplifies and strengthens previous justification results of Schneider and Wayne. Rather than using analytic norms, as was done by Schneider and Wayne, we construct a modified energy functional so that the approximation holds for the full interval of existence of the approximate NLS solution as opposed to a subinterval (as is seen in the analytic case). Furthermore, the proof avoids problems associated with inverting the normal form transform by working with a modified energy functional motivated by Craig and Hunter et al. We then consider the Klein-Gordon-Zakharov system and prove a long wave approximation result. In this case there is a non-trivial resonance that cannot be eliminated via a normal form transform. By combining the normal form transform for small Fourier modes and using analytic norms elsewhere, we can get a justification result on the order 1 over epsilon squared time scale.

  12. A Novel Iterative Scheme for the Very Fast and Accurate Solution of Non-LTE Radiative Transfer Problems

    NASA Astrophysics Data System (ADS)

    Trujillo Bueno, J.; Fabiani Bendicho, P.

    1995-12-01

    Iterative schemes based on Gauss-Seidel (G-S) and optimal successive over-relaxation (SOR) iteration are shown to provide a dramatic increase in the speed with which non-LTE radiation transfer (RT) problems can be solved. The convergence rates of these new RT methods are identical to those of upper triangular nonlocal approximate operator splitting techniques, but the computing time per iteration and the memory requirements are similar to those of a local operator splitting method. In addition to these properties, both methods are particularly suitable for multidimensional geometry, since they neither require the actual construction of nonlocal approximate operators nor the application of any matrix inversion procedure. Compared with the currently used Jacobi technique, which is based on the optimal local approximate operator (see Olson, Auer, & Buchler 1986), the G-S method presented here is faster by a factor 2. It gives excellent smoothing of the high-frequency error components, which makes it the iterative scheme of choice for multigrid radiative transfer. This G-S method can also be suitably combined with standard acceleration techniques to achieve even higher performance. Although the convergence rate of the optimal SOR scheme developed here for solving non-LTE RT problems is much higher than G-S, the computing time per iteration is also minimal, i.e., virtually identical to that of a local operator splitting method. While the conventional optimal local operator scheme provides the converged solution after a total CPU time (measured in arbitrary units) approximately equal to the number n of points per decade of optical depth, the time needed by this new method based on the optimal SOR iterations is only √n/2√2. This method is competitive with those that result from combining the above-mentioned Jacobi and G-S schemes with the best acceleration techniques. Contrary to what happens with the local operator splitting strategy currently in use, these novel

  13. Laplace transform homotopy perturbation method for the approximation of variational problems.

    PubMed

    Filobello-Nino, U; Vazquez-Leal, H; Rashidi, M M; Sedighi, H M; Perez-Sesma, A; Sandoval-Hernandez, M; Sarmiento-Reyes, A; Contreras-Hernandez, A D; Pereyra-Diaz, D; Hoyos-Reyes, C; Jimenez-Fernandez, V M; Huerta-Chua, J; Castro-Gonzalez, F; Laguna-Camacho, J R

    2016-01-01

    This article proposes the application of Laplace Transform-Homotopy Perturbation Method and some of its modifications in order to find analytical approximate solutions for the linear and nonlinear differential equations which arise from some variational problems. As case study we will solve four ordinary differential equations, and we will show that the proposed solutions have good accuracy, even we will obtain an exact solution. In the sequel, we will see that the square residual error for the approximate solutions, belongs to the interval [0.001918936920, 0.06334882582], which confirms the accuracy of the proposed methods, taking into account the complexity and difficulty of variational problems.

  14. Influence of the Numerical Scheme on the Solution Quality of the SWE for Tsunami Numerical Codes: The Tohoku-Oki, 2011Example.

    NASA Astrophysics Data System (ADS)

    Reis, C.; Clain, S.; Figueiredo, J.; Baptista, M. A.; Miranda, J. M. A.

    2015-12-01

    Numerical tools turn to be very important for scenario evaluations of hazardous phenomena such as tsunami. Nevertheless, the predictions highly depends on the numerical tool quality and the design of efficient numerical schemes still receives important attention to provide robust and accurate solutions. In this study we propose a comparative study between the efficiency of two volume finite numerical codes with second-order discretization implemented with different method to solve the non-conservative shallow water equations, the MUSCL (Monotonic Upstream-Centered Scheme for Conservation Laws) and the MOOD methods (Multi-dimensional Optimal Order Detection) which optimize the accuracy of the approximation in function of the solution local smoothness. The MUSCL is based on a priori criteria where the limiting procedure is performed before updated the solution to the next time-step leading to non-necessary accuracy reduction. On the contrary, the new MOOD technique uses a posteriori detectors to prevent the solution from oscillating in the vicinity of the discontinuities. Indeed, a candidate solution is computed and corrections are performed only for the cells where non-physical oscillations are detected. Using a simple one-dimensional analytical benchmark, 'Single wave on a sloping beach', we show that the classical 1D shallow-water system can be accurately solved with the finite volume method equipped with the MOOD technique and provide better approximation with sharper shock and less numerical diffusion. For the code validation, we also use the Tohoku-Oki 2011 tsunami and reproduce two DART records, demonstrating that the quality of the solution may deeply interfere with the scenario one can assess. This work is funded by the Portugal-France research agreement, through the research project GEONUM FCT-ANR/MAT-NAN/0122/2012.Numerical tools turn to be very important for scenario evaluations of hazardous phenomena such as tsunami. Nevertheless, the predictions highly

  15. Loglinear Approximate Solutions to Real-Business-Cycle Models: Some Observations

    ERIC Educational Resources Information Center

    Lau, Sau-Him Paul; Ng, Philip Hoi-Tak

    2007-01-01

    Following the analytical approach suggested in Campbell, the authors consider a baseline real-business-cycle (RBC) model with endogenous labor supply. They observe that the coefficients in the loglinear approximation of the dynamic equations characterizing the equilibrium are related to the fundamental parameters in a relatively simple manner.…

  16. Hartree-Fock theory of the inhomogeneous electron gas at a jellium metal surface: Rigorous upper bounds to the surface energy and accurate work functions

    NASA Astrophysics Data System (ADS)

    Sahni, V.; Ma, C. Q.

    1980-12-01

    The inhomogeneous electron gas at a jellium metal surface is studied in the Hartree-Fock approximation by Kohn-Sham density functional theory. Rigorous upper bounds to the surface energy are derived by application of the Rayleigh-Ritz variational principle for the energy, the surface kinetic, electrostatic, and nonlocal exchange energy functionals being determined exactly for the accurate linear-potential model electronic wave functions. The densities obtained by the energy minimization constraint are then employed to determine work-function results via the variationally accurate "displaced-profile change-in-self-consistent-field" expression. The theoretical basis of this non-self-consistent procedure and its demonstrated accuracy for the fully correlated system (as treated within the local-density approximation for exchange and correlation) leads us to conclude these results for the surface energies and work functions to be essentially exact. Work-function values are also determined by the Koopmans'-theorem expression, both for these densities as well as for those obtained by satisfaction of the constraint set on the electrostatic potential by the Budd-Vannimenus theorem. The use of the Hartree-Fock results in the accurate estimation of correlation-effect contributions to these surface properties of the nonuniform electron gas is also indicated. In addition, the original work and approximations made by Bardeen in this attempt at a solution of the Hartree-Fock problem are briefly reviewed in order to contrast with the present work.

  17. Parameter estimation in nonlinear distributed systems - Approximation theory and convergence results

    NASA Technical Reports Server (NTRS)

    Banks, H. T.; Reich, Simeon; Rosen, I. G.

    1988-01-01

    An abstract approximation framework and convergence theory is described for Galerkin approximations applied to inverse problems involving nonlinear distributed parameter systems. Parameter estimation problems are considered and formulated as the minimization of a least-squares-like performance index over a compact admissible parameter set subject to state constraints given by an inhomogeneous nonlinear distributed system. The theory applies to systems whose dynamics can be described by either time-independent or nonstationary strongly maximal monotonic operators defined on a reflexive Banach space which is densely and continuously embedded in a Hilbert space. It is demonstrated that if readily verifiable conditions on the system's dependence on the unknown parameters are satisfied, and the usual Galerkin approximation assumption holds, then solutions to the approximating problems exist and approximate a solution to the original infinite-dimensional identification problem.

  18. Approximate registration of point clouds with large scale differences

    NASA Astrophysics Data System (ADS)

    Novak, D.; Schindler, K.

    2013-10-01

    3D reconstruction of objects is a basic task in many fields, including surveying, engineering, entertainment and cultural heritage. The task is nowadays often accomplished with a laser scanner, which produces dense point clouds, but lacks accurate colour information, and lacks per-point accuracy measures. An obvious solution is to combine laser scanning with photogrammetric recording. In that context, the problem arises to register the two datasets, which feature large scale, translation and rotation differences. The absence of approximate registration parameters (3D translation, 3D rotation and scale) precludes the use of fine-registration methods such as ICP. Here, we present a method to register realistic photogrammetric and laser point clouds in a fully automated fashion. The proposed method decomposes the registration into a sequence of simpler steps: first, two rotation angles are determined by finding dominant surface normal directions, then the remaining parameters are found with RANSAC followed by ICP and scale refinement. These two steps are carried out at low resolution, before computing a precise final registration at higher resolution.

  19. Approximation of traveling wave solutions in wall-bounded flows using resolvent modes

    NASA Astrophysics Data System (ADS)

    McKeon, Beverley; Graham, Michael; Moarref, Rashad; Park, Jae Sung; Sharma, Ati; Willis, Ashley

    2014-11-01

    Significant recent attention has been devoted to computing and understanding exact traveling wave solutions of the Navier-Stokes equations. These solutions can be interpreted as the state-space skeleton of turbulence and are attractive benchmarks for studying low-order models of wall turbulence. Here, we project such solutions onto the velocity response (or resolvent) modes supplied by the gain-based resolvent analysis outlined by McKeon & Sharma (JFM, 2010). We demonstrate that in both pipe (Pringle et al., Phil. Trans. R. Soc. A, 2009) and channel (Waleffe, JFM, 2001) flows, the solutions can be well-described by a small number of resolvent modes. Analysis of the nonlinear forcing modes sustaining these solutions reveals the importance of small amplitude forcing, consistent with the large amplifications admitted by the resolvent operator. We investigate the use of resolvent modes as computationally cheap ``seeds'' for the identification of further traveling wave solutions. The support of AFOSR under Grants FA9550-09-1-0701, FA9550-12-1-0469, FA9550-11-1-0094 and FA9550-14-1-0042 (program managers Rengasamy Ponnappan, Doug Smith and Gregg Abate) is gratefully acknowledged.

  20. Continuous supercritical synthesis and dielectric behaviour of the whole BST solid solution.

    PubMed

    Reverón, H; Elissalde, C; Aymonier, C; Bousquet, C; Maglione, M; Cansell, F

    2006-07-28

    In this study we show that pure and well crystallized nanoparticles of Ba(x)Sr(1-x)TiO(3) (BST) can be synthesized over the entire range of composition through the hydrolysis and further crystallization of alkoxide precursors under supercritical conditions. To our knowledge, this is the first time that the whole ferroelectric solid solution has been produced in a continuous way, using the same experimental conditions. The composition of the powder can be easily controlled by adjusting the feed solution composition. The powders consist of soft-aggregated monocrystalline nanoparticles with an average particle size ranging from approximately 20 to 40 nm. Ferroelectric ceramics with accurately adjustable Curie temperature (100-390 K) can thus be obtained by sintering.

  1. High order accurate solutions of viscous problems

    NASA Technical Reports Server (NTRS)

    Hayder, M. Ehtesham; Turkel, Eli

    1993-01-01

    We consider a fourth order extension to MacCormack's scheme. The original extension was fourth order only for the inviscid terms but was second order for the viscous terms. We show how to modify the viscous terms so that the scheme is uniformly fourth order in the spatial derivatives. Applications are given to some boundary layer flows. In addition, for applications to shear flows the effect of the outflow boundary conditions are very important. We compare the accuracy of several of these different boundary conditions for both boundary layer and shear flows. Stretching at the outflow usually increases the oscillations in the numerical solution but the addition of a filtered sponge layer (with or without stretching) reduces such oscillations. The oscillations are generated by insufficient resolution of the shear layer. When the shear layer is sufficiently resolved then oscillations are not generated and there is less of a need for a nonreflecting boundary condition.

  2. First and second order approximations to stage numbers in multicomponent enrichment cascades

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

    Scopatz, A.

    2013-07-01

    This paper describes closed form, Taylor series approximations to the number product stages in a multicomponent enrichment cascade. Such closed form approximations are required when a symbolic, rather than a numeric, algorithm is used to compute the optimal cascade state. Both first and second order approximations were implemented. The first order solution was found to be grossly incorrect, having the wrong functional form over the entire domain. On the other hand, the second order solution shows excellent agreement with the 'true' solution over the domain of interest. An implementation of the symbolic, second order solver is available in the freemore » and open source PyNE library. (authors)« less

  3. Validity of the Born approximation for beyond Gaussian weak lensing observables

    DOE PAGES

    Petri, Andrea; Haiman, Zoltan; May, Morgan

    2017-06-06

    Accurate forward modeling of weak lensing (WL) observables from cosmological parameters is necessary for upcoming galaxy surveys. Because WL probes structures in the nonlinear regime, analytical forward modeling is very challenging, if not impossible. Numerical simulations of WL features rely on ray tracing through the outputs of N-body simulations, which requires knowledge of the gravitational potential and accurate solvers for light ray trajectories. A less accurate procedure, based on the Born approximation, only requires knowledge of the density field, and can be implemented more efficiently and at a lower computational cost. In this work, we use simulations to show thatmore » deviations of the Born-approximated convergence power spectrum, skewness and kurtosis from their fully ray-traced counterparts are consistent with the smallest nontrivial O(Φ 3) post-Born corrections (so-called geodesic and lens-lens terms). Our results imply a cancellation among the larger O(Φ 4) (and higher order) terms, consistent with previous analytic work. We also find that cosmological parameter bias induced by the Born-approximated power spectrum is negligible even for a LSST-like survey, once galaxy shape noise is considered. When considering higher order statistics such as the κ skewness and kurtosis, however, we find significant bias of up to 2.5σ. Using the LensTools software suite, we show that the Born approximation saves a factor of 4 in computing time with respect to the full ray tracing in reconstructing the convergence.« less

  4. Validity of the Born approximation for beyond Gaussian weak lensing observables

    NASA Astrophysics Data System (ADS)

    Petri, Andrea; Haiman, Zoltán; May, Morgan

    2017-06-01

    Accurate forward modeling of weak lensing (WL) observables from cosmological parameters is necessary for upcoming galaxy surveys. Because WL probes structures in the nonlinear regime, analytical forward modeling is very challenging, if not impossible. Numerical simulations of WL features rely on ray tracing through the outputs of N -body simulations, which requires knowledge of the gravitational potential and accurate solvers for light ray trajectories. A less accurate procedure, based on the Born approximation, only requires knowledge of the density field, and can be implemented more efficiently and at a lower computational cost. In this work, we use simulations to show that deviations of the Born-approximated convergence power spectrum, skewness and kurtosis from their fully ray-traced counterparts are consistent with the smallest nontrivial O (Φ3) post-Born corrections (so-called geodesic and lens-lens terms). Our results imply a cancellation among the larger O (Φ4) (and higher order) terms, consistent with previous analytic work. We also find that cosmological parameter bias induced by the Born-approximated power spectrum is negligible even for a LSST-like survey, once galaxy shape noise is considered. When considering higher order statistics such as the κ skewness and kurtosis, however, we find significant bias of up to 2.5 σ . Using the LensTools software suite, we show that the Born approximation saves a factor of 4 in computing time with respect to the full ray tracing in reconstructing the convergence.

  5. A hybrid solution using computational prediction and measured data to accurately determine process corrections with reduced overlay sampling

    NASA Astrophysics Data System (ADS)

    Noyes, Ben F.; Mokaberi, Babak; Mandoy, Ram; Pate, Alex; Huijgen, Ralph; McBurney, Mike; Chen, Owen

    2017-03-01

    Reducing overlay error via an accurate APC feedback system is one of the main challenges in high volume production of the current and future nodes in the semiconductor industry. The overlay feedback system directly affects the number of dies meeting overlay specification and the number of layers requiring dedicated exposure tools through the fabrication flow. Increasing the former number and reducing the latter number is beneficial for the overall efficiency and yield of the fabrication process. An overlay feedback system requires accurate determination of the overlay error, or fingerprint, on exposed wafers in order to determine corrections to be automatically and dynamically applied to the exposure of future wafers. Since current and future nodes require correction per exposure (CPE), the resolution of the overlay fingerprint must be high enough to accommodate CPE in the overlay feedback system, or overlay control module (OCM). Determining a high resolution fingerprint from measured data requires extremely dense overlay sampling that takes a significant amount of measurement time. For static corrections this is acceptable, but in an automated dynamic correction system this method creates extreme bottlenecks for the throughput of said system as new lots have to wait until the previous lot is measured. One solution is using a less dense overlay sampling scheme and employing computationally up-sampled data to a dense fingerprint. That method uses a global fingerprint model over the entire wafer; measured localized overlay errors are therefore not always represented in its up-sampled output. This paper will discuss a hybrid system shown in Fig. 1 that combines a computationally up-sampled fingerprint with the measured data to more accurately capture the actual fingerprint, including local overlay errors. Such a hybrid system is shown to result in reduced modelled residuals while determining the fingerprint, and better on-product overlay performance.

  6. An exact closed form solution for constant area compressible flow with friction and heat transfer

    NASA Technical Reports Server (NTRS)

    Sturas, J. I.

    1971-01-01

    The well-known differential equation for the one-dimensional flow of a compressible fluid with heat transfer and wall friction has no known solution in closed form for the general case. This report presents a closed form solution for the special case of constant heat flux per unit length and constant specific heat. The solution was obtained by choosing the square of a dimensionless flow parameter as one of the independent variables to describe the flow. From this exact solution, an approximate simplified form is derived that is applicable for predicting subsonic flow performance characteristics for many types of constant area passages in internal flow. The data included in this report are considered sufficiently accurate for use as a guide in analyzing and designing internal gas flow systems.

  7. Modeling boundary measurements of scattered light using the corrected diffusion approximation

    PubMed Central

    Lehtikangas, Ossi; Tarvainen, Tanja; Kim, Arnold D.

    2012-01-01

    We study the modeling and simulation of steady-state measurements of light scattered by a turbid medium taken at the boundary. In particular, we implement the recently introduced corrected diffusion approximation in two spatial dimensions to model these boundary measurements. This implementation uses expansions in plane wave solutions to compute boundary conditions and the additive boundary layer correction, and a finite element method to solve the diffusion equation. We show that this corrected diffusion approximation models boundary measurements substantially better than the standard diffusion approximation in comparison to numerical solutions of the radiative transport equation. PMID:22435102

  8. The TSP-approach to approximate solving the m-Cycles Cover Problem

    NASA Astrophysics Data System (ADS)

    Gimadi, Edward Kh.; Rykov, Ivan; Tsidulko, Oxana

    2016-10-01

    In the m-Cycles Cover problem it is required to find a collection of m vertex-disjoint cycles that covers all vertices of the graph and the total weight of edges in the cover is minimum (or maximum). The problem is a generalization of the Traveling salesmen problem. It is strongly NP-hard. We discuss a TSP-approach that gives polynomial approximate solutions for this problem. It transforms an approximation TSP algorithm into an approximation m-CCP algorithm. In this paper we present a number of successful transformations with proven performance guarantees for the obtained solutions.

  9. The closure approximation in the hierarchy equations.

    NASA Technical Reports Server (NTRS)

    Adomian, G.

    1971-01-01

    The expectation of the solution process in a stochastic operator equation can be obtained from averaged equations only under very special circumstances. Conditions for validity are given and the significance and validity of the approximation in widely used hierarchy methods and the ?self-consistent field' approximation in nonequilibrium statistical mechanics are clarified. The error at any level of the hierarchy can be given and can be avoided by the use of the iterative method.

  10. Applying the Zel'dovich approximation to general relativity

    NASA Astrophysics Data System (ADS)

    Croudace, K. M.; Parry, J.; Salopek, D. S.; Stewart, J. M.

    1994-03-01

    Starting from general relativity, we give a systematic derivation of the Zel'dovich approximation describing the nonlinear evolution of collisionless dust. We begin by evolving dust along world lines, and we demonstrate that the Szekeres line element is an exact but apparently unstable solution of the evolution equations describing pancake collapse. Next, we solve the Einstein field equations by employing Hamilton-Jacobi techniques and a spatial gradient expansion. We give a prescription for evolving a primordial or 'seed' metric up to the formation of pancakes, and demonstrate its validity by rederiving the Szekeres solution approximately at third order and exactly at fifth order in spatial gradients. Finally we show that the range of validity of the expansion can be improved quite significantly if one notes that the 3-metric must have nonnegative eigenvalues. With this improvement the exact Szekeres solution is obtained after only one iteration.

  11. The sagitta and lens thickness: the exact solution and a matrix approximation for lenses with toric, spherical, and cylindrical surfaces.

    PubMed

    Harris, W F

    1989-03-01

    The exact equation for sagitta of spherical surfaces is generalized to toric surfaces which include spherical and cylindrical surfaces as special cases. Lens thickness, therefore, can be calculated accurately anywhere on a lens even in cases of extreme spherical and cylindrical powers and large diameters. The sagittae of tire- and barrel-form toric surfaces differ off the principal meridians, as is shown by a numerical example. The same holds for pulley- and capstan-form toric surfaces. A general expression is given for thickness at an arbitrary point on a toric lens. Approximate expressions are derived and re-expressed in terms of matrices. The matrix provides an elegant means of generalizing equations for spherical surfaces and lenses to toric surfaces and lenses.

  12. The measurement of psychological literacy: a first approximation

    PubMed Central

    Roberts, Lynne D.; Heritage, Brody; Gasson, Natalie

    2015-01-01

    Psychological literacy, the ability to apply psychological knowledge to personal, family, occupational, community and societal challenges, is promoted as the primary outcome of an undergraduate education in psychology. As the concept of psychological literacy becomes increasingly adopted as the core business of undergraduate psychology training courses world-wide, there is urgent need for the construct to be accurately measured so that student and institutional level progress can be assessed and monitored. Key to the measurement of psychological literacy is determining the underlying factor-structure of psychological literacy. In this paper we provide a first approximation of the measurement of psychological literacy by identifying and evaluating self-report measures for psychological literacy. Multi-item and single-item self-report measures of each of the proposed nine dimensions of psychological literacy were completed by two samples (N = 218 and N = 381) of undergraduate psychology students at an Australian university. Single and multi-item measures of each dimension were weakly to moderately correlated. Exploratory and confirmatory factor analyses of multi-item measures indicated a higher order three factor solution best represented the construct of psychological literacy. The three factors were reflective processes, generic graduate attributes, and psychology as a helping profession. For the measurement of psychological literacy to progress there is a need to further develop self-report measures and to identify/develop and evaluate objective measures of psychological literacy. Further approximations of the measurement of psychological literacy remain an imperative, given the construct's ties to measuring institutional efficacy in teaching psychology to an undergraduate audience. PMID:25741300

  13. Spline approximation, Part 1: Basic methodology

    NASA Astrophysics Data System (ADS)

    Ezhov, Nikolaj; Neitzel, Frank; Petrovic, Svetozar

    2018-04-01

    In engineering geodesy point clouds derived from terrestrial laser scanning or from photogrammetric approaches are almost never used as final results. For further processing and analysis a curve or surface approximation with a continuous mathematical function is required. In this paper the approximation of 2D curves by means of splines is treated. Splines offer quite flexible and elegant solutions for interpolation or approximation of "irregularly" distributed data. Depending on the problem they can be expressed as a function or as a set of equations that depend on some parameter. Many different types of splines can be used for spline approximation and all of them have certain advantages and disadvantages depending on the approximation problem. In a series of three articles spline approximation is presented from a geodetic point of view. In this paper (Part 1) the basic methodology of spline approximation is demonstrated using splines constructed from ordinary polynomials and splines constructed from truncated polynomials. In the forthcoming Part 2 the notion of B-spline will be explained in a unique way, namely by using the concept of convex combinations. The numerical stability of all spline approximation approaches as well as the utilization of splines for deformation detection will be investigated on numerical examples in Part 3.

  14. Galerkin approximation for inverse problems for nonautonomous nonlinear distributed systems

    NASA Technical Reports Server (NTRS)

    Banks, H. T.; Reich, Simeon; Rosen, I. G.

    1988-01-01

    An abstract framework and convergence theory is developed for Galerkin approximation for inverse problems involving the identification of nonautonomous nonlinear distributed parameter systems. A set of relatively easily verified conditions is provided which are sufficient to guarantee the existence of optimal solutions and their approximation by a sequence of solutions to a sequence of approximating finite dimensional identification problems. The approach is based on the theory of monotone operators in Banach spaces and is applicable to a reasonably broad class of nonlinear distributed systems. Operator theoretic and variational techniques are used to establish a fundamental convergence result. An example involving evolution systems with dynamics described by nonstationary quasilinear elliptic operators along with some applications are presented and discussed.

  15. Dynamic simulation of concentrated macromolecular solutions with screened long-range hydrodynamic interactions: Algorithm and limitations

    PubMed Central

    Ando, Tadashi; Chow, Edmond; Skolnick, Jeffrey

    2013-01-01

    Hydrodynamic interactions exert a critical effect on the dynamics of macromolecules. As the concentration of macromolecules increases, by analogy to the behavior of semidilute polymer solutions or the flow in porous media, one might expect hydrodynamic screening to occur. Hydrodynamic screening would have implications both for the understanding of macromolecular dynamics as well as practical implications for the simulation of concentrated macromolecular solutions, e.g., in cells. Stokesian dynamics (SD) is one of the most accurate methods for simulating the motions of N particles suspended in a viscous fluid at low Reynolds number, in that it considers both far-field and near-field hydrodynamic interactions. This algorithm traditionally involves an O(N3) operation to compute Brownian forces at each time step, although asymptotically faster but more complex SD methods are now available. Motivated by the idea of hydrodynamic screening, the far-field part of the hydrodynamic matrix in SD may be approximated by a diagonal matrix, which is equivalent to assuming that long range hydrodynamic interactions are completely screened. This approximation allows sparse matrix methods to be used, which can reduce the apparent computational scaling to O(N). Previously there were several simulation studies using this approximation for monodisperse suspensions. Here, we employ newly designed preconditioned iterative methods for both the computation of Brownian forces and the solution of linear systems, and consider the validity of this approximation in polydisperse suspensions. We evaluate the accuracy of the diagonal approximation method using an intracellular-like suspension. The diffusivities of particles obtained with this approximation are close to those with the original method. However, this approximation underestimates intermolecular correlated motions, which is a trade-off between accuracy and computing efficiency. The new method makes it possible to perform large-scale and

  16. An approximate methods approach to probabilistic structural analysis

    NASA Technical Reports Server (NTRS)

    Mcclung, R. C.; Millwater, H. R.; Wu, Y.-T.; Thacker, B. H.; Burnside, O. H.

    1989-01-01

    A major research and technology program in Probabilistic Structural Analysis Methods (PSAM) is currently being sponsored by the NASA Lewis Research Center with Southwest Research Institute as the prime contractor. This program is motivated by the need to accurately predict structural response in an environment where the loadings, the material properties, and even the structure may be considered random. The heart of PSAM is a software package which combines advanced structural analysis codes with a fast probability integration (FPI) algorithm for the efficient calculation of stochastic structural response. The basic idea of PAAM is simple: make an approximate calculation of system response, including calculation of the associated probabilities, with minimal computation time and cost, based on a simplified representation of the geometry, loads, and material. The deterministic solution resulting should give a reasonable and realistic description of performance-limiting system responses, although some error will be inevitable. If the simple model has correctly captured the basic mechanics of the system, however, including the proper functional dependence of stress, frequency, etc. on design parameters, then the response sensitivities calculated may be of significantly higher accuracy.

  17. An accurate coarse-grained model for chitosan polysaccharides in aqueous solution.

    PubMed

    Tsereteli, Levan; Grafmüller, Andrea

    2017-01-01

    Computational models can provide detailed information about molecular conformations and interactions in solution, which is currently inaccessible by other means in many cases. Here we describe an efficient and precise coarse-grained model for long polysaccharides in aqueous solution at different physico-chemical conditions such as pH and ionic strength. The Model is carefully constructed based on all-atom simulations of small saccharides and metadynamics sampling of the dihedral angles in the glycosidic links, which represent the most flexible degrees of freedom of the polysaccharides. The model is validated against experimental data for Chitosan molecules in solution with various degree of deacetylation, and is shown to closely reproduce the available experimental data. For long polymers, subtle differences of the free energy maps of the glycosidic links are found to significantly affect the measurable polymer properties. Therefore, for titratable monomers the free energy maps of the corresponding links are updated according to the current charge of the monomers. We then characterize the microscopic and mesoscopic structural properties of large chitosan polysaccharides in solution for a wide range of solvent pH and ionic strength, and investigate the effect of polymer length and degree and pattern of deacetylation on the polymer properties.

  18. An accurate coarse-grained model for chitosan polysaccharides in aqueous solution

    PubMed Central

    Tsereteli, Levan

    2017-01-01

    Computational models can provide detailed information about molecular conformations and interactions in solution, which is currently inaccessible by other means in many cases. Here we describe an efficient and precise coarse-grained model for long polysaccharides in aqueous solution at different physico-chemical conditions such as pH and ionic strength. The Model is carefully constructed based on all-atom simulations of small saccharides and metadynamics sampling of the dihedral angles in the glycosidic links, which represent the most flexible degrees of freedom of the polysaccharides. The model is validated against experimental data for Chitosan molecules in solution with various degree of deacetylation, and is shown to closely reproduce the available experimental data. For long polymers, subtle differences of the free energy maps of the glycosidic links are found to significantly affect the measurable polymer properties. Therefore, for titratable monomers the free energy maps of the corresponding links are updated according to the current charge of the monomers. We then characterize the microscopic and mesoscopic structural properties of large chitosan polysaccharides in solution for a wide range of solvent pH and ionic strength, and investigate the effect of polymer length and degree and pattern of deacetylation on the polymer properties. PMID:28732036

  19. A second-order accurate finite volume scheme with the discrete maximum principle for solving Richards’ equation on unstructured meshes

    DOE PAGES

    Svyatsky, Daniil; Lipnikov, Konstantin

    2017-03-18

    Richards’s equation describes steady-state or transient flow in a variably saturated medium. For a medium having multiple layers of soils that are not aligned with coordinate axes, a mesh fitted to these layers is no longer orthogonal and the classical two-point flux approximation finite volume scheme is no longer accurate. Here, we propose new second-order accurate nonlinear finite volume (NFV) schemes for the head and pressure formulations of Richards’ equation. We prove that the discrete maximum principles hold for both formulations at steady-state which mimics similar properties of the continuum solution. The second-order accuracy is achieved using high-order upwind algorithmsmore » for the relative permeability. Numerical simulations of water infiltration into a dry soil show significant advantage of the second-order NFV schemes over the first-order NFV schemes even on coarse meshes. Since explicit calculation of the Jacobian matrix becomes prohibitively expensive for high-order schemes due to build-in reconstruction and slope limiting algorithms, we study numerically the preconditioning strategy introduced recently in Lipnikov et al. (2016) that uses a stable approximation of the continuum Jacobian. Lastly, numerical simulations show that the new preconditioner reduces computational cost up to 2–3 times in comparison with the conventional preconditioners.« less

  20. A second-order accurate finite volume scheme with the discrete maximum principle for solving Richards’ equation on unstructured meshes

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

    Svyatsky, Daniil; Lipnikov, Konstantin

    Richards’s equation describes steady-state or transient flow in a variably saturated medium. For a medium having multiple layers of soils that are not aligned with coordinate axes, a mesh fitted to these layers is no longer orthogonal and the classical two-point flux approximation finite volume scheme is no longer accurate. Here, we propose new second-order accurate nonlinear finite volume (NFV) schemes for the head and pressure formulations of Richards’ equation. We prove that the discrete maximum principles hold for both formulations at steady-state which mimics similar properties of the continuum solution. The second-order accuracy is achieved using high-order upwind algorithmsmore » for the relative permeability. Numerical simulations of water infiltration into a dry soil show significant advantage of the second-order NFV schemes over the first-order NFV schemes even on coarse meshes. Since explicit calculation of the Jacobian matrix becomes prohibitively expensive for high-order schemes due to build-in reconstruction and slope limiting algorithms, we study numerically the preconditioning strategy introduced recently in Lipnikov et al. (2016) that uses a stable approximation of the continuum Jacobian. Lastly, numerical simulations show that the new preconditioner reduces computational cost up to 2–3 times in comparison with the conventional preconditioners.« less

  1. Partially Coherent Scattering in Stellar Chromospheres. Part 4; Analytic Wing Approximations

    NASA Technical Reports Server (NTRS)

    Gayley, K. G.

    1993-01-01

    Simple analytic expressions are derived to understand resonance-line wings in stellar chromospheres and similar astrophysical plasmas. The results are approximate, but compare well with accurate numerical simulations. The redistribution is modeled using an extension of the partially coherent scattering approximation (PCS) which we term the comoving-frame partially coherent scattering approximation (CPCS). The distinction is made here because Doppler diffusion is included in the coherent/noncoherent decomposition, in a form slightly improved from the earlier papers in this series.

  2. Analytical inversions in remote sensing of particle size distributions. IV - Comparison of Fymat and Box-McKellar solutions in the anomalous diffraction approximation

    NASA Technical Reports Server (NTRS)

    Fymat, A. L.; Smith, C. B.

    1979-01-01

    It is shown that the inverse analytical solutions, provided separately by Fymat and Box-McKellar, for reconstructing particle size distributions from remote spectral transmission measurements under the anomalous diffraction approximation can be derived using a cosine and a sine transform, respectively. Sufficient conditions of validity of the two formulas are established. Their comparison shows that the former solution is preferable to the latter in that it requires less a priori information (knowledge of the particle number density is not needed) and has wider applicability. For gamma-type distributions, and either a real or a complex refractive index, explicit expressions are provided for retrieving the distribution parameters; such expressions are, interestingly, proportional to the geometric area of the polydispersion.

  3. An evaluation of solution algorithms and numerical approximation methods for modeling an ion exchange process

    NASA Astrophysics Data System (ADS)

    Bu, Sunyoung; Huang, Jingfang; Boyer, Treavor H.; Miller, Cass T.

    2010-07-01

    The focus of this work is on the modeling of an ion exchange process that occurs in drinking water treatment applications. The model formulation consists of a two-scale model in which a set of microscale diffusion equations representing ion exchange resin particles that vary in size and age are coupled through a boundary condition with a macroscopic ordinary differential equation (ODE), which represents the concentration of a species in a well-mixed reactor. We introduce a new age-averaged model (AAM) that averages all ion exchange particle ages for a given size particle to avoid the expensive Monte-Carlo simulation associated with previous modeling applications. We discuss two different numerical schemes to approximate both the original Monte-Carlo algorithm and the new AAM for this two-scale problem. The first scheme is based on the finite element formulation in space coupled with an existing backward difference formula-based ODE solver in time. The second scheme uses an integral equation based Krylov deferred correction (KDC) method and a fast elliptic solver (FES) for the resulting elliptic equations. Numerical results are presented to validate the new AAM algorithm, which is also shown to be more computationally efficient than the original Monte-Carlo algorithm. We also demonstrate that the higher order KDC scheme is more efficient than the traditional finite element solution approach and this advantage becomes increasingly important as the desired accuracy of the solution increases. We also discuss issues of smoothness, which affect the efficiency of the KDC-FES approach, and outline additional algorithmic changes that would further improve the efficiency of these developing methods for a wide range of applications.

  4. Computational modeling of fully-ionized, magnetized plasmas using the fluid approximation

    NASA Astrophysics Data System (ADS)

    Schnack, Dalton

    2005-10-01

    Strongly magnetized plasmas are rich in spatial and temporal scales, making a computational approach useful for studying these systems. The most accurate model of a magnetized plasma is based on a kinetic equation that describes the evolution of the distribution function for each species in six-dimensional phase space. However, the high dimensionality renders this approach impractical for computations for long time scales in relevant geometry. Fluid models, derived by taking velocity moments of the kinetic equation [1] and truncating (closing) the hierarchy at some level, are an approximation to the kinetic model. The reduced dimensionality allows a wider range of spatial and/or temporal scales to be explored. Several approximations have been used [2-5]. Successful computational modeling requires understanding the ordering and closure approximations, the fundamental waves supported by the equations, and the numerical properties of the discretization scheme. We review and discuss several ordering schemes, their normal modes, and several algorithms that can be applied to obtain a numerical solution. The implementation of kinetic parallel closures is also discussed [6].[1] S. Chapman and T.G. Cowling, ``The Mathematical Theory of Non-Uniform Gases'', Cambridge University Press, Cambridge, UK (1939).[2] R.D. Hazeltine and J.D. Meiss, ``Plasma Confinement'', Addison-Wesley Publishing Company, Redwood City, CA (1992).[3] L.E. Sugiyama and W. Park, Physics of Plasmas 7, 4644 (2000).[4] J.J. Ramos, Physics of Plasmas, 10, 3601 (2003).[5] P.J. Catto and A.N. Simakov, Physics of Plasmas, 11, 90 (2004).[6] E.D. Held et al., Phys. Plasmas 11, 2419 (2004)

  5. Initial conditions for accurate N-body simulations of massive neutrino cosmologies

    NASA Astrophysics Data System (ADS)

    Zennaro, M.; Bel, J.; Villaescusa-Navarro, F.; Carbone, C.; Sefusatti, E.; Guzzo, L.

    2017-04-01

    The set-up of the initial conditions in cosmological N-body simulations is usually implemented by rescaling the desired low-redshift linear power spectrum to the required starting redshift consistently with the Newtonian evolution of the simulation. The implementation of this practical solution requires more care in the context of massive neutrino cosmologies, mainly because of the non-trivial scale-dependence of the linear growth that characterizes these models. In this work, we consider a simple two-fluid, Newtonian approximation for cold dark matter and massive neutrinos perturbations that can reproduce the cold matter linear evolution predicted by Boltzmann codes such as CAMB or CLASS with a 0.1 per cent accuracy or below for all redshift relevant to non-linear structure formation. We use this description, in the first place, to quantify the systematic errors induced by several approximations often assumed in numerical simulations, including the typical set-up of the initial conditions for massive neutrino cosmologies adopted in previous works. We then take advantage of the flexibility of this approach to rescale the late-time linear power spectra to the simulation initial redshift, in order to be as consistent as possible with the dynamics of the N-body code and the approximations it assumes. We implement our method in a public code (REPS rescaled power spectra for initial conditions with massive neutrinos https://github.com/matteozennaro/reps) providing the initial displacements and velocities for cold dark matter and neutrino particles that will allow accurate, I.e. 1 per cent level, numerical simulations for this cosmological scenario.

  6. Applying the method of fundamental solutions to harmonic problems with singular boundary conditions

    NASA Astrophysics Data System (ADS)

    Valtchev, Svilen S.; Alves, Carlos J. S.

    2017-07-01

    The method of fundamental solutions (MFS) is known to produce highly accurate numerical results for elliptic boundary value problems (BVP) with smooth boundary conditions, posed in analytic domains. However, due to the analyticity of the shape functions in its approximation basis, the MFS is usually disregarded when the boundary functions possess singularities. In this work we present a modification of the classical MFS which can be applied for the numerical solution of the Laplace BVP with Dirichlet boundary conditions exhibiting jump discontinuities. In particular, a set of harmonic functions with discontinuous boundary traces is added to the MFS basis. The accuracy of the proposed method is compared with the results form the classical MFS.

  7. Approximations to the exact exchange potential: KLI versus semilocal

    NASA Astrophysics Data System (ADS)

    Tran, Fabien; Blaha, Peter; Betzinger, Markus; Blügel, Stefan

    2016-10-01

    In the search for an accurate and computationally efficient approximation to the exact exchange potential of Kohn-Sham density functional theory, we recently compared various semilocal exchange potentials to the exact one [F. Tran et al., Phys. Rev. B 91, 165121 (2015), 10.1103/PhysRevB.91.165121]. It was concluded that the Becke-Johnson (BJ) potential is a very good starting point, but requires the use of empirical parameters to obtain good agreement with the exact exchange potential. In this work, we extend the comparison by considering the Krieger-Li-Iafrate (KLI) approximation, which is a beyond-semilocal approximation. It is shown that overall the KLI- and BJ-based potentials are the most reliable approximations to the exact exchange potential, however, sizable differences, especially for the antiferromagnetic transition-metal oxides, can be obtained.

  8. Homogenization of one-dimensional draining through heterogeneous porous media including higher-order approximations

    NASA Astrophysics Data System (ADS)

    Anderson, Daniel M.; McLaughlin, Richard M.; Miller, Cass T.

    2018-02-01

    We examine a mathematical model of one-dimensional draining of a fluid through a periodically-layered porous medium. A porous medium, initially saturated with a fluid of a high density is assumed to drain out the bottom of the porous medium with a second lighter fluid replacing the draining fluid. We assume that the draining layer is sufficiently dense that the dynamics of the lighter fluid can be neglected with respect to the dynamics of the heavier draining fluid and that the height of the draining fluid, represented as a free boundary in the model, evolves in time. In this context, we neglect interfacial tension effects at the boundary between the two fluids. We show that this problem admits an exact solution. Our primary objective is to develop a homogenization theory in which we find not only leading-order, or effective, trends but also capture higher-order corrections to these effective draining rates. The approximate solution obtained by this homogenization theory is compared to the exact solution for two cases: (1) the permeability of the porous medium varies smoothly but rapidly and (2) the permeability varies as a piecewise constant function representing discrete layers of alternating high/low permeability. In both cases we are able to show that the corrections in the homogenization theory accurately predict the position of the free boundary moving through the porous medium.

  9. Infinite order sudden approximation for rotational energy transfer in gaseous mixtures

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

    Goldflam, R.; Green, S.; Kouri, D.J.

    1977-11-01

    Rotational energy transfer in gaseous mixtures has been considered within the framework of the infinite order sudden (IOS) approximation. A new derivation of the IOS from the coupled states Lippmann--Schwinger equation is given. This approach shows the relation between the IOS and CS T matrices and also shows in a rather transparent fashion Sencrest's result that the IOS method does not truncate closed channels but rather employs a closure relation to sum over all rotor states. The general CS effective cross section formula for relaxation processes is used, along with the IOS approximation to the CS T matrix, to derivemore » the general IOS effctive cross section.Factorization permits one to calculate other types of cross sections if any one type of cross section has been obtained by some procedure. The functional form can also be used to compact data. This formalism has been applied to calculate pressure broadening for the systems HD--He, HCl--He, CO--He, HCN--He, HCl--Ar, and CO/sub 2/--Ar. To test the IOS approximation, comparisons have been made to the CS results, which are known to be accurate for all these systems. The IOS approximation is found to be very accurate whenever the rotor spacings are small compared to the kinetic energy, provided closed channels do not play too great a role. For the systems CO--He, HCN--He, and CO/sub 2/--Ar, these conditions are well satisfied and the IOS is found to yield results accurate to within 10%--15%.« less

  10. Reduced-Order Direct Numerical Simulation of Solute Transport in Porous Media

    NASA Astrophysics Data System (ADS)

    Mehmani, Yashar; Tchelepi, Hamdi

    2017-11-01

    Pore-scale models are an important tool for analyzing fluid dynamics in porous materials (e.g., rocks, soils, fuel cells). Current direct numerical simulation (DNS) techniques, while very accurate, are computationally prohibitive for sample sizes that are statistically representative of the porous structure. Reduced-order approaches such as pore-network models (PNM) aim to approximate the pore-space geometry and physics to remedy this problem. Predictions from current techniques, however, have not always been successful. This work focuses on single-phase transport of a passive solute under advection-dominated regimes and delineates the minimum set of approximations that consistently produce accurate PNM predictions. Novel network extraction (discretization) and particle simulation techniques are developed and compared to high-fidelity DNS simulations for a wide range of micromodel heterogeneities and a single sphere pack. Moreover, common modeling assumptions in the literature are analyzed and shown that they can lead to first-order errors under advection-dominated regimes. This work has implications for optimizing material design and operations in manufactured (electrodes) and natural (rocks) porous media pertaining to energy systems. This work was supported by the Stanford University Petroleum Research Institute for Reservoir Simulation (SUPRI-B).

  11. Approximate optimal guidance for the advanced launch system

    NASA Technical Reports Server (NTRS)

    Feeley, T. S.; Speyer, J. L.

    1993-01-01

    A real-time guidance scheme for the problem of maximizing the payload into orbit subject to the equations of motion for a rocket over a spherical, non-rotating earth is presented. An approximate optimal launch guidance law is developed based upon an asymptotic expansion of the Hamilton - Jacobi - Bellman or dynamic programming equation. The expansion is performed in terms of a small parameter, which is used to separate the dynamics of the problem into primary and perturbation dynamics. For the zeroth-order problem the small parameter is set to zero and a closed-form solution to the zeroth-order expansion term of Hamilton - Jacobi - Bellman equation is obtained. Higher-order terms of the expansion include the effects of the neglected perturbation dynamics. These higher-order terms are determined from the solution of first-order linear partial differential equations requiring only the evaluation of quadratures. This technique is preferred as a real-time, on-line guidance scheme to alternative numerical iterative optimization schemes because of the unreliable convergence properties of these iterative guidance schemes and because the quadratures needed for the approximate optimal guidance law can be performed rapidly and by parallel processing. Even if the approximate solution is not nearly optimal, when using this technique the zeroth-order solution always provides a path which satisfies the terminal constraints. Results for two-degree-of-freedom simulations are presented for the simplified problem of flight in the equatorial plane and compared to the guidance scheme generated by the shooting method which is an iterative second-order technique.

  12. Approximate Bayesian evaluations of measurement uncertainty

    NASA Astrophysics Data System (ADS)

    Possolo, Antonio; Bodnar, Olha

    2018-04-01

    The Guide to the Expression of Uncertainty in Measurement (GUM) includes formulas that produce an estimate of a scalar output quantity that is a function of several input quantities, and an approximate evaluation of the associated standard uncertainty. This contribution presents approximate, Bayesian counterparts of those formulas for the case where the output quantity is a parameter of the joint probability distribution of the input quantities, also taking into account any information about the value of the output quantity available prior to measurement expressed in the form of a probability distribution on the set of possible values for the measurand. The approximate Bayesian estimates and uncertainty evaluations that we present have a long history and illustrious pedigree, and provide sufficiently accurate approximations in many applications, yet are very easy to implement in practice. Differently from exact Bayesian estimates, which involve either (analytical or numerical) integrations, or Markov Chain Monte Carlo sampling, the approximations that we describe involve only numerical optimization and simple algebra. Therefore, they make Bayesian methods widely accessible to metrologists. We illustrate the application of the proposed techniques in several instances of measurement: isotopic ratio of silver in a commercial silver nitrate; odds of cryptosporidiosis in AIDS patients; height of a manometer column; mass fraction of chromium in a reference material; and potential-difference in a Zener voltage standard.

  13. Unsteady fluid flow in a slightly curved pipe: A comparative study of a matched asymptotic expansions solution with a single analytical solution

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

    Messaris, Gerasimos A. T., E-mail: messaris@upatras.gr; School of Science and Technology, Hellenic Open University, 11 Sahtouri Street, GR 262 22 Patras; Hadjinicolaou, Maria

    The present work is motivated by the fact that blood flow in the aorta and the main arteries is governed by large finite values of the Womersley number α and for such values of α there is not any analytical solution in the literature. The existing numerical solutions, although accurate, give limited information about the factors that affect the flow, whereas an analytical approach has an advantage in that it can provide physical insight to the flow mechanism. Having this in mind, we seek analytical solution to the equations of the fluid flow driven by a sinusoidal pressure gradient inmore » a slightly curved pipe of circular cross section when the Womersley number varies from small finite to infinite values. Initially the equations of motion are expanded in terms of the curvature ratio δ and the resulting linearized equations are solved analytically in two ways. In the first, we match the solution for the main core to that for the Stokes boundary layer. This solution is valid for very large values of α. In the second, we derive a straightforward single solution valid to the entire flow region and for 8 ≤ α < ∞, a range which includes the values of α that refer to the physiological flows. Each solution contains expressions for the axial velocity, the stream function, and the wall stresses and is compared to the analogous forms presented in other studies. The two solutions give identical results to each other regarding the axial flow but differ in the secondary flow and the circumferential wall stress, due to the approximations employed in the matched asymptotic expansion process. The results on the stream function from the second solution are in agreement with analogous results from other numerical solutions. The second solution predicts that the atherosclerotic plaques may develop in any location around the cross section of the aortic wall unlike to the prescribed locations predicted by the first solution. In addition, it gives circumferential wall

  14. Unsteady fluid flow in a slightly curved pipe: A comparative study of a matched asymptotic expansions solution with a single analytical solution

    NASA Astrophysics Data System (ADS)

    Messaris, Gerasimos A. T.; Hadjinicolaou, Maria; Karahalios, George T.

    2016-08-01

    The present work is motivated by the fact that blood flow in the aorta and the main arteries is governed by large finite values of the Womersley number α and for such values of α there is not any analytical solution in the literature. The existing numerical solutions, although accurate, give limited information about the factors that affect the flow, whereas an analytical approach has an advantage in that it can provide physical insight to the flow mechanism. Having this in mind, we seek analytical solution to the equations of the fluid flow driven by a sinusoidal pressure gradient in a slightly curved pipe of circular cross section when the Womersley number varies from small finite to infinite values. Initially the equations of motion are expanded in terms of the curvature ratio δ and the resulting linearized equations are solved analytically in two ways. In the first, we match the solution for the main core to that for the Stokes boundary layer. This solution is valid for very large values of α. In the second, we derive a straightforward single solution valid to the entire flow region and for 8 ≤ α < ∞, a range which includes the values of α that refer to the physiological flows. Each solution contains expressions for the axial velocity, the stream function, and the wall stresses and is compared to the analogous forms presented in other studies. The two solutions give identical results to each other regarding the axial flow but differ in the secondary flow and the circumferential wall stress, due to the approximations employed in the matched asymptotic expansion process. The results on the stream function from the second solution are in agreement with analogous results from other numerical solutions. The second solution predicts that the atherosclerotic plaques may develop in any location around the cross section of the aortic wall unlike to the prescribed locations predicted by the first solution. In addition, it gives circumferential wall stresses

  15. Solution of linear systems by a singular perturbation technique

    NASA Technical Reports Server (NTRS)

    Ardema, M. D.

    1976-01-01

    An approximate solution is obtained for a singularly perturbed system of initial valued, time invariant, linear differential equations with multiple boundary layers. Conditions are stated under which the approximate solution converges uniformly to the exact solution as the perturbation parameter tends to zero. The solution is obtained by the method of matched asymptotic expansions. Use of the results for obtaining approximate solutions of general linear systems is discussed. An example is considered to illustrate the method and it is shown that the formulas derived give a readily computed uniform approximation.

  16. Approximate nonlinear multiparameter inversion for multicomponent single and double P-wave scattering in isotropic elastic media

    NASA Astrophysics Data System (ADS)

    Ouyang, Wei; Mao, Weijian

    2018-07-01

    An asymptotic quadratic true-amplitude inversion method for isotropic elastic P waves is proposed to invert medium parameters. The multicomponent P-wave scattered wavefield is computed based on a forward relationship using second-order Born approximation and corresponding high-frequency ray theoretical methods. Within the local double scattering mechanism, the P-wave transmission factors are elaborately calculated, which results in the radiation pattern for P-wave scattering being a quadratic combination of the density and Lamé's moduli perturbation parameters. We further express the elastic P-wave scattered wavefield in a form of generalized Radon transform. After introducing classical backprojection operators, we obtain an approximate solution of the inverse problem by solving a quadratic nonlinear system. Numerical tests with synthetic data computed by finite-differences scheme demonstrate that our quadratic inversion can accurately invert perturbation parameters for strong perturbations, compared with the P-wave single-scattering linear inversion method. Although our inversion strategy here is only syncretized with P-wave scattering, it can be extended to invert multicomponent elastic data containing both P- and S-wave information.

  17. Metaheuristic optimisation methods for approximate solving of singular boundary value problems

    NASA Astrophysics Data System (ADS)

    Sadollah, Ali; Yadav, Neha; Gao, Kaizhou; Su, Rong

    2017-07-01

    This paper presents a novel approximation technique based on metaheuristics and weighted residual function (WRF) for tackling singular boundary value problems (BVPs) arising in engineering and science. With the aid of certain fundamental concepts of mathematics, Fourier series expansion, and metaheuristic optimisation algorithms, singular BVPs can be approximated as an optimisation problem with boundary conditions as constraints. The target is to minimise the WRF (i.e. error function) constructed in approximation of BVPs. The scheme involves generational distance metric for quality evaluation of the approximate solutions against exact solutions (i.e. error evaluator metric). Four test problems including two linear and two non-linear singular BVPs are considered in this paper to check the efficiency and accuracy of the proposed algorithm. The optimisation task is performed using three different optimisers including the particle swarm optimisation, the water cycle algorithm, and the harmony search algorithm. Optimisation results obtained show that the suggested technique can be successfully applied for approximate solving of singular BVPs.

  18. On the arbitrary l-wave solutions of the deformed hyperbolic manning-rosen potential including an improved approximation to the orbital centrifugal term

    NASA Astrophysics Data System (ADS)

    Xu, Chun-Long; Zhang, Min-Cang

    2017-01-01

    The arbitrary l-wave solutions to the Schrödinger equation for the deformed hyperbolic Manning-Rosen potential is investigated analytically by using the Nikiforov-Uvarov method, the centrifugal term is treated with an improved Greene and Aldrich's approximation scheme. The wavefunctions depend on the deformation parameter q, which is expressed in terms of the Jocobi polynomial or the hypergeometric function. The bound state energy is obtained, and the discrete spectrum is shown to be independent of the deformation parameter q.

  19. A Gaussian Approximation Potential for Silicon

    NASA Astrophysics Data System (ADS)

    Bernstein, Noam; Bartók, Albert; Kermode, James; Csányi, Gábor

    We present an interatomic potential for silicon using the Gaussian Approximation Potential (GAP) approach, which uses the Gaussian process regression method to approximate the reference potential energy surface as a sum of atomic energies. Each atomic energy is approximated as a function of the local environment around the atom, which is described with the smooth overlap of atomic environments (SOAP) descriptor. The potential is fit to a database of energies, forces, and stresses calculated using density functional theory (DFT) on a wide range of configurations from zero and finite temperature simulations. These include crystalline phases, liquid, amorphous, and low coordination structures, and diamond-structure point defects, dislocations, surfaces, and cracks. We compare the results of the potential to DFT calculations, as well as to previously published models including Stillinger-Weber, Tersoff, modified embedded atom method (MEAM), and ReaxFF. We show that it is very accurate as compared to the DFT reference results for a wide range of properties, including low energy bulk phases, liquid structure, as well as point, line, and plane defects in the diamond structure.

  20. An accurate two-dimensional LBIC solution for bipolar transistors

    NASA Astrophysics Data System (ADS)

    Benarab, A.; Baudrand, H.; Lescure, M.; Boucher, J.

    1988-05-01

    A complete solution of the diffusion problem of carriers generated by a located light beam in the emitter and base region of a bipolar structure is presented. Green's function method and moment method are used to solve the 2-D diffusion equation in these regions. From the Green's functions solution of these equations, the light beam induced currents (LBIC) in the different junctions of the structure due to an extended generation represented by a rectangular light spot; are thus decided. The equations of these currents depend both on the parameters which characterise the structure, surface states, dimensions of the emitter and the base region, and the characteristics of the light spot, that is to say, the width and the wavelength. Curves illustrating the variation of the various LBIC in the base region junctions as a function of the impact point of the light beam ( x0) for different values of these parameters are discussed. In particular, the study of the base-emitter currents when the light beam is swept right across the sample illustrates clearly a good geometrical definition of the emitter region up to base end of the emitter-base space-charge areas and a "whirl" lateral diffusion beneath this region, (i.e. the diffusion of the generated carriers near the surface towards the horizontal base-emitter junction and those created beneath this junction towards the lateral (B-E) junctions).

  1. When is the Anelastic Approximation a Valid Model for Compressible Convection?

    NASA Astrophysics Data System (ADS)

    Alboussiere, T.; Curbelo, J.; Labrosse, S.; Ricard, Y. R.; Dubuffet, F.

    2017-12-01

    Compressible convection is ubiquitous in large natural systems such Planetary atmospheres, stellar and planetary interiors. Its modelling is notoriously more difficult than the case when the Boussinesq approximation applies. One reason for that difficulty has been put forward by Ogura and Phillips (1961): the compressible equations generate sound waves with very short time scales which need to be resolved. This is why they introduced an anelastic model, based on an expansion of the solution around an isentropic hydrostatic profile. How accurate is that anelastic model? What are the conditions for its validity? To answer these questions, we have developed a numerical model for the full set of compressible equations and compared its solutions with those of the corresponding anelastic model. We considered a simple rectangular 2D Rayleigh-Bénard configuration and decided to restrict the analysis to infinite Prandtl numbers. This choice is valid for convection in the mantles of rocky planets, but more importantly lead to a zero Mach number. So we got rid of the question of the interference of acoustic waves with convection. In that simplified context, we used the entropy balances (that of the full set of equations and that of the anelastic model) to investigate the differences between exact and anelastic solutions. We found that the validity of the anelastic model is dictated by two conditions: first, the superadiabatic temperature difference must be small compared with the adiabatic temperature difference (as expected) ɛ = Δ TSA / delta Ta << 1, and secondly that the product of ɛ with the Nusselt number must be small.

  2. Approximate convective heating equations for hypersonic flows

    NASA Technical Reports Server (NTRS)

    Zoby, E. V.; Moss, J. N.; Sutton, K.

    1979-01-01

    Laminar and turbulent heating-rate equations appropriate for engineering predictions of the convective heating rates about blunt reentry spacecraft at hypersonic conditions are developed. The approximate methods are applicable to both nonreacting and reacting gas mixtures for either constant or variable-entropy edge conditions. A procedure which accounts for variable-entropy effects and is not based on mass balancing is presented. Results of the approximate heating methods are in good agreement with existing experimental results as well as boundary-layer and viscous-shock-layer solutions.

  3. Numerical solution of the unsteady Navier-Stokes equation

    NASA Technical Reports Server (NTRS)

    Osher, Stanley J.; Engquist, Bjoern

    1985-01-01

    The construction and the analysis of nonoscillatory shock capturing methods for the approximation of hyperbolic conservation laws are discussed. These schemes share many desirable properties with total variation diminishing schemes, but TVD schemes have at most first-order accuracy, in the sense of truncation error, at extrema of the solution. In this paper a uniformly second-order approximation is constructed, which is nonoscillatory in the sense that the number of extrema of the discrete solution is not increasing in time. This is achieved via a nonoscillatory piecewise linear reconstruction of the solution from its cell averages, time evolution through an approximate solution of the resulting initial value problem, and averaging of this approximate solution over each cell.

  4. Integral approximations to classical diffusion and smoothed particle hydrodynamics

    DOE PAGES

    Du, Qiang; Lehoucq, R. B.; Tartakovsky, A. M.

    2014-12-31

    The contribution of the paper is the approximation of a classical diffusion operator by an integral equation with a volume constraint. A particular focus is on classical diffusion problems associated with Neumann boundary conditions. By exploiting this approximation, we can also approximate other quantities such as the flux out of a domain. Our analysis of the model equation on the continuum level is closely related to the recent work on nonlocal diffusion and peridynamic mechanics. In particular, we elucidate the role of a volumetric constraint as an approximation to a classical Neumann boundary condition in the presence of physical boundary.more » The volume-constrained integral equation then provides the basis for accurate and robust discretization methods. As a result, an immediate application is to the understanding and improvement of the Smoothed Particle Hydrodynamics (SPH) method.« less

  5. An approximate method for calculating three-dimensional inviscid hypersonic flow fields

    NASA Technical Reports Server (NTRS)

    Riley, Christopher J.; Dejarnette, Fred R.

    1990-01-01

    An approximate solution technique was developed for 3-D inviscid, hypersonic flows. The method employs Maslen's explicit pressure equation in addition to the assumption of approximate stream surfaces in the shock layer. This approximation represents a simplification to Maslen's asymmetric method. The present method presents a tractable procedure for computing the inviscid flow over 3-D surfaces at angle of attack. The solution procedure involves iteratively changing the shock shape in the subsonic-transonic region until the correct body shape is obtained. Beyond this region, the shock surface is determined using a marching procedure. Results are presented for a spherically blunted cone, paraboloid, and elliptic cone at angle of attack. The calculated surface pressures are compared with experimental data and finite difference solutions of the Euler equations. Shock shapes and profiles of pressure are also examined. Comparisons indicate the method adequately predicts shock layer properties on blunt bodies in hypersonic flow. The speed of the calculations makes the procedure attractive for engineering design applications.

  6. A highly accurate analytical solution for the surface fields of a short vertical wire antenna lying on a multilayer ground

    NASA Astrophysics Data System (ADS)

    Parise, M.

    2018-01-01

    A highly accurate analytical solution is derived to the electromagnetic problem of a short vertical wire antenna located on a stratified ground. The derivation consists of three steps. First, the integration path of the integrals describing the fields of the dipole is deformed and wrapped around the pole singularities and the two vertical branch cuts of the integrands located in the upper half of the complex plane. This allows to decompose the radiated field into its three contributions, namely the above-surface ground wave, the lateral wave, and the trapped surface waves. Next, the square root terms responsible for the branch cuts are extracted from the integrands of the branch-cut integrals. Finally, the extracted square roots are replaced with their rational representations according to Newton's square root algorithm, and residue theorem is applied to give explicit expressions, in series form, for the fields. The rigorous integration procedure and the convergence of square root algorithm ensure that the obtained formulas converge to the exact solution. Numerical simulations are performed to show the validity and robustness of the developed formulation, as well as its advantages in terms of time cost over standard numerical integration procedures.

  7. Localized solutions of Lugiato-Lefever equations with focused pump.

    PubMed

    Cardoso, Wesley B; Salasnich, Luca; Malomed, Boris A

    2017-12-04

    Lugiato-Lefever (LL) equations in one and two dimensions (1D and 2D) accurately describe the dynamics of optical fields in pumped lossy cavities with the intrinsic Kerr nonlinearity. The external pump is usually assumed to be uniform, but it can be made tightly focused too-in particular, for building small pixels. We obtain solutions of the LL equations, with both the focusing and defocusing intrinsic nonlinearity, for 1D and 2D confined modes supported by the localized pump. In the 1D setting, we first develop a simple perturbation theory, based in the sech ansatz, in the case of weak pump and loss. Then, a family of exact analytical solutions for spatially confined modes is produced for the pump focused in the form of a delta-function, with a nonlinear loss (two-photon absorption) added to the LL model. Numerical findings demonstrate that these exact solutions are stable, both dynamically and structurally (the latter means that stable numerical solutions close to the exact ones are found when a specific condition, necessary for the existence of the analytical solution, does not hold). In 2D, vast families of stable confined modes are produced by means of a variational approximation and full numerical simulations.

  8. The complex variable boundary element method: Applications in determining approximative boundaries

    USGS Publications Warehouse

    Hromadka, T.V.

    1984-01-01

    The complex variable boundary element method (CVBEM) is used to determine approximation functions for boundary value problems of the Laplace equation such as occurs in potential theory. By determining an approximative boundary upon which the CVBEM approximator matches the desired constant (level curves) boundary conditions, the CVBEM is found to provide the exact solution throughout the interior of the transformed problem domain. Thus, the acceptability of the CVBEM approximation is determined by the closeness-of-fit of the approximative boundary to the study problem boundary. ?? 1984.

  9. Approximate non-linear multiparameter inversion for multicomponent single and double P-wave scattering in isotropic elastic media

    NASA Astrophysics Data System (ADS)

    Ouyang, Wei; Mao, Weijian

    2018-03-01

    An asymptotic quadratic true-amplitude inversion method for isotropic elastic P waves is proposed to invert medium parameters. The multicomponent P-wave scattered wavefield is computed based on a forward relationship using second-order Born approximation and corresponding high-frequency ray theoretical methods. Within the local double scattering mechanism, the P-wave transmission factors are elaborately calculated, which results in the radiation pattern for P-waves scattering being a quadratic combination of the density and Lamé's moduli perturbation parameters. We further express the elastic P-wave scattered wavefield in a form of generalized Radon transform (GRT). After introducing classical backprojection operators, we obtain an approximate solution of the inverse problem by solving a quadratic non-linear system. Numerical tests with synthetic data computed by finite-differences scheme demonstrate that our quadratic inversion can accurately invert perturbation parameters for strong perturbations, compared with the P-wave single-scattering linear inversion method. Although our inversion strategy here is only syncretized with P-wave scattering, it can be extended to invert multicomponent elastic data containing both P-wave and S-wave information.

  10. On the loop approximation in nucleon QCD sum rules

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

    Drukarev, E. G., E-mail: drukarev@thd.pnpi.spb.ru; Ryskin, M. G.; Sadovnikova, V. A.

    There was a general belief that the nucleon QCD sum rules which include only the quark loops and thus contain only the condensates of dimension d = 3 and d = 4 have only a trivial solution. We demonstrate that there is also a nontrivial solution. We show that it can be treated as the lowest order approximation to the solution which includes the higher terms of the Operator Product Expansion. Inclusion of the radiative corrections improves the convergence of the series.

  11. Simple, Fast and Accurate Implementation of the Diffusion Approximation Algorithm for Stochastic Ion Channels with Multiple States

    PubMed Central

    Orio, Patricio; Soudry, Daniel

    2012-01-01

    Background The phenomena that emerge from the interaction of the stochastic opening and closing of ion channels (channel noise) with the non-linear neural dynamics are essential to our understanding of the operation of the nervous system. The effects that channel noise can have on neural dynamics are generally studied using numerical simulations of stochastic models. Algorithms based on discrete Markov Chains (MC) seem to be the most reliable and trustworthy, but even optimized algorithms come with a non-negligible computational cost. Diffusion Approximation (DA) methods use Stochastic Differential Equations (SDE) to approximate the behavior of a number of MCs, considerably speeding up simulation times. However, model comparisons have suggested that DA methods did not lead to the same results as in MC modeling in terms of channel noise statistics and effects on excitability. Recently, it was shown that the difference arose because MCs were modeled with coupled gating particles, while the DA was modeled using uncoupled gating particles. Implementations of DA with coupled particles, in the context of a specific kinetic scheme, yielded similar results to MC. However, it remained unclear how to generalize these implementations to different kinetic schemes, or whether they were faster than MC algorithms. Additionally, a steady state approximation was used for the stochastic terms, which, as we show here, can introduce significant inaccuracies. Main Contributions We derived the SDE explicitly for any given ion channel kinetic scheme. The resulting generic equations were surprisingly simple and interpretable – allowing an easy, transparent and efficient DA implementation, avoiding unnecessary approximations. The algorithm was tested in a voltage clamp simulation and in two different current clamp simulations, yielding the same results as MC modeling. Also, the simulation efficiency of this DA method demonstrated considerable superiority over MC methods, except when

  12. Approximation algorithms for planning and control

    NASA Technical Reports Server (NTRS)

    Boddy, Mark; Dean, Thomas

    1989-01-01

    A control system operating in a complex environment will encounter a variety of different situations, with varying amounts of time available to respond to critical events. Ideally, such a control system will do the best possible with the time available. In other words, its responses should approximate those that would result from having unlimited time for computation, where the degree of the approximation depends on the amount of time it actually has. There exist approximation algorithms for a wide variety of problems. Unfortunately, the solution to any reasonably complex control problem will require solving several computationally intensive problems. Algorithms for successive approximation are a subclass of the class of anytime algorithms, algorithms that return answers for any amount of computation time, where the answers improve as more time is allotted. An architecture is described for allocating computation time to a set of anytime algorithms, based on expectations regarding the value of the answers they return. The architecture described is quite general, producing optimal schedules for a set of algorithms under widely varying conditions.

  13. A Time-Accurate Upwind Unstructured Finite Volume Method for Compressible Flow with Cure of Pathological Behaviors

    NASA Technical Reports Server (NTRS)

    Loh, Ching Y.; Jorgenson, Philip C. E.

    2007-01-01

    A time-accurate, upwind, finite volume method for computing compressible flows on unstructured grids is presented. The method is second order accurate in space and time and yields high resolution in the presence of discontinuities. For efficiency, the Roe approximate Riemann solver with an entropy correction is employed. In the basic Euler/Navier-Stokes scheme, many concepts of high order upwind schemes are adopted: the surface flux integrals are carefully treated, a Cauchy-Kowalewski time-stepping scheme is used in the time-marching stage, and a multidimensional limiter is applied in the reconstruction stage. However even with these up-to-date improvements, the basic upwind scheme is still plagued by the so-called "pathological behaviors," e.g., the carbuncle phenomenon, the expansion shock, etc. A solution to these limitations is presented which uses a very simple dissipation model while still preserving second order accuracy. This scheme is referred to as the enhanced time-accurate upwind (ETAU) scheme in this paper. The unstructured grid capability renders flexibility for use in complex geometry; and the present ETAU Euler/Navier-Stokes scheme is capable of handling a broad spectrum of flow regimes from high supersonic to subsonic at very low Mach number, appropriate for both CFD (computational fluid dynamics) and CAA (computational aeroacoustics). Numerous examples are included to demonstrate the robustness of the methods.

  14. On Hilbert-Schmidt norm convergence of Galerkin approximation for operator Riccati equations

    NASA Technical Reports Server (NTRS)

    Rosen, I. G.

    1988-01-01

    An abstract approximation framework for the solution of operator algebraic Riccati equations is developed. The approach taken is based on a formulation of the Riccati equation as an abstract nonlinear operator equation on the space of Hilbert-Schmidt operators. Hilbert-Schmidt norm convergence of solutions to generic finite dimensional Galerkin approximations to the Riccati equation to the solution of the original infinite dimensional problem is argued. The application of the general theory is illustrated via an operator Riccati equation arising in the linear-quadratic design of an optimal feedback control law for a 1-D heat/diffusion equation. Numerical results demonstrating the convergence of the associated Hilbert-Schmidt kernels are included.

  15. Accurate Vehicle Location System Using RFID, an Internet of Things Approach.

    PubMed

    Prinsloo, Jaco; Malekian, Reza

    2016-06-04

    Modern infrastructure, such as dense urban areas and underground tunnels, can effectively block all GPS signals, which implies that effective position triangulation will not be achieved. The main problem that is addressed in this project is the design and implementation of an accurate vehicle location system using radio-frequency identification (RFID) technology in combination with GPS and the Global system for Mobile communication (GSM) technology, in order to provide a solution to the limitation discussed above. In essence, autonomous vehicle tracking will be facilitated with the use of RFID technology where GPS signals are non-existent. The design of the system and the results are reflected in this paper. An extensive literature study was done on the field known as the Internet of Things, as well as various topics that covered the integration of independent technology in order to address a specific challenge. The proposed system is then designed and implemented. An RFID transponder was successfully designed and a read range of approximately 31 cm was obtained in the low frequency communication range (125 kHz to 134 kHz). The proposed system was designed, implemented, and field tested and it was found that a vehicle could be accurately located and tracked. It is also found that the antenna size of both the RFID reader unit and RFID transponder plays a critical role in the maximum communication range that can be achieved.

  16. Accurate Vehicle Location System Using RFID, an Internet of Things Approach

    PubMed Central

    Prinsloo, Jaco; Malekian, Reza

    2016-01-01

    Modern infrastructure, such as dense urban areas and underground tunnels, can effectively block all GPS signals, which implies that effective position triangulation will not be achieved. The main problem that is addressed in this project is the design and implementation of an accurate vehicle location system using radio-frequency identification (RFID) technology in combination with GPS and the Global system for Mobile communication (GSM) technology, in order to provide a solution to the limitation discussed above. In essence, autonomous vehicle tracking will be facilitated with the use of RFID technology where GPS signals are non-existent. The design of the system and the results are reflected in this paper. An extensive literature study was done on the field known as the Internet of Things, as well as various topics that covered the integration of independent technology in order to address a specific challenge. The proposed system is then designed and implemented. An RFID transponder was successfully designed and a read range of approximately 31 cm was obtained in the low frequency communication range (125 kHz to 134 kHz). The proposed system was designed, implemented, and field tested and it was found that a vehicle could be accurately located and tracked. It is also found that the antenna size of both the RFID reader unit and RFID transponder plays a critical role in the maximum communication range that can be achieved. PMID:27271638

  17. Approximate Algorithms for Computing Spatial Distance Histograms with Accuracy Guarantees

    PubMed Central

    Grupcev, Vladimir; Yuan, Yongke; Tu, Yi-Cheng; Huang, Jin; Chen, Shaoping; Pandit, Sagar; Weng, Michael

    2014-01-01

    Particle simulation has become an important research tool in many scientific and engineering fields. Data generated by such simulations impose great challenges to database storage and query processing. One of the queries against particle simulation data, the spatial distance histogram (SDH) query, is the building block of many high-level analytics, and requires quadratic time to compute using a straightforward algorithm. Previous work has developed efficient algorithms that compute exact SDHs. While beating the naive solution, such algorithms are still not practical in processing SDH queries against large-scale simulation data. In this paper, we take a different path to tackle this problem by focusing on approximate algorithms with provable error bounds. We first present a solution derived from the aforementioned exact SDH algorithm, and this solution has running time that is unrelated to the system size N. We also develop a mathematical model to analyze the mechanism that leads to errors in the basic approximate algorithm. Our model provides insights on how the algorithm can be improved to achieve higher accuracy and efficiency. Such insights give rise to a new approximate algorithm with improved time/accuracy tradeoff. Experimental results confirm our analysis. PMID:24693210

  18. Cold pasta phase in the extended Thomas-Fermi approximation

    NASA Astrophysics Data System (ADS)

    Avancini, S. S.; Bertolino, B. P.

    2015-10-01

    In this paper, we aim to obtain more accurate values for the transition density to the homogenous phase in the nuclear pasta that occurs in the inner crust of neutron stars. To that end, we use the nonlinear Walecka model at zero temperature and an approach based on the extended Thomas-Fermi (ETF) approximation.

  19. Toward more accurate loss tangent measurements in reentrant cavities

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

    Moyer, R. D.

    1980-05-01

    Karpova has described an absolute method for measurement of dielectric properties of a solid in a coaxial reentrant cavity. His cavity resonance equation yields very accurate results for dielectric constants. However, he presented only approximate expressions for the loss tangent. This report presents more exact expressions for that quantity and summarizes some experimental results.

  20. Extending the Utility of the Parabolic Approximation in Medical Ultrasound Using Wide-Angle Diffraction Modeling.

    PubMed

    Soneson, Joshua E

    2017-04-01

    Wide-angle parabolic models are commonly used in geophysics and underwater acoustics but have seen little application in medical ultrasound. Here, a wide-angle model for continuous-wave high-intensity ultrasound beams is derived, which approximates the diffraction process more accurately than the commonly used Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation without increasing implementation complexity or computing time. A method for preventing the high spatial frequencies often present in source boundary conditions from corrupting the solution is presented. Simulations of shallowly focused axisymmetric beams using both the wide-angle and standard parabolic models are compared to assess the accuracy with which they model diffraction effects. The wide-angle model proposed here offers improved focusing accuracy and less error throughout the computational domain than the standard parabolic model, offering a facile method for extending the utility of existing KZK codes.

  1. Finite element approximation of an optimal control problem for the von Karman equations

    NASA Technical Reports Server (NTRS)

    Hou, L. Steven; Turner, James C.

    1994-01-01

    This paper is concerned with optimal control problems for the von Karman equations with distributed controls. We first show that optimal solutions exist. We then show that Lagrange multipliers may be used to enforce the constraints and derive an optimality system from which optimal states and controls may be deduced. Finally we define finite element approximations of solutions for the optimality system and derive error estimates for the approximations.

  2. On the accuracy of the 'decoupled l-dominant' approximation for atom-molecule scattering

    NASA Technical Reports Server (NTRS)

    Green, S.

    1976-01-01

    Cross sections for rotational excitation and spectral pressure broadening of HD, HCl, CO, and HCN due to collisions with low energy He atoms have been computed within the 'decoupled l-dominant' (DLD) approximation and are compared with accurate close coupling results and also with two similar approximations, the effective potential of Rabitz and the coupled states of McGuire and Kouri. DLD predictions of state-to-state cross sections are rather good, being only slightly less accurate than coupled states results. DLD is far superior to either the coupled states or effective potential methods for pressure broadening calculations, although it may not be uniformly of the quantitative accuracy desirable for obtaining intermolecular potentials from experimental data.

  3. Unified Approximations: A New Approach for Monoprotic Weak Acid-Base Equilibria

    ERIC Educational Resources Information Center

    Pardue, Harry; Odeh, Ihab N.; Tesfai, Teweldemedhin M.

    2004-01-01

    The unified approximations reduce the conceptual complexity by combining solutions for a relatively large number of different situations into just two similar sets of processes. Processes used to solve problems by either the unified or classical approximations require similar degrees of understanding of the underlying chemical processes.

  4. Approximate analytical solution for non-Darcian flow toward a partially penetrating well in a confined aquifer

    NASA Astrophysics Data System (ADS)

    Wen, Zhang; Liu, Kai; Chen, Xiaolian

    2013-08-01

    In this study, non-Darcian flow to a partially penetrating well in a confined aquifer was investigated. The flow in the horizontal direction was assumed to be non-Darcian, while the flow in the vertical direction was assumed to be Darcian. The Izbash equation was employed to describe the non-Darcian flow in the horizontal direction of the aquifer. We used a linearization procedure to approximate the non-linear term in the governing equation enabling the mathematical model to be solved using a combination of Laplace and Fourier cosine transforms. Approximate analytical solutions for the drawdown were obtained and the impacts of different parameters on the drawdown were analyzed. The results indicated that a larger power index n in the Izbash equation leads to a larger drawdown at early times, while a larger n results in a smaller drawdown at late times. The drawdowns along the vertical direction z are symmetric if the well screen is located in the center of the aquifer, and the drawdown at the center of the aquifer is the largest along the vertical direction for this case. The length of the well screen w has little impact on the drawdown at early times, while a larger length of the well screen results in a smaller drawdown at late times. The drawdown increases with Kr at early times, while it decreases as Kr increases at late times, in which Kr is the apparent radial hydraulic conductivity. A sensitivity analysis of the parameters, i.e., the specific storage Ss, w, n and Kr, indicated that the drawdown is not sensitive to them at early times, while it is very sensitive to these parameters at late times especially to the power index n.

  5. Inverse eigenproblem for R-symmetric matrices and their approximation

    NASA Astrophysics Data System (ADS)

    Yuan, Yongxin

    2009-11-01

    Let be a nontrivial involution, i.e., R=R-1[not equal to]±In. We say that is R-symmetric if RGR=G. The set of all -symmetric matrices is denoted by . In this paper, we first give the solvability condition for the following inverse eigenproblem (IEP): given a set of vectors in and a set of complex numbers , find a matrix such that and are, respectively, the eigenvalues and eigenvectors of A. We then consider the following approximation problem: Given an n×n matrix , find such that , where is the solution set of IEP and ||[dot operator]|| is the Frobenius norm. We provide an explicit formula for the best approximation solution by means of the canonical correlation decomposition.

  6. An approximation theory for the identification of nonlinear distributed parameter systems

    NASA Technical Reports Server (NTRS)

    Banks, H. T.; Reich, Simeon; Rosen, I. G.

    1988-01-01

    An abstract approximation framework for the identification of nonlinear distributed parameter systems is developed. Inverse problems for nonlinear systems governed by strongly maximal monotone operators (satisfying a mild continuous dependence condition with respect to the unknown parameters to be identified) are treated. Convergence of Galerkin approximations and the corresponding solutions of finite dimensional approximating identification problems to a solution of the original finite dimensional identification problem is demonstrated using the theory of nonlinear evolution systems and a nonlinear analog of the Trotter-Kato approximation result for semigroups of bounded linear operators. The nonlinear theory developed here is shown to subsume an existing linear theory as a special case. It is also shown to be applicable to a broad class of nonlinear elliptic operators and the corresponding nonlinear parabolic partial differential equations to which they lead. An application of the theory to a quasilinear model for heat conduction or mass transfer is discussed.

  7. Finite volume solution of the compressible boundary-layer equations

    NASA Technical Reports Server (NTRS)

    Loyd, B.; Murman, E. M.

    1986-01-01

    A box-type finite volume discretization is applied to the integral form of the compressible boundary layer equations. Boundary layer scaling is introduced through the grid construction: streamwise grid lines follow eta = y/h = const., where y is the normal coordinate and h(x) is a scale factor proportional to the boundary layer thickness. With this grid, similarity can be applied explicity to calculate initial conditions. The finite volume method preserves the physical transparency of the integral equations in the discrete approximation. The resulting scheme is accurate, efficient, and conceptually simple. Computations for similar and non-similar flows show excellent agreement with tabulated results, solutions computed with Keller's Box scheme, and experimental data.

  8. Solution algorithms for nonlinear transient heat conduction analysis employing element-by-element iterative strategies

    NASA Technical Reports Server (NTRS)

    Winget, J. M.; Hughes, T. J. R.

    1985-01-01

    The particular problems investigated in the present study arise from nonlinear transient heat conduction. One of two types of nonlinearities considered is related to a material temperature dependence which is frequently needed to accurately model behavior over the range of temperature of engineering interest. The second nonlinearity is introduced by radiation boundary conditions. The finite element equations arising from the solution of nonlinear transient heat conduction problems are formulated. The finite element matrix equations are temporally discretized, and a nonlinear iterative solution algorithm is proposed. Algorithms for solving the linear problem are discussed, taking into account the form of the matrix equations, Gaussian elimination, cost, and iterative techniques. Attention is also given to approximate factorization, implementational aspects, and numerical results.

  9. Rapid perfusion quantification using Welch-Satterthwaite approximation and analytical spectral filtering

    NASA Astrophysics Data System (ADS)

    Krishnan, Karthik; Reddy, Kasireddy V.; Ajani, Bhavya; Yalavarthy, Phaneendra K.

    2017-02-01

    CT and MR perfusion weighted imaging (PWI) enable quantification of perfusion parameters in stroke studies. These parameters are calculated from the residual impulse response function (IRF) based on a physiological model for tissue perfusion. The standard approach for estimating the IRF is deconvolution using oscillatory-limited singular value decomposition (oSVD) or Frequency Domain Deconvolution (FDD). FDD is widely recognized as the fastest approach currently available for deconvolution of CT Perfusion/MR PWI. In this work, three faster methods are proposed. The first is a direct (model based) crude approximation to the final perfusion quantities (Blood flow, Blood volume, Mean Transit Time and Delay) using the Welch-Satterthwaite approximation for gamma fitted concentration time curves (CTC). The second method is a fast accurate deconvolution method, we call Analytical Fourier Filtering (AFF). The third is another fast accurate deconvolution technique using Showalter's method, we call Analytical Showalter's Spectral Filtering (ASSF). Through systematic evaluation on phantom and clinical data, the proposed methods are shown to be computationally more than twice as fast as FDD. The two deconvolution based methods, AFF and ASSF, are also shown to be quantitatively accurate compared to FDD and oSVD.

  10. High-Order Accurate Solutions to the Helmholtz Equation in the Presence of Boundary Singularities

    NASA Astrophysics Data System (ADS)

    Britt, Darrell Steven, Jr.

    Problems of time-harmonic wave propagation arise in important fields of study such as geological surveying, radar detection/evasion, and aircraft design. These often involve highfrequency waves, which demand high-order methods to mitigate the dispersion error. We propose a high-order method for computing solutions to the variable-coefficient inhomogeneous Helmholtz equation in two dimensions on domains bounded by piecewise smooth curves of arbitrary shape with a finite number of boundary singularities at known locations. We utilize compact finite difference (FD) schemes on regular structured grids to achieve highorder accuracy due to their efficiency and simplicity, as well as the capability to approximate variable-coefficient differential operators. In this work, a 4th-order compact FD scheme for the variable-coefficient Helmholtz equation on a Cartesian grid in 2D is derived and tested. The well known limitation of finite differences is that they lose accuracy when the boundary curve does not coincide with the discretization grid, which is a severe restriction on the geometry of the computational domain. Therefore, the algorithm presented in this work combines high-order FD schemes with the method of difference potentials (DP), which retains the efficiency of FD while allowing for boundary shapes that are not aligned with the grid without sacrificing the accuracy of the FD scheme. Additionally, the theory of DP allows for the universal treatment of the boundary conditions. One of the significant contributions of this work is the development of an implementation that accommodates general boundary conditions (BCs). In particular, Robin BCs with discontinuous coefficients are studied, for which we introduce a piecewise parameterization of the boundary curve. Problems with discontinuities in the boundary data itself are also studied. We observe that the design convergence rate suffers whenever the solution loses regularity due to the boundary conditions. This is

  11. Transverse signal decay under the weak field approximation: Theory and validation.

    PubMed

    Berman, Avery J L; Pike, G Bruce

    2018-07-01

    To derive an expression for the transverse signal time course from systems in the motional narrowing regime, such as water diffusing in blood. This was validated in silico and experimentally with ex vivo blood samples. A closed-form solution (CFS) for transverse signal decay under any train of refocusing pulses was derived using the weak field approximation. The CFS was validated via simulations of water molecules diffusing in the presence of spherical perturbers, with a range of sizes and under various pulse sequences. The CFS was compared with more conventional fits assuming monoexponential decay, including chemical exchange, using ex vivo blood Carr-Purcell-Meiboom-Gill data. From simulations, the CFS was shown to be valid in the motional narrowing regime and partially into the intermediate dephasing regime, with increased accuracy with increasing Carr-Purcell-Meiboom-Gill refocusing rate. In theoretical calculations of the CFS, fitting for the transverse relaxation rate (R 2 ) gave excellent agreement with the weak field approximation expression for R 2 for Carr-Purcell-Meiboom-Gill sequences, but diverged for free induction decay. These same results were confirmed in the ex vivo analysis. Transverse signal decay in the motional narrowing regime can be accurately described analytically. This theory has applications in areas such as tissue iron imaging, relaxometry of blood, and contrast agent imaging. Magn Reson Med 80:341-350, 2018. © 2017 International Society for Magnetic Resonance in Medicine. © 2017 International Society for Magnetic Resonance in Medicine.

  12. An approximate method for solution to variable moment of inertia problems

    NASA Technical Reports Server (NTRS)

    Beans, E. W.

    1981-01-01

    An approximation method is presented for reducing a nonlinear differential equation (for the 'weather vaning' motion of a wind turbine) to an equivalent constant moment of inertia problem. The integrated average of the moment of inertia is determined. Cycle time was found to be the equivalent cycle time if the rotating speed is 4 times greater than the system's minimum natural frequency.

  13. The neural network approximation method for solving multidimensional nonlinear inverse problems of geophysics

    NASA Astrophysics Data System (ADS)

    Shimelevich, M. I.; Obornev, E. A.; Obornev, I. E.; Rodionov, E. A.

    2017-07-01

    The iterative approximation neural network method for solving conditionally well-posed nonlinear inverse problems of geophysics is presented. The method is based on the neural network approximation of the inverse operator. The inverse problem is solved in the class of grid (block) models of the medium on a regularized parameterization grid. The construction principle of this grid relies on using the calculated values of the continuity modulus of the inverse operator and its modifications determining the degree of ambiguity of the solutions. The method provides approximate solutions of inverse problems with the maximal degree of detail given the specified degree of ambiguity with the total number of the sought parameters n × 103 of the medium. The a priori and a posteriori estimates of the degree of ambiguity of the approximated solutions are calculated. The work of the method is illustrated by the example of the three-dimensional (3D) inversion of the synthesized 2D areal geoelectrical (audio magnetotelluric sounding, AMTS) data corresponding to the schematic model of a kimberlite pipe.

  14. H2+, HeH and H2: Approximating potential curves, calculating rovibrational states

    NASA Astrophysics Data System (ADS)

    Olivares-Pilón, Horacio; Turbiner, Alexander V.

    2018-06-01

    Analytic consideration of the Bohr-Oppenheimer (BO) potential curves for diatomic molecules is proposed: accurate analytic interpolation for a potential curve consistent with its rovibrational spectra is found. It is shown that in the BO approximation for four lowest electronic states 1 sσg and 2 pσu, 2 pπu and 3 dπg of H2+, the ground state X2Σ+ of HeH and the two lowest states 1 Σg+ and 3 Σu+ of H2, the potential curves can be analytically interpolated in full range of internuclear distances R with not less than 4-5-6 s.d. Approximation based on matching the Laurant-type expansion at small R and a combination of the multipole expansion with one-instanton type contribution at large distances R is given by two-point Padé approximant. The position of minimum, when exists, is predicted within 1% or better. For the molecular ion H2+ in the Lagrange mesh method, the spectra of vibrational, rotational and rovibrational states (ν , L) associated with 1 sσg and 2 pσu, 2 pπu and 3 dπg potential curves are calculated. In general, it coincides with spectra found via numerical solution of the Schrödinger equation (when available) within six s.d. It is shown that 1 sσg curve contains 19 vibrational states (ν , 0) , while 2 pσu curve contains a single one (0 , 0) and 2 pπu state contains 12 vibrational states (ν , 0) . In general, 1 sσg electronic curve contains 420 rovibrational states, which increases up to 423 when we are beyond BO approximation. For the state 2 pσu the total number of rovibrational states (all with ν = 0) is equal to 3, within or beyond Bohr-Oppenheimer approximation. As for the state 2 pπu within the Bohr-Oppenheimer approximation the total number of the rovibrational bound states is equal to 284. The state 3 dπg is repulsive, no rovibrational state is found. It is confirmed in Lagrange mesh formalism the statement that the ground state potential curve of the heteronuclear molecule HeH does not support rovibrational states. Accurate

  15. 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

  16. Accurate solutions for transonic viscous flow over finite wings

    NASA Technical Reports Server (NTRS)

    Vatsa, V. N.

    1986-01-01

    An explicit multistage Runge-Kutta type time-stepping scheme is used for solving the three-dimensional, compressible, thin-layer Navier-Stokes equations. A finite-volume formulation is employed to facilitate treatment of complex grid topologies encountered in three-dimensional calculations. Convergence to steady state is expedited through usage of acceleration techniques. Further numerical efficiency is achieved through vectorization of the computer code. The accuracy of the overall scheme is evaluated by comparing the computed solutions with the experimental data for a finite wing under different test conditions in the transonic regime. A grid refinement study ir conducted to estimate the grid requirements for adequate resolution of salient features of such flows.

  17. Simulation of solute transport across low-permeability barrier walls

    USGS Publications Warehouse

    Harte, P.T.; Konikow, Leonard F.; Hornberger, G.Z.

    2006-01-01

    Low-permeability, non-reactive barrier walls are often used to contain contaminants in an aquifer. Rates of solute transport through such barriers are typically many orders of magnitude slower than rates through the aquifer. Nevertheless, the success of remedial actions may be sensitive to these low rates of transport. Two numerical simulation methods for representing low-permeability barriers in a finite-difference groundwater-flow and transport model were tested. In the first method, the hydraulic properties of the barrier were represented directly on grid cells and in the second method, the intercell hydraulic-conductance values were adjusted to approximate the reduction in horizontal flow, allowing use of a coarser and computationally efficient grid. The alternative methods were tested and evaluated on the basis of hypothetical test problems and a field case involving tetrachloroethylene (PCE) contamination at a Superfund site in New Hampshire. For all cases, advective transport across the barrier was negligible, but preexisting numerical approaches to calculate dispersion yielded dispersive fluxes that were greater than expected. A transport model (MODFLOW-GWT) was modified to (1) allow different dispersive and diffusive properties to be assigned to the barrier than the adjacent aquifer and (2) more accurately calculate dispersion from concentration gradients and solute fluxes near barriers. The new approach yields reasonable and accurate concentrations for the test cases. ?? 2006.

  18. The slow-scale linear noise approximation: an accurate, reduced stochastic description of biochemical networks under timescale separation conditions

    PubMed Central

    2012-01-01

    Background It is well known that the deterministic dynamics of biochemical reaction networks can be more easily studied if timescale separation conditions are invoked (the quasi-steady-state assumption). In this case the deterministic dynamics of a large network of elementary reactions are well described by the dynamics of a smaller network of effective reactions. Each of the latter represents a group of elementary reactions in the large network and has associated with it an effective macroscopic rate law. A popular method to achieve model reduction in the presence of intrinsic noise consists of using the effective macroscopic rate laws to heuristically deduce effective probabilities for the effective reactions which then enables simulation via the stochastic simulation algorithm (SSA). The validity of this heuristic SSA method is a priori doubtful because the reaction probabilities for the SSA have only been rigorously derived from microscopic physics arguments for elementary reactions. Results We here obtain, by rigorous means and in closed-form, a reduced linear Langevin equation description of the stochastic dynamics of monostable biochemical networks in conditions characterized by small intrinsic noise and timescale separation. The slow-scale linear noise approximation (ssLNA), as the new method is called, is used to calculate the intrinsic noise statistics of enzyme and gene networks. The results agree very well with SSA simulations of the non-reduced network of elementary reactions. In contrast the conventional heuristic SSA is shown to overestimate the size of noise for Michaelis-Menten kinetics, considerably under-estimate the size of noise for Hill-type kinetics and in some cases even miss the prediction of noise-induced oscillations. Conclusions A new general method, the ssLNA, is derived and shown to correctly describe the statistics of intrinsic noise about the macroscopic concentrations under timescale separation conditions. The ssLNA provides a

  19. Polynomial compensation, inversion, and approximation of discrete time linear systems

    NASA Technical Reports Server (NTRS)

    Baram, Yoram

    1987-01-01

    The least-squares transformation of a discrete-time multivariable linear system into a desired one by convolving the first with a polynomial system yields optimal polynomial solutions to the problems of system compensation, inversion, and approximation. The polynomial coefficients are obtained from the solution to a so-called normal linear matrix equation, whose coefficients are shown to be the weighting patterns of certain linear systems. These, in turn, can be used in the recursive solution of the normal equation.

  20. The application of Legendre-tau approximation to parameter identification for delay and partial differential equations

    NASA Technical Reports Server (NTRS)

    Ito, K.

    1983-01-01

    Approximation schemes based on Legendre-tau approximation are developed for application to parameter identification problem for delay and partial differential equations. The tau method is based on representing the approximate solution as a truncated series of orthonormal functions. The characteristic feature of the Legendre-tau approach is that when the solution to a problem is infinitely differentiable, the rate of convergence is faster than any finite power of 1/N; higher accuracy is thus achieved, making the approach suitable for small N.

  1. Jacobi-Gauss-Lobatto collocation method for the numerical solution of 1+1 nonlinear Schrödinger equations

    NASA Astrophysics Data System (ADS)

    Doha, E. H.; Bhrawy, A. H.; Abdelkawy, M. A.; Van Gorder, Robert A.

    2014-03-01

    A Jacobi-Gauss-Lobatto collocation (J-GL-C) method, used in combination with the implicit Runge-Kutta method of fourth order, is proposed as a numerical algorithm for the approximation of solutions to nonlinear Schrödinger equations (NLSE) with initial-boundary data in 1+1 dimensions. Our procedure is implemented in two successive steps. In the first one, the J-GL-C is employed for approximating the functional dependence on the spatial variable, using (N-1) nodes of the Jacobi-Gauss-Lobatto interpolation which depends upon two general Jacobi parameters. The resulting equations together with the two-point boundary conditions induce a system of 2(N-1) first-order ordinary differential equations (ODEs) in time. In the second step, the implicit Runge-Kutta method of fourth order is applied to solve this temporal system. The proposed J-GL-C method, used in combination with the implicit Runge-Kutta method of fourth order, is employed to obtain highly accurate numerical approximations to four types of NLSE, including the attractive and repulsive NLSE and a Gross-Pitaevskii equation with space-periodic potential. The numerical results obtained by this algorithm have been compared with various exact solutions in order to demonstrate the accuracy and efficiency of the proposed method. Indeed, for relatively few nodes used, the absolute error in our numerical solutions is sufficiently small.

  2. Elastic parabolic equation and normal mode solutions for seismo-acoustic propagation in underwater environments with ice covers.

    PubMed

    Collis, Jon M; Frank, Scott D; Metzler, Adam M; Preston, Kimberly S

    2016-05-01

    Sound propagation predictions for ice-covered ocean acoustic environments do not match observational data: received levels in nature are less than expected, suggesting that the effects of the ice are substantial. Effects due to elasticity in overlying ice can be significant enough that low-shear approximations, such as effective complex density treatments, may not be appropriate. Building on recent elastic seafloor modeling developments, a range-dependent parabolic equation solution that treats the ice as an elastic medium is presented. The solution is benchmarked against a derived elastic normal mode solution for range-independent underwater acoustic propagation. Results from both solutions accurately predict plate flexural modes that propagate in the ice layer, as well as Scholte interface waves that propagate at the boundary between the water and the seafloor. The parabolic equation solution is used to model a scenario with range-dependent ice thickness and a water sound speed profile similar to those observed during the 2009 Ice Exercise (ICEX) in the Beaufort Sea.

  3. A numerical analysis of the Born approximation for image formation modeling of differential interference contrast microscopy for human embryos

    NASA Astrophysics Data System (ADS)

    Trattner, Sigal; Feigin, Micha; Greenspan, Hayit; Sochen, Nir

    2008-03-01

    The differential interference contrast (DIC) microscope is commonly used for the visualization of live biological specimens. It enables the view of the transparent specimens while preserving their viability, being a non-invasive modality. Fertility clinics often use the DIC microscope for evaluation of human embryos quality. Towards quantification and reconstruction of the visualized specimens, an image formation model for DIC imaging is sought and the interaction of light waves with biological matter is examined. In many image formation models the light-matter interaction is expressed via the first Born approximation. The validity region of this approximation is defined in a theoretical bound which limits its use to very small specimens with low dielectric contrast. In this work the Born approximation is investigated via the Helmholtz equation, which describes the interaction between the specimen and light. A solution on the lens field is derived using the Gaussian Legendre quadrature formulation. This numerical scheme is considered both accurate and efficient and has shortened significantly the computation time as compared to integration methods that required a great amount of sampling for satisfying the Whittaker - Shannon sampling theorem. By comparing the numerical results with the theoretical values it is shown that the theoretical bound is not directly relevant to microscopic imaging and is far too limiting. The numerical exhaustive experiments show that the Born approximation is inappropriate for modeling the visualization of thick human embryos.

  4. An implicit higher-order spatially accurate scheme for solving time dependent flows on unstructured meshes

    NASA Astrophysics Data System (ADS)

    Tomaro, Robert F.

    1998-07-01

    The present research is aimed at developing a higher-order, spatially accurate scheme for both steady and unsteady flow simulations using unstructured meshes. The resulting scheme must work on a variety of general problems to ensure the creation of a flexible, reliable and accurate aerodynamic analysis tool. To calculate the flow around complex configurations, unstructured grids and the associated flow solvers have been developed. Efficient simulations require the minimum use of computer memory and computational times. Unstructured flow solvers typically require more computer memory than a structured flow solver due to the indirect addressing of the cells. The approach taken in the present research was to modify an existing three-dimensional unstructured flow solver to first decrease the computational time required for a solution and then to increase the spatial accuracy. The terms required to simulate flow involving non-stationary grids were also implemented. First, an implicit solution algorithm was implemented to replace the existing explicit procedure. Several test cases, including internal and external, inviscid and viscous, two-dimensional, three-dimensional and axi-symmetric problems, were simulated for comparison between the explicit and implicit solution procedures. The increased efficiency and robustness of modified code due to the implicit algorithm was demonstrated. Two unsteady test cases, a plunging airfoil and a wing undergoing bending and torsion, were simulated using the implicit algorithm modified to include the terms required for a moving and/or deforming grid. Secondly, a higher than second-order spatially accurate scheme was developed and implemented into the baseline code. Third- and fourth-order spatially accurate schemes were implemented and tested. The original dissipation was modified to include higher-order terms and modified near shock waves to limit pre- and post-shock oscillations. The unsteady cases were repeated using the higher

  5. Approximate Dispersion Relations for Waves on Arbitrary Shear Flows

    NASA Astrophysics Data System (ADS)

    Ellingsen, S. À.; Li, Y.

    2017-12-01

    An approximate dispersion relation is derived and presented for linear surface waves atop a shear current whose magnitude and direction can vary arbitrarily with depth. The approximation, derived to first order of deviation from potential flow, is shown to produce good approximations at all wavelengths for a wide range of naturally occuring shear flows as well as widely used model flows. The relation reduces in many cases to a 3-D generalization of the much used approximation by Skop (1987), developed further by Kirby and Chen (1989), but is shown to be more robust, succeeding in situations where the Kirby and Chen model fails. The two approximations incur the same numerical cost and difficulty. While the Kirby and Chen approximation is excellent for a wide range of currents, the exact criteria for its applicability have not been known. We explain the apparently serendipitous success of the latter and derive proper conditions of applicability for both approximate dispersion relations. Our new model has a greater range of applicability. A second order approximation is also derived. It greatly improves accuracy, which is shown to be important in difficult cases. It has an advantage over the corresponding second-order expression proposed by Kirby and Chen that its criterion of accuracy is explicitly known, which is not currently the case for the latter to our knowledge. Our second-order term is also arguably significantly simpler to implement, and more physically transparent, than its sibling due to Kirby and Chen.Plain Language SummaryIn order to answer key questions such as how the ocean surface affects the climate, erodes the coastline and transports nutrients, we must understand how waves move. This is not so easy when depth varying currents are present, as they often are in coastal waters. We have developed a modeling tool for <span class="hlt">accurately</span> predicting wave properties in such situations, ready for use, for example, in the</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/23582485','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/23582485"><span>Configurable hardware integrate and fire neurons for sparse <span class="hlt">approximation</span>.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Shapero, Samuel; Rozell, Christopher; Hasler, Paul</p> <p>2013-09-01</p> <p>Sparse <span class="hlt">approximation</span> is an important optimization problem in signal and image processing applications. A Hopfield-Network-like system of integrate and fire (IF) neurons is proposed as a <span class="hlt">solution</span>, using the Locally Competitive Algorithm (LCA) to solve an overcomplete L1 sparse <span class="hlt">approximation</span> problem. A scalable system architecture is described, including IF neurons with a nonlinear firing function, and current-based synapses to provide linear computation. A network of 18 neurons with 12 inputs is implemented on the RASP 2.9v chip, a Field Programmable Analog Array (FPAA) with directly programmable floating gate elements. Said system uses over 1400 floating gates, the largest system programmed on a FPAA to date. The circuit successfully reproduced the outputs of a digital optimization program, converging to within 4.8% RMS, and an objective cost only 1.7% higher on average. The active circuit consumed 559 μA of current at 2.4 V and converges on <span class="hlt">solutions</span> in 25 μs, with measurement of the converged spike rate taking an additional 1 ms. Extrapolating the scaling trends to a N=1000 node system, the spiking LCA compares favorably with state-of-the-art digital <span class="hlt">solutions</span>, and analog <span class="hlt">solutions</span> using a non-spiking approach. Copyright © 2013 Elsevier Ltd. All rights reserved.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19800006556','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19800006556"><span><span class="hlt">Approximate</span> and exact numerical integration of the gas dynamic equations</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Lewis, T. S.; Sirovich, L.</p> <p>1979-01-01</p> <p>A highly <span class="hlt">accurate</span> <span class="hlt">approximation</span> and a rapidly convergent numerical procedure are developed for two dimensional steady supersonic flow over an airfoil. Examples are given for a symmetric airfoil over a range of Mach numbers. Several interesting features are found in the calculation of the tail shock and the flow behind the airfoil.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/8373601','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/8373601"><span><span class="hlt">Accurate</span> high-speed liquid handling of very small biological samples.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Schober, A; Günther, R; Schwienhorst, A; Döring, M; Lindemann, B F</p> <p>1993-08-01</p> <p>Molecular biology techniques require the <span class="hlt">accurate</span> pipetting of buffers and <span class="hlt">solutions</span> with volumes in the microliter range. Traditionally, hand-held pipetting devices are used to fulfill these requirements, but many laboratories have also introduced robotic workstations for the handling of liquids. Piston-operated pumps are commonly used in manually as well as automatically operated pipettors. These devices cannot meet the demands for extremely <span class="hlt">accurate</span> pipetting of very small volumes at the high speed that would be necessary for certain applications (e.g., in sequencing projects with high throughput). In this paper we describe a technique for the <span class="hlt">accurate</span> microdispensation of biochemically relevant <span class="hlt">solutions</span> and suspensions with the aid of a piezoelectric transducer. It is suitable for liquids of a viscosity between 0.5 and 500 milliPascals. The obtainable drop sizes range from 5 picoliters to a few nanoliters with up to 10,000 drops per second. Liquids can be dispensed in single or accumulated drops to handle a wide volume range. The system proved to be excellently suitable for the handling of biological samples. It did not show any detectable negative impact on the biological function of dissolved or suspended molecules or particles.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/20060027797','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/20060027797"><span>Time-<span class="hlt">Accurate</span>, Unstructured-Mesh Navier-Stokes Computations with the Space-Time CESE Method</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Chang, Chau-Lyan</p> <p>2006-01-01</p> <p>Application of the newly emerged space-time conservation element <span class="hlt">solution</span> element (CESE) method to compressible Navier-Stokes equations is studied. In contrast to Euler equations solvers, several issues such as boundary conditions, numerical dissipation, and grid stiffness warrant systematic investigations and validations. Non-reflecting boundary conditions applied at the truncated boundary are also investigated from the stand point of acoustic wave propagation. Validations of the numerical <span class="hlt">solutions</span> are performed by comparing with exact <span class="hlt">solutions</span> for steady-state as well as time-<span class="hlt">accurate</span> viscous flow problems. The test cases cover a broad speed regime for problems ranging from acoustic wave propagation to 3D hypersonic configurations. Model problems pertinent to hypersonic configurations demonstrate the effectiveness of the CESE method in treating flows with shocks, unsteady waves, and separations. Good agreement with exact <span class="hlt">solutions</span> suggests that the space-time CESE method provides a viable alternative for time-<span class="hlt">accurate</span> Navier-Stokes calculations of a broad range of problems.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/pages/biblio/1414139-analytic-saddlepoint-approximation-ionization-energy-loss-distributions','SCIGOV-DOEP'); return false;" href="https://www.osti.gov/pages/biblio/1414139-analytic-saddlepoint-approximation-ionization-energy-loss-distributions"><span>Analytic saddlepoint <span class="hlt">approximation</span> for ionization energy loss distributions</span></a></p> <p><a target="_blank" href="http://www.osti.gov/pages">DOE PAGES</a></p> <p>Sjue, Sky K. L.; George, Jr., Richard Neal; Mathews, David Gregory</p> <p>2017-07-27</p> <p>Here, we present a saddlepoint <span class="hlt">approximation</span> for ionization energy loss distributions, valid for arbitrary relativistic velocities of the incident particle 0 < v/c < 1, provided that ionizing collisions are still the dominant energy loss mechanism. We derive a closed form <span class="hlt">solution</span> closely related to Moyal’s distribution. This distribution is intended for use in simulations with relatively low computational overhead. The <span class="hlt">approximation</span> generally reproduces the Vavilov most probable energy loss and full width at half maximum to better than 1% and 10%, respectively, with significantly better agreement as Vavilov’s κ approaches 1.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017NIMPB.407..270S','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017NIMPB.407..270S"><span>Analytic saddlepoint <span class="hlt">approximation</span> for ionization energy loss distributions</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Sjue, S. K. L.; George, R. N.; Mathews, D. G.</p> <p>2017-09-01</p> <p>We present a saddlepoint <span class="hlt">approximation</span> for ionization energy loss distributions, valid for arbitrary relativistic velocities of the incident particle 0 < v / c < 1 , provided that ionizing collisions are still the dominant energy loss mechanism. We derive a closed form <span class="hlt">solution</span> closely related to Moyal's distribution. This distribution is intended for use in simulations with relatively low computational overhead. The <span class="hlt">approximation</span> generally reproduces the Vavilov most probable energy loss and full width at half maximum to better than 1% and 10%, respectively, with significantly better agreement as Vavilov's κ approaches 1.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/biblio/1414139-analytic-saddlepoint-approximation-ionization-energy-loss-distributions','SCIGOV-STC'); return false;" href="https://www.osti.gov/biblio/1414139-analytic-saddlepoint-approximation-ionization-energy-loss-distributions"><span>Analytic saddlepoint <span class="hlt">approximation</span> for ionization energy loss distributions</span></a></p> <p><a target="_blank" href="http://www.osti.gov/search">DOE Office of Scientific and Technical Information (OSTI.GOV)</a></p> <p>Sjue, Sky K. L.; George, Jr., Richard Neal; Mathews, David Gregory</p> <p></p> <p>Here, we present a saddlepoint <span class="hlt">approximation</span> for ionization energy loss distributions, valid for arbitrary relativistic velocities of the incident particle 0 < v/c < 1, provided that ionizing collisions are still the dominant energy loss mechanism. We derive a closed form <span class="hlt">solution</span> closely related to Moyal’s distribution. This distribution is intended for use in simulations with relatively low computational overhead. The <span class="hlt">approximation</span> generally reproduces the Vavilov most probable energy loss and full width at half maximum to better than 1% and 10%, respectively, with significantly better agreement as Vavilov’s κ approaches 1.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19820046619&hterms=kernel&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D60%26Ntt%3Dkernel','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19820046619&hterms=kernel&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D60%26Ntt%3Dkernel"><span>An <span class="hlt">accurate</span> and efficient method for evaluating the kernel of the integral equation relating pressure to normalwash in unsteady potential flow</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Desmarais, R. N.</p> <p>1982-01-01</p> <p>This paper describes an <span class="hlt">accurate</span> economical method for generating <span class="hlt">approximations</span> to the kernel of the integral equation relating unsteady pressure to normalwash in nonplanar flow. The method is capable of generating <span class="hlt">approximations</span> of arbitrary accuracy. It is based on <span class="hlt">approximating</span> the algebraic part of the non elementary integrals in the kernel by exponential <span class="hlt">approximations</span> and then integrating termwise. The exponent spacing in the <span class="hlt">approximation</span> is a geometric sequence. The coefficients and exponent multiplier of the exponential <span class="hlt">approximation</span> are computed by least squares so the method is completely automated. Exponential <span class="hlt">approximates</span> generated in this manner are two orders of magnitude more <span class="hlt">accurate</span> than the exponential <span class="hlt">approximation</span> that is currently most often used for this purpose. Coefficients for 8, 12, 24, and 72 term <span class="hlt">approximations</span> are tabulated in the report. Also, since the method is automated, it can be used to generate <span class="hlt">approximations</span> to attain any desired trade-off between accuracy and computing cost.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2010JCAMD..24..591G','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2010JCAMD..24..591G"><span>Towards the comprehensive, rapid, and <span class="hlt">accurate</span> prediction of the favorable tautomeric states of drug-like molecules in aqueous <span class="hlt">solution</span></span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Greenwood, Jeremy R.; Calkins, David; Sullivan, Arron P.; Shelley, John C.</p> <p>2010-06-01</p> <p>Generating the appropriate protonation states of drug-like molecules in <span class="hlt">solution</span> is important for success in both ligand- and structure-based virtual screening. Screening collections of millions of compounds requires a method for determining tautomers and their energies that is sufficiently rapid, <span class="hlt">accurate</span>, and comprehensive. To maximise enrichment, the lowest energy tautomers must be determined from heterogeneous input, without over-enumerating unfavourable states. While computationally expensive, the density functional theory (DFT) method M06-2X/aug-cc-pVTZ(-f) [PB-SCRF] provides <span class="hlt">accurate</span> energies for enumerated model tautomeric systems. The empirical Hammett-Taft methodology can very rapidly extrapolate substituent effects from model systems to drug-like molecules via the relationship between pKT and pKa. Combining the two complementary approaches transforms the tautomer problem from a scientific challenge to one of engineering scale-up, and avoids issues that arise due to the very limited number of measured pKT values, especially for the complicated heterocycles often favoured by medicinal chemists for their novelty and versatility. Several hundreds of pre-calculated tautomer energies and substituent pKa effects are tabulated in databases for use in structural adjustment by the program Epik, which treats tautomers as a subset of the larger problem of the protonation states in aqueous ensembles and their energy penalties. Accuracy and coverage is continually improved and expanded by parameterizing new systems of interest using DFT and experimental data. Recommendations are made for how to best incorporate tautomers in molecular design and virtual screening workflows.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19920021650','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19920021650"><span>Direct Coupling Method for Time-<span class="hlt">Accurate</span> <span class="hlt">Solution</span> of Incompressible Navier-Stokes Equations</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Soh, Woo Y.</p> <p>1992-01-01</p> <p>A noniterative finite difference numerical method is presented for the <span class="hlt">solution</span> of the incompressible Navier-Stokes equations with second order accuracy in time and space. Explicit treatment of convection and diffusion terms and implicit treatment of the pressure gradient give a single pressure Poisson equation when the discretized momentum and continuity equations are combined. A pressure boundary condition is not needed on solid boundaries in the staggered mesh system. The <span class="hlt">solution</span> of the pressure Poisson equation is obtained directly by Gaussian elimination. This method is tested on flow problems in a driven cavity and a curved duct.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/26170553','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/26170553"><span>Fast and <span class="hlt">accurate</span> Monte Carlo modeling of a kilovoltage X-ray therapy unit using a photon-source <span class="hlt">approximation</span> for treatment planning in complex media.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Zeinali-Rafsanjani, B; Mosleh-Shirazi, M A; Faghihi, R; Karbasi, S; Mosalaei, A</p> <p>2015-01-01</p> <p>To <span class="hlt">accurately</span> recompute dose distributions in chest-wall radiotherapy with 120 kVp kilovoltage X-rays, an MCNP4C Monte Carlo model is presented using a fast method that obviates the need to fully model the tube components. To validate the model, half-value layer (HVL), percentage depth doses (PDDs) and beam profiles were measured. Dose measurements were performed for a more complex situation using thermoluminescence dosimeters (TLDs) placed within a Rando phantom. The measured and computed first and second HVLs were 3.8, 10.3 mm Al and 3.8, 10.6 mm Al, respectively. The differences between measured and calculated PDDs and beam profiles in water were within 2 mm/2% for all data points. In the Rando phantom, differences for majority of data points were within 2%. The proposed model offered an <span class="hlt">approximately</span> 9500-fold reduced run time compared to the conventional full simulation. The acceptable agreement, based on international criteria, between the simulations and the measurements validates the accuracy of the model for its use in treatment planning and radiobiological modeling studies of superficial therapies including chest-wall irradiation using kilovoltage beam.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19880031324&hterms=equations+quadratics&qs=N%3D0%26Ntk%3DAll%26Ntx%3Dmode%2Bmatchall%26Ntt%3Dequations%2Bquadratics','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19880031324&hterms=equations+quadratics&qs=N%3D0%26Ntk%3DAll%26Ntx%3Dmode%2Bmatchall%26Ntt%3Dequations%2Bquadratics"><span>Legendre-tau <span class="hlt">approximation</span> for functional differential equations. II - The linear quadratic optimal control problem</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Ito, Kazufumi; Teglas, Russell</p> <p>1987-01-01</p> <p>The numerical scheme based on the Legendre-tau <span class="hlt">approximation</span> is proposed to <span class="hlt">approximate</span> the feedback <span class="hlt">solution</span> to the linear quadratic optimal control problem for hereditary differential systems. The convergence property is established using Trotter ideas. The method yields very good <span class="hlt">approximations</span> at low orders and provides an <span class="hlt">approximation</span> technique for computing closed-loop eigenvalues of the feedback system. A comparison with existing methods (based on averaging and spline <span class="hlt">approximations</span>) is made.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/biblio/22617041-numerical-approximation-framework-stochastic-linear-quadratic-regulator-hilbert-spaces','SCIGOV-STC'); return false;" href="https://www.osti.gov/biblio/22617041-numerical-approximation-framework-stochastic-linear-quadratic-regulator-hilbert-spaces"><span>A Numerical <span class="hlt">Approximation</span> Framework for the Stochastic Linear Quadratic Regulator on Hilbert Spaces</span></a></p> <p><a target="_blank" href="http://www.osti.gov/search">DOE Office of Scientific and Technical Information (OSTI.GOV)</a></p> <p>Levajković, Tijana, E-mail: tijana.levajkovic@uibk.ac.at, E-mail: t.levajkovic@sf.bg.ac.rs; Mena, Hermann, E-mail: hermann.mena@uibk.ac.at; Tuffaha, Amjad, E-mail: atufaha@aus.edu</p> <p></p> <p>We present an <span class="hlt">approximation</span> framework for computing the <span class="hlt">solution</span> of the stochastic linear quadratic control problem on Hilbert spaces. We focus on the finite horizon case and the related differential Riccati equations (DREs). Our <span class="hlt">approximation</span> framework is concerned with the so-called “singular estimate control systems” (Lasiecka in Optimal control problems and Riccati equations for systems with unbounded controls and partially analytic generators: applications to boundary and point control problems, 2004) which model certain coupled systems of parabolic/hyperbolic mixed partial differential equations with boundary or point control. We prove that the <span class="hlt">solutions</span> of the <span class="hlt">approximate</span> finite-dimensional DREs converge to the solutionmore » of the infinite-dimensional DRE. In addition, we prove that the optimal state and control of the <span class="hlt">approximate</span> finite-dimensional problem converge to the optimal state and control of the corresponding infinite-dimensional problem.« less</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2013JAP...113u3707Z','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2013JAP...113u3707Z"><span>Low rank <span class="hlt">approximation</span> method for efficient Green's function calculation of dissipative quantum transport</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Zeng, Lang; He, Yu; Povolotskyi, Michael; Liu, XiaoYan; Klimeck, Gerhard; Kubis, Tillmann</p> <p>2013-06-01</p> <p>In this work, the low rank <span class="hlt">approximation</span> concept is extended to the non-equilibrium Green's function (NEGF) method to achieve a very efficient <span class="hlt">approximated</span> algorithm for coherent and incoherent electron transport. This new method is applied to inelastic transport in various semiconductor nanodevices. Detailed benchmarks with exact NEGF <span class="hlt">solutions</span> show (1) a very good agreement between <span class="hlt">approximated</span> and exact NEGF results, (2) a significant reduction of the required memory, and (3) a large reduction of the computational time (a factor of speed up as high as 150 times is observed). A non-recursive <span class="hlt">solution</span> of the inelastic NEGF transport equations of a 1000 nm long resistor on standard hardware illustrates nicely the capability of this new method.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19850012840','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19850012840"><span>Alternative <span class="hlt">approximation</span> concepts for space frame synthesis</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Lust, R. V.; Schmit, L. A.</p> <p>1985-01-01</p> <p>A structural synthesis methodology for the minimum mass design of 3-dimensionall frame-truss structures under multiple static loading conditions and subject to limits on displacements, rotations, stresses, local buckling, and element cross-sectional dimensions is presented. A variety of <span class="hlt">approximation</span> concept options are employed to yield near optimum designs after no more than 10 structural analyses. Available options include: (A) formulation of the nonlinear mathematcal programming problem in either reciprocal section property (RSP) or cross-sectional dimension (CSD) space; (B) two alternative <span class="hlt">approximate</span> problem structures in each design space; and (C) three distinct assumptions about element end-force variations. Fixed element, design element linking, and temporary constraint deletion features are also included. The <span class="hlt">solution</span> of each <span class="hlt">approximate</span> problem, in either its primal or dual form, is obtained using CONMIN, a feasible directions program. The frame-truss synthesis methodology is implemented in the COMPASS computer program and is used to solve a variety of problems. These problems were chosen so that, in addition to exercising the various <span class="hlt">approximation</span> concepts options, the results could be compared with previously published work.</p> </li> </ol> <div class="pull-right"> <ul class="pagination"> <li><a href="#" onclick='return showDiv("page_1");'>«</a></li> <li><a href="#" onclick='return showDiv("page_17");'>17</a></li> <li><a href="#" onclick='return showDiv("page_18");'>18</a></li> <li class="active"><span>19</span></li> <li><a href="#" onclick='return showDiv("page_20");'>20</a></li> <li><a href="#" onclick='return showDiv("page_21");'>21</a></li> <li><a href="#" onclick='return showDiv("page_25");'>»</a></li> </ul> </div> </div><!-- col-sm-12 --> </div><!-- row --> </div><!-- page_19 --> <div id="page_20" class="hiddenDiv"> <div class="row"> <div class="col-sm-12"> <div class="pull-right"> <ul class="pagination"> <li><a href="#" onclick='return showDiv("page_1");'>«</a></li> <li><a href="#" onclick='return showDiv("page_18");'>18</a></li> <li><a href="#" onclick='return showDiv("page_19");'>19</a></li> <li class="active"><span>20</span></li> <li><a href="#" onclick='return showDiv("page_21");'>21</a></li> <li><a href="#" onclick='return showDiv("page_22");'>22</a></li> <li><a href="#" onclick='return showDiv("page_25");'>»</a></li> </ul> </div> </div> </div> <div class="row"> <div class="col-sm-12"> <ol class="result-class" start="381"> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017PhRvD..95j4054G','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017PhRvD..95j4054G"><span>Deriving analytic <span class="hlt">solutions</span> for compact binary inspirals without recourse to adiabatic <span class="hlt">approximations</span></span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Galley, Chad R.; Rothstein, Ira Z.</p> <p>2017-05-01</p> <p>We utilize the dynamical renormalization group formalism to calculate the real space trajectory of a compact binary inspiral for long times via a systematic resummation of secularly growing terms. This method generates closed form <span class="hlt">solutions</span> without orbit averaging, and the accuracy can be systematically improved. The expansion parameter is v5ν Ω (t -t0) where t0 is the initial time, t is the time elapsed, and Ω and v are the angular orbital frequency and initial speed, respectively. ν is the binary's symmetric mass ratio. We demonstrate how to apply the renormalization group method to resum <span class="hlt">solutions</span> beyond leading order in two ways. First, we calculate the second-order corrections of the leading radiation reaction force, which involves highly nontrivial checks of the formalism (i.e., its renormalizability). Second, we show how to systematically include post-Newtonian corrections to the radiation reaction force. By avoiding orbit averaging, we gain predictive power and eliminate ambiguities in the initial conditions. Finally, we discuss how this methodology can be used to find analytic <span class="hlt">solutions</span> to the spin equations of motion that are valid over long times.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://pubs.er.usgs.gov/publication/70171530','USGSPUBS'); return false;" href="https://pubs.er.usgs.gov/publication/70171530"><span>Annual estimates of water and <span class="hlt">solute</span> export from 42 tributaries to the Yukon River</span></a></p> <p><a target="_blank" href="http://pubs.er.usgs.gov/pubs/index.jsp?view=adv">USGS Publications Warehouse</a></p> <p>Frederick Zanden,; Suzanne P. Anderson,; Striegl, Robert G.</p> <p>2012-01-01</p> <p>Annual export of 11 major and trace <span class="hlt">solutes</span> for the Yukon River is found to be <span class="hlt">accurately</span> determined based on summing 42 tributary contributions. These findings provide the first published estimates of tributary specific distribution of <span class="hlt">solutes</span> within the Yukon River basin. First, we show that annual discharge of the Yukon River can be computed by summing calculated annual discharges from 42 tributaries. Annual discharge for the tributaries is calculated from the basin area and average annual precipitation over that area using a previously published regional regression equation. Based on tributary inputs, we estimate an average annual discharge for the Yukon River of 210 km3 year–1. This value is within 1% of the average measured annual discharge at the U.S. Geological Survey gaging station near the river terminus at Pilot Station, AK, for water years 2001 through 2005. Next, annual loads for 11 <span class="hlt">solutes</span> are determined by combining annual discharge with point measurements of <span class="hlt">solute</span> concentrations in tributary river water. Based on the sum of <span class="hlt">solutes</span> in tributary water, we find that the Yukon River discharges <span class="hlt">approximately</span> 33 million metric tons of dissolved solids each year at Pilot Station. Discharged <span class="hlt">solutes</span> are dominated by cations calcium and magnesium (5.65 × 109 and 1.42 × 109 g year–1) and anions bicarbonate and sulphate (17.3 × 109 and 5.40 × 109 g year–1). These loads compare well with loads calculated independently at the three continuous gaging stations along the Yukon River. These findings show how annual <span class="hlt">solute</span> yields vary throughout a major subarctic river basin and that <span class="hlt">accurate</span> estimates of total river export can be determined from calculated tributary contributions.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19940017095','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19940017095"><span>Rational trigonometric <span class="hlt">approximations</span> using Fourier series partial sums</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Geer, James F.</p> <p>1993-01-01</p> <p>A class of <span class="hlt">approximations</span> (S(sub N,M)) to a periodic function f which uses the ideas of Pade, or rational function, <span class="hlt">approximations</span> based on the Fourier series representation of f, rather than on the Taylor series representation of f, is introduced and studied. Each <span class="hlt">approximation</span> S(sub N,M) is the quotient of a trigonometric polynomial of degree N and a trigonometric polynomial of degree M. The coefficients in these polynomials are determined by requiring that an appropriate number of the Fourier coefficients of S(sub N,M) agree with those of f. Explicit expressions are derived for these coefficients in terms of the Fourier coefficients of f. It is proven that these 'Fourier-Pade' <span class="hlt">approximations</span> converge point-wise to (f(x(exp +))+f(x(exp -)))/2 more rapidly (in some cases by a factor of 1/k(exp 2M)) than the Fourier series partial sums on which they are based. The <span class="hlt">approximations</span> are illustrated by several examples and an application to the <span class="hlt">solution</span> of an initial, boundary value problem for the simple heat equation is presented.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2014CEJPh..12..111D','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2014CEJPh..12..111D"><span>A Jacobi collocation <span class="hlt">approximation</span> for nonlinear coupled viscous Burgers' equation</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Doha, Eid H.; Bhrawy, Ali H.; Abdelkawy, Mohamed A.; Hafez, Ramy M.</p> <p>2014-02-01</p> <p>This article presents a numerical <span class="hlt">approximation</span> of the initial-boundary nonlinear coupled viscous Burgers' equation based on spectral methods. A Jacobi-Gauss-Lobatto collocation (J-GL-C) scheme in combination with the implicit Runge-Kutta-Nyström (IRKN) scheme are employed to obtain highly <span class="hlt">accurate</span> <span class="hlt">approximations</span> to the mentioned problem. This J-GL-C method, based on Jacobi polynomials and Gauss-Lobatto quadrature integration, reduces solving the nonlinear coupled viscous Burgers' equation to a system of nonlinear ordinary differential equation which is far easier to solve. The given examples show, by selecting relatively few J-GL-C points, the accuracy of the <span class="hlt">approximations</span> and the utility of the approach over other analytical or numerical methods. The illustrative examples demonstrate the accuracy, efficiency, and versatility of the proposed algorithm.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://www.ars.usda.gov/research/publications/publication/?seqNo115=321693','TEKTRAN'); return false;" href="http://www.ars.usda.gov/research/publications/publication/?seqNo115=321693"><span><span class="hlt">Approximate</span> furrow infiltration model for time-variable ponding depth</span></a></p> <p><a target="_blank" href="https://www.ars.usda.gov/research/publications/find-a-publication/">USDA-ARS?s Scientific Manuscript database</a></p> <p></p> <p></p> <p>A methodology is proposed for estimating furrow infiltration under time-variable ponding depth conditions. The methodology <span class="hlt">approximates</span> the <span class="hlt">solution</span> to the two-dimensional Richards equation, and is a modification of a procedure that was originally proposed for computing infiltration under constant ...</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=1300360','PMC'); return false;" href="https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=1300360"><span>Conformational equilibria of alkanes in aqueous <span class="hlt">solution</span>: relationship to water structure near hydrophobic <span class="hlt">solutes</span>.</span></a></p> <p><a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pmc">PubMed Central</a></p> <p>Ashbaugh, H S; Garde, S; Hummer, G; Kaler, E W; Paulaitis, M E</p> <p>1999-01-01</p> <p>Conformational free energies of butane, pentane, and hexane in water are calculated from molecular simulations with explicit waters and from a simple molecular theory in which the local hydration structure is estimated based on a proximity <span class="hlt">approximation</span>. This proximity <span class="hlt">approximation</span> uses only the two nearest carbon atoms on the alkane to predict the local water density at a given point in space. Conformational free energies of hydration are subsequently calculated using a free energy perturbation method. Quantitative agreement is found between the free energies obtained from simulations and theory. Moreover, free energy calculations using this proximity <span class="hlt">approximation</span> are <span class="hlt">approximately</span> four orders of magnitude faster than those based on explicit water simulations. Our results demonstrate the accuracy and utility of the proximity <span class="hlt">approximation</span> for predicting water structure as the basis for a quantitative description of n-alkane conformational equilibria in water. In addition, the proximity <span class="hlt">approximation</span> provides a molecular foundation for extending predictions of water structure and hydration thermodynamic properties of simple hydrophobic <span class="hlt">solutes</span> to larger clusters or assemblies of hydrophobic <span class="hlt">solutes</span>. PMID:10423414</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19970041606','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19970041606"><span>Exponential <span class="hlt">Approximations</span> Using Fourier Series Partial Sums</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Banerjee, Nana S.; Geer, James F.</p> <p>1997-01-01</p> <p>The problem of <span class="hlt">accurately</span> reconstructing a piece-wise smooth, 2(pi)-periodic function f and its first few derivatives, given only a truncated Fourier series representation of f, is studied and solved. The reconstruction process is divided into two steps. In the first step, the first 2N + 1 Fourier coefficients of f are used to <span class="hlt">approximate</span> the locations and magnitudes of the discontinuities in f and its first M derivatives. This is accomplished by first finding initial estimates of these quantities based on certain properties of Gibbs phenomenon, and then refining these estimates by fitting the asymptotic form of the Fourier coefficients to the given coefficients using a least-squares approach. It is conjectured that the locations of the singularities are <span class="hlt">approximated</span> to within O(N(sup -M-2), and the associated jump of the k(sup th) derivative of f is <span class="hlt">approximated</span> to within O(N(sup -M-l+k), as N approaches infinity, and the method is robust. These estimates are then used with a class of singular basis functions, which have certain 'built-in' singularities, to construct a new sequence of <span class="hlt">approximations</span> to f. Each of these new <span class="hlt">approximations</span> is the sum of a piecewise smooth function and a new Fourier series partial sum. When N is proportional to M, it is shown that these new <span class="hlt">approximations</span>, and their derivatives, converge exponentially in the maximum norm to f, and its corresponding derivatives, except in the union of a finite number of small open intervals containing the points of singularity of f. The total measure of these intervals decreases exponentially to zero as M approaches infinity. The technique is illustrated with several examples.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/16854002','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/16854002"><span>Experimental determination of water activity for binary aqueous cerium(III) ionic <span class="hlt">solutions</span>: application to an assessment of the predictive capability of the binding mean spherical <span class="hlt">approximation</span> model.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Ruas, Alexandre; Simonin, Jean-Pierre; Turq, Pierre; Moisy, Philippe</p> <p>2005-12-08</p> <p>This work is aimed at a description of the thermodynamic properties of actinide salt <span class="hlt">solutions</span> at high concentration. The predictive capability of the binding mean spherical <span class="hlt">approximation</span> (BIMSA) theory to describe the thermodynamic properties of electrolytes is assessed in the case of aqueous <span class="hlt">solutions</span> of lanthanide(III) nitrate and chloride salts. Osmotic coefficients of cerium(III) nitrate and chloride were calculated from other lanthanide(III) salts properties. In parallel, concentrated binary <span class="hlt">solutions</span> of cerium nitrate were prepared in order to measure experimentally its water activity and density as a function of concentration, at 25 degrees C. Water activities of several binary <span class="hlt">solutions</span> of cerium chloride were also measured to check existing data on this salt. Then, the properties of cerium chloride and cerium nitrate <span class="hlt">solutions</span> were compared within the BIMSA model. Osmotic coefficient values for promethium nitrate and promethium chloride given by this theory are proposed. Finally, water activity measurements were made to examine the fact that the ternary system Ce(NO3)3/HNO3/H2O and the quaternary system Ce(NO3)3/HNO3/N2H5NO3/H2O may be regarded as "simple <span class="hlt">solutions</span>" (in the sense of Zdanovskii and Mikulin).</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/26991943','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/26991943"><span><span class="hlt">Solute</span> Migration from the Aquifer Matrix into a <span class="hlt">Solution</span> Conduit and the Reverse.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Li, Guangquan; Field, Malcolm S</p> <p>2016-09-01</p> <p>A <span class="hlt">solution</span> conduit has a permeable wall allowing for water exchange and <span class="hlt">solute</span> transfer between the conduit and its surrounding aquifer matrix. In this paper, we use Laplace Transform to solve a one-dimensional equation constructed using the Euler approach to describe advective transport of <span class="hlt">solute</span> in a conduit, a production-value problem. Both nonuniform cross-section of the conduit and nonuniform seepage at the conduit wall are considered in the <span class="hlt">solution</span>. Physical analysis using the Lagrangian approach and a lumping method is performed to verify the <span class="hlt">solution</span>. Two-way transfer between conduit water and matrix water is also investigated by using the <span class="hlt">solution</span> for the production-value problem as a first-order <span class="hlt">approximation</span>. The <span class="hlt">approximate</span> <span class="hlt">solution</span> agrees well with the exact <span class="hlt">solution</span> if dimensionless travel time in the conduit is an order of magnitude smaller than unity. Our analytical <span class="hlt">solution</span> is based on the assumption that the spatial and/or temporal heterogeneity in the wall <span class="hlt">solute</span> flux is the dominant factor in the spreading of spring-breakthrough curves, and conduit dispersion is only a secondary mechanism. Such an approach can lead to the better understanding of water exchange and <span class="hlt">solute</span> transfer between conduits and aquifer matrix. Euler and Lagrangian approaches are used to solve transport in conduit. Two-way transfer between conduit and matrix is investigated. The <span class="hlt">solution</span> is applicable to transport in conduit of persisting <span class="hlt">solute</span> from matrix. © 2016, National Ground Water Association.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19870020664','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19870020664"><span>UNAERO: A package of FORTRAN subroutines for <span class="hlt">approximating</span> unsteady aerodynamics in the time domain</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Dunn, H. J.</p> <p>1985-01-01</p> <p>This report serves as an instruction and maintenance manual for a collection of CDC CYBER FORTRAN IV subroutines for <span class="hlt">approximating</span> the unsteady aerodynamic forces in the time domain. The result is a set of constant-coefficient first-order differential equations that <span class="hlt">approximate</span> the dynamics of the vehicle. Provisions are included for adjusting the number of modes used for calculating the <span class="hlt">approximations</span> so that an <span class="hlt">accurate</span> <span class="hlt">approximation</span> is generated. The number of data points at different values of reduced frequency can also be varied to adjust the accuracy of the <span class="hlt">approximation</span> over the reduced-frequency range. The denominator coefficients of the <span class="hlt">approximation</span> may be calculated by means of a gradient method or a least-squares <span class="hlt">approximation</span> technique. Both the <span class="hlt">approximation</span> methods use weights on the residual error. A new set of system equations, at a different dynamic pressure, can be generated without the <span class="hlt">approximations</span> being recalculated.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/29055349','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/29055349"><span>A finite state projection algorithm for the stationary <span class="hlt">solution</span> of the chemical master equation.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Gupta, Ankit; Mikelson, Jan; Khammash, Mustafa</p> <p>2017-10-21</p> <p>The chemical master equation (CME) is frequently used in systems biology to quantify the effects of stochastic fluctuations that arise due to biomolecular species with low copy numbers. The CME is a system of ordinary differential equations that describes the evolution of probability density for each population vector in the state-space of the stochastic reaction dynamics. For many examples of interest, this state-space is infinite, making it difficult to obtain exact <span class="hlt">solutions</span> of the CME. To deal with this problem, the Finite State Projection (FSP) algorithm was developed by Munsky and Khammash [J. Chem. Phys. 124(4), 044104 (2006)], to provide <span class="hlt">approximate</span> <span class="hlt">solutions</span> to the CME by truncating the state-space. The FSP works well for finite time-periods but it cannot be used for estimating the stationary <span class="hlt">solutions</span> of CMEs, which are often of interest in systems biology. The aim of this paper is to develop a version of FSP which we refer to as the stationary FSP (sFSP) that allows one to obtain <span class="hlt">accurate</span> <span class="hlt">approximations</span> of the stationary <span class="hlt">solutions</span> of a CME by solving a finite linear-algebraic system that yields the stationary distribution of a continuous-time Markov chain over the truncated state-space. We derive bounds for the <span class="hlt">approximation</span> error incurred by sFSP and we establish that under certain stability conditions, these errors can be made arbitrarily small by appropriately expanding the truncated state-space. We provide several examples to illustrate our sFSP method and demonstrate its efficiency in estimating the stationary distributions. In particular, we show that using a quantized tensor-train implementation of our sFSP method, problems admitting more than 100 × 10 6 states can be efficiently solved.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017JChPh.147o4101G','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017JChPh.147o4101G"><span>A finite state projection algorithm for the stationary <span class="hlt">solution</span> of the chemical master equation</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Gupta, Ankit; Mikelson, Jan; Khammash, Mustafa</p> <p>2017-10-01</p> <p>The chemical master equation (CME) is frequently used in systems biology to quantify the effects of stochastic fluctuations that arise due to biomolecular species with low copy numbers. The CME is a system of ordinary differential equations that describes the evolution of probability density for each population vector in the state-space of the stochastic reaction dynamics. For many examples of interest, this state-space is infinite, making it difficult to obtain exact <span class="hlt">solutions</span> of the CME. To deal with this problem, the Finite State Projection (FSP) algorithm was developed by Munsky and Khammash [J. Chem. Phys. 124(4), 044104 (2006)], to provide <span class="hlt">approximate</span> <span class="hlt">solutions</span> to the CME by truncating the state-space. The FSP works well for finite time-periods but it cannot be used for estimating the stationary <span class="hlt">solutions</span> of CMEs, which are often of interest in systems biology. The aim of this paper is to develop a version of FSP which we refer to as the stationary FSP (sFSP) that allows one to obtain <span class="hlt">accurate</span> <span class="hlt">approximations</span> of the stationary <span class="hlt">solutions</span> of a CME by solving a finite linear-algebraic system that yields the stationary distribution of a continuous-time Markov chain over the truncated state-space. We derive bounds for the <span class="hlt">approximation</span> error incurred by sFSP and we establish that under certain stability conditions, these errors can be made arbitrarily small by appropriately expanding the truncated state-space. We provide several examples to illustrate our sFSP method and demonstrate its efficiency in estimating the stationary distributions. In particular, we show that using a quantized tensor-train implementation of our sFSP method, problems admitting more than 100 × 106 states can be efficiently solved.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.cs.cornell.edu/~yexiang/publications/Xiaojian-Wu-et-al-aaai-2018-final.pdf','USGSPUBS'); return false;" href="https://www.cs.cornell.edu/~yexiang/publications/Xiaojian-Wu-et-al-aaai-2018-final.pdf"><span>Efficiently <span class="hlt">approximating</span> the Pareto frontier: Hydropower dam placement in the Amazon basin</span></a></p> <p><a target="_blank" href="http://pubs.er.usgs.gov/pubs/index.jsp?view=adv">USGS Publications Warehouse</a></p> <p>Wu, Xiaojian; Gomes-Selman, Jonathan; Shi, Qinru; Xue, Yexiang; Garcia-Villacorta, Roosevelt; Anderson, Elizabeth; Sethi, Suresh; Steinschneider, Scott; Flecker, Alexander; Gomes, Carla P.</p> <p>2018-01-01</p> <p>Real–world problems are often not fully characterized by a single optimal <span class="hlt">solution</span>, as they frequently involve multiple competing objectives; it is therefore important to identify the so-called Pareto frontier, which captures <span class="hlt">solution</span> trade-offs. We propose a fully polynomial-time <span class="hlt">approximation</span> scheme based on Dynamic Programming (DP) for computing a polynomially succinct curve that <span class="hlt">approximates</span> the Pareto frontier to within an arbitrarily small > 0 on treestructured networks. Given a set of objectives, our <span class="hlt">approximation</span> scheme runs in time polynomial in the size of the instance and 1/. We also propose a Mixed Integer Programming (MIP) scheme to <span class="hlt">approximate</span> the Pareto frontier. The DP and MIP Pareto frontier approaches have complementary strengths and are surprisingly effective. We provide empirical results showing that our methods outperform other approaches in efficiency and accuracy. Our work is motivated by a problem in computational sustainability concerning the proliferation of hydropower dams throughout the Amazon basin. Our goal is to support decision-makers in evaluating impacted ecosystem services on the full scale of the Amazon basin. Our work is general and can be applied to <span class="hlt">approximate</span> the Pareto frontier of a variety of multiobjective problems on tree-structured networks.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19910027253&hterms=ito&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D80%26Ntt%3Dito','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19910027253&hterms=ito&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D80%26Ntt%3Dito"><span>Finite-dimensional compensators for infinite-dimensional systems via Galerkin-type <span class="hlt">approximation</span></span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Ito, Kazufumi</p> <p>1990-01-01</p> <p>In this paper existence and construction of stabilizing compensators for linear time-invariant systems defined on Hilbert spaces are discussed. An existence result is established using Galkerin-type <span class="hlt">approximations</span> in which independent basis elements are used instead of the complete set of eigenvectors. A design procedure based on <span class="hlt">approximate</span> <span class="hlt">solutions</span> of the optimal regulator and optimal observer via Galerkin-type <span class="hlt">approximation</span> is given and the Schumacher approach is used to reduce the dimension of compensators. A detailed discussion for parabolic and hereditary differential systems is included.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/20140008919','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/20140008919"><span>Trajectory Optimization Using Adjoint Method and Chebyshev Polynomial <span class="hlt">Approximation</span> for Minimizing Fuel Consumption During Climb</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Nguyen, Nhan T.; Hornby, Gregory; Ishihara, Abe</p> <p>2013-01-01</p> <p>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 <span class="hlt">approximation</span> and cubic spine <span class="hlt">approximation</span>. The <span class="hlt">approximate</span> optimal trajectory will be compared with the adjoint-based optimal trajectory which is considered as the true optimal <span class="hlt">solution</span> of the trajectory optimization problem. The adjoint-based optimization problem leads to a singular optimal control <span class="hlt">solution</span> which results in a bang-singular-bang optimal control.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/servlets/purl/1427020','SCIGOV-STC'); return false;" href="https://www.osti.gov/servlets/purl/1427020"><span>An Investigation into <span class="hlt">Solution</span> Verification for CFD-DEM</span></a></p> <p><a target="_blank" href="http://www.osti.gov/search">DOE Office of Scientific and Technical Information (OSTI.GOV)</a></p> <p>Fullmer, William D.; Musser, Jordan</p> <p></p> <p> different randomized particle configurations of the same general problem (for the fictitious case) or different instances of freezing a transient simulation, the numerical uncertainties appeared to be on the same order of magnitude as ensemble or time averaging uncertainties. By testing different drag laws, almost all cases studied show that model form uncertainty in this one, very important closure relation was larger than the numerical uncertainty, at least with a reasonable CFD grid, roughly five particle diameters. In this study, the diffusion width (filtering length scale) was mostly set at a constant of six particle diameters. A few exploratory tests were performed to show that similar convergence behavior was observed for diffusion widths greater than <span class="hlt">approximately</span> two particle diameters. However, this subject was not investigated in great detail because determining an appropriate filter size is really a validation question which must be determined by comparison to experimental or highly <span class="hlt">accurate</span> numerical data. Future studies are being considered targeting <span class="hlt">solution</span> verification of transient simulations as well as validation of the filter size with direct numerical simulation data.« less</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://www.ars.usda.gov/research/publications/publication/?seqNo115=316103','TEKTRAN'); return false;" href="http://www.ars.usda.gov/research/publications/publication/?seqNo115=316103"><span>Closure to new results for an <span class="hlt">approximate</span> method for calculating two-dimensional furrow infiltration</span></a></p> <p><a target="_blank" href="https://www.ars.usda.gov/research/publications/find-a-publication/">USDA-ARS?s Scientific Manuscript database</a></p> <p></p> <p></p> <p>In a discussion paper, Ebrahimian and Noury (2015) raised several concerns about an <span class="hlt">approximate</span> <span class="hlt">solution</span> to the two-dimensional Richards equation presented by Bautista et al (2014). The <span class="hlt">solution</span> is based on a procedure originally proposed by Warrick et al. (2007). Such a <span class="hlt">solution</span> is of practical i...</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/biblio/22255061-double-hybrid-density-functional-theory-meta-generalized-gradient-approximations','SCIGOV-STC'); return false;" href="https://www.osti.gov/biblio/22255061-double-hybrid-density-functional-theory-meta-generalized-gradient-approximations"><span>Double-hybrid density-functional theory with meta-generalized-gradient <span class="hlt">approximations</span></span></a></p> <p><a target="_blank" href="http://www.osti.gov/search">DOE Office of Scientific and Technical Information (OSTI.GOV)</a></p> <p>Souvi, Sidi M. O., E-mail: sidi.souvi@irsn.fr; Sharkas, Kamal; Toulouse, Julien, E-mail: julien.toulouse@upmc.fr</p> <p>2014-02-28</p> <p>We extend the previously proposed one-parameter double-hybrid density-functional theory [K. Sharkas, J. Toulouse, and A. Savin, J. Chem. Phys. 134, 064113 (2011)] to meta-generalized-gradient-<span class="hlt">approximation</span> (meta-GGA) exchange-correlation density functionals. We construct several variants of one-parameter double-hybrid <span class="hlt">approximations</span> using the Tao-Perdew-Staroverov-Scuseria (TPSS) meta-GGA functional and test them on test sets of atomization energies and reaction barrier heights. The most <span class="hlt">accurate</span> variant uses the uniform coordinate scaling of the density and of the kinetic energy density in the correlation functional, and improves over both standard Kohn-Sham TPSS and second-order Møller-Plesset calculations.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/20060051747','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/20060051747"><span>Comparison of the Radiative Two-Flux and Diffusion <span class="hlt">Approximations</span></span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Spuckler, Charles M.</p> <p>2006-01-01</p> <p><span class="hlt">Approximate</span> <span class="hlt">solutions</span> are sometimes used to determine the heat transfer and temperatures in a semitransparent material in which conduction and thermal radiation are acting. A comparison of the Milne-Eddington two-flux <span class="hlt">approximation</span> and the diffusion <span class="hlt">approximation</span> for combined conduction and radiation heat transfer in a ceramic material was preformed to determine the accuracy of the diffusion <span class="hlt">solution</span>. A plane gray semitransparent layer without a substrate and a non-gray semitransparent plane layer on an opaque substrate were considered. For the plane gray layer the material is semitransparent for all wavelengths and the scattering and absorption coefficients do not vary with wavelength. For the non-gray plane layer the material is semitransparent with constant absorption and scattering coefficients up to a specified wavelength. At higher wavelengths the non-gray plane layer is assumed to be opaque. The layers are heated on one side and cooled on the other by diffuse radiation and convection. The scattering and absorption coefficients were varied. The error in the diffusion <span class="hlt">approximation</span> compared to the Milne-Eddington two flux <span class="hlt">approximation</span> was obtained as a function of scattering coefficient and absorption coefficient. The percent difference in interface temperatures and heat flux through the layer obtained using the Milne-Eddington two-flux and diffusion <span class="hlt">approximations</span> are presented as a function of scattering coefficient and absorption coefficient. The largest errors occur for high scattering and low absorption except for the back surface temperature of the plane gray layer where the error is also larger at low scattering and low absorption. It is shown that the accuracy of the diffusion <span class="hlt">approximation</span> can be improved for some scattering and absorption conditions if a reflectance obtained from a Kubelka-Munk type two flux theory is used instead of a reflection obtained from the Fresnel equation. The Kubelka-Munk reflectance accounts for surface reflection and</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2011JCoPh.230.4811A','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2011JCoPh.230.4811A"><span>Data mining techniques for scientific computing: Application to asymptotic paraxial <span class="hlt">approximations</span> to model ultrarelativistic particles</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Assous, Franck; Chaskalovic, Joël</p> <p>2011-06-01</p> <p>We propose a new approach that consists in using data mining techniques for scientific computing. Indeed, data mining has proved to be efficient in other contexts which deal with huge data like in biology, medicine, marketing, advertising and communications. Our aim, here, is to deal with the important problem of the exploitation of the results produced by any numerical method. Indeed, more and more data are created today by numerical simulations. Thus, it seems necessary to look at efficient tools to analyze them. In this work, we focus our presentation to a test case dedicated to an asymptotic paraxial <span class="hlt">approximation</span> to model ultrarelativistic particles. Our method directly deals with numerical results of simulations and try to understand what each order of the asymptotic expansion brings to the simulation results over what could be obtained by other lower-order or less <span class="hlt">accurate</span> means. This new heuristic approach offers new potential applications to treat numerical <span class="hlt">solutions</span> to mathematical models.</p> </li> </ol> <div class="pull-right"> <ul class="pagination"> <li><a href="#" onclick='return showDiv("page_1");'>«</a></li> <li><a href="#" onclick='return showDiv("page_18");'>18</a></li> <li><a href="#" onclick='return showDiv("page_19");'>19</a></li> <li class="active"><span>20</span></li> <li><a href="#" onclick='return showDiv("page_21");'>21</a></li> <li><a href="#" onclick='return showDiv("page_22");'>22</a></li> <li><a href="#" onclick='return showDiv("page_25");'>»</a></li> </ul> </div> </div><!-- col-sm-12 --> </div><!-- row --> </div><!-- page_20 --> <div id="page_21" class="hiddenDiv"> <div class="row"> <div class="col-sm-12"> <div class="pull-right"> <ul class="pagination"> <li><a href="#" onclick='return showDiv("page_1");'>«</a></li> <li><a href="#" onclick='return showDiv("page_19");'>19</a></li> <li><a href="#" onclick='return showDiv("page_20");'>20</a></li> <li class="active"><span>21</span></li> <li><a href="#" onclick='return showDiv("page_22");'>22</a></li> <li><a href="#" onclick='return showDiv("page_23");'>23</a></li> <li><a href="#" onclick='return showDiv("page_25");'>»</a></li> </ul> </div> </div> </div> <div class="row"> <div class="col-sm-12"> <ol class="result-class" start="401"> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/17316725','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/17316725"><span>Probability distribution of haplotype frequencies under the two-locus Wright-Fisher model by diffusion <span class="hlt">approximation</span>.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Boitard, Simon; Loisel, Patrice</p> <p>2007-05-01</p> <p>The probability distribution of haplotype frequencies in a population, and the way it is influenced by genetical forces such as recombination, selection, random drift ...is a question of fundamental interest in population genetics. For large populations, the distribution of haplotype frequencies for two linked loci under the classical Wright-Fisher model is almost impossible to compute because of numerical reasons. However the Wright-Fisher process can in such cases be <span class="hlt">approximated</span> by a diffusion process and the transition density can then be deduced from the Kolmogorov equations. As no exact <span class="hlt">solution</span> has been found for these equations, we developed a numerical method based on finite differences to solve them. It applies to transient states and models including selection or mutations. We show by several tests that this method is <span class="hlt">accurate</span> for computing the conditional joint density of haplotype frequencies given that no haplotype has been lost. We also prove that it is far less time consuming than other methods such as Monte Carlo simulations.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/25916918','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/25916918"><span>Fast and <span class="hlt">Accurate</span> Circuit Design Automation through Hierarchical Model Switching.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Huynh, Linh; Tagkopoulos, Ilias</p> <p>2015-08-21</p> <p>In computer-aided biological design, the trifecta of characterized part libraries, <span class="hlt">accurate</span> models and optimal design parameters is crucial for producing reliable designs. As the number of parts and model complexity increase, however, it becomes exponentially more difficult for any optimization method to search the <span class="hlt">solution</span> space, hence creating a trade-off that hampers efficient design. To address this issue, we present a hierarchical computer-aided design architecture that uses a two-step approach for biological design. First, a simple model of low computational complexity is used to predict circuit behavior and assess candidate circuit branches through branch-and-bound methods. Then, a complex, nonlinear circuit model is used for a fine-grained search of the reduced <span class="hlt">solution</span> space, thus achieving more <span class="hlt">accurate</span> results. Evaluation with a benchmark of 11 circuits and a library of 102 experimental designs with known characterization parameters demonstrates a speed-up of 3 orders of magnitude when compared to other design methods that provide optimality guarantees.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19840022749','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19840022749"><span>Legendre-tau <span class="hlt">approximation</span> for functional differential equations. Part 2: The linear quadratic optimal control problem</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Ito, K.; Teglas, R.</p> <p>1984-01-01</p> <p>The numerical scheme based on the Legendre-tau <span class="hlt">approximation</span> is proposed to <span class="hlt">approximate</span> the feedback <span class="hlt">solution</span> to the linear quadratic optimal control problem for hereditary differential systems. The convergence property is established using Trotter ideas. The method yields very good <span class="hlt">approximations</span> at low orders and provides an <span class="hlt">approximation</span> technique for computing closed-loop eigenvalues of the feedback system. A comparison with existing methods (based on averaging and spline <span class="hlt">approximations</span>) is made.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/25637925','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/25637925"><span><span class="hlt">Approximating</span> high-dimensional dynamics by barycentric coordinates with linear programming.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Hirata, Yoshito; Shiro, Masanori; Takahashi, Nozomu; Aihara, Kazuyuki; Suzuki, Hideyuki; Mas, Paloma</p> <p>2015-01-01</p> <p>The increasing development of novel methods and techniques facilitates the measurement of high-dimensional time series but challenges our ability for <span class="hlt">accurate</span> modeling and predictions. The use of a general mathematical model requires the inclusion of many parameters, which are difficult to be fitted for relatively short high-dimensional time series observed. Here, we propose a novel method to <span class="hlt">accurately</span> model a high-dimensional time series. Our method extends the barycentric coordinates to high-dimensional phase space by employing linear programming, and allowing the <span class="hlt">approximation</span> errors explicitly. The extension helps to produce free-running time-series predictions that preserve typical topological, dynamical, and/or geometric characteristics of the underlying attractors more <span class="hlt">accurately</span> than the radial basis function model that is widely used. The method can be broadly applied, from helping to improve weather forecasting, to creating electronic instruments that sound more natural, and to comprehensively understanding complex biological data.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/28861954','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/28861954"><span><span class="hlt">Accurate</span> viscosity measurements of flowing aqueous glucose <span class="hlt">solutions</span> with suspended scatterers using a dynamic light scattering approach with optical coherence tomography.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Weatherbee, Andrew; Popov, Ivan; Vitkin, Alex</p> <p>2017-08-01</p> <p>The viscosity of turbid colloidal glucose <span class="hlt">solutions</span> has been <span class="hlt">accurately</span> determined from spectral domain optical coherence tomography (OCT) M-mode measurements and our recently developed OCT dynamic light scattering model. Results for various glucose concentrations, flow speeds, and flow angles are reported. The relative "combined standard uncertainty" uc(η) on the viscosity measurements was ±1% for the no-flow case and ±5% for the flow cases, a significant improvement in measurement robustness over previously published reports. The available literature data for the viscosity of pure water and our measurements differ by 1% (stagnant case) and 1.5% (flow cases), demonstrating good accuracy; similar agreement is seen across the measured glucose concentration range when compared to interpolated literature values. The developed technique may contribute toward eventual noninvasive glucose measurements in medicine. (2017) COPYRIGHT Society of Photo-Optical Instrumentation Engineers (SPIE).</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19950013594','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19950013594"><span>Finite dimensional <span class="hlt">approximation</span> of a class of constrained nonlinear optimal control problems</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Gunzburger, Max D.; Hou, L. S.</p> <p>1994-01-01</p> <p>An abstract framework for the analysis and <span class="hlt">approximation</span> of a class of nonlinear optimal control and optimization problems is constructed. Nonlinearities occur in both the objective functional and in the constraints. The framework includes an abstract nonlinear optimization problem posed on infinite dimensional spaces, and <span class="hlt">approximate</span> problem posed on finite dimensional spaces, together with a number of hypotheses concerning the two problems. The framework is used to show that optimal <span class="hlt">solutions</span> exist, to show that Lagrange multipliers may be used to enforce the constraints, to derive an optimality system from which optimal states and controls may be deduced, and to derive existence results and error estimates for <span class="hlt">solutions</span> of the <span class="hlt">approximate</span> problem. The abstract framework and the results derived from that framework are then applied to three concrete control or optimization problems and their <span class="hlt">approximation</span> by finite element methods. The first involves the von Karman plate equations of nonlinear elasticity, the second, the Ginzburg-Landau equations of superconductivity, and the third, the Navier-Stokes equations for incompressible, viscous flows.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19760012070','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19760012070"><span>The Investigation of Optimal Discrete <span class="hlt">Approximations</span> for Real Time Flight Simulations</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Parrish, E. A.; Mcvey, E. S.; Cook, G.; Henderson, K. C.</p> <p>1976-01-01</p> <p>The results are presented of an investigation of discrete <span class="hlt">approximations</span> for real time flight simulation. Major topics discussed include: (1) consideration of the particular problem of <span class="hlt">approximation</span> of continuous autopilots by digital autopilots; (2) use of Bode plots and synthesis of transfer functions by asymptotic fits in a warped frequency domain; (3) an investigation of the various substitution formulas, including the effects of nonlinearities; (4) use of pade <span class="hlt">approximation</span> to the <span class="hlt">solution</span> of the matrix exponential arising from the discrete state equations; and (5) an analytical integration of the state equation using interpolated input.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2018AJ....155..253S','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2018AJ....155..253S"><span>A Fast and <span class="hlt">Accurate</span> Method of Radiation Hydrodynamics Calculation in Spherical Symmetry</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Stamer, Torsten; Inutsuka, Shu-ichiro</p> <p>2018-06-01</p> <p>We develop a new numerical scheme for solving the radiative transfer equation in a spherically symmetric system. This scheme does not rely on any kind of diffusion <span class="hlt">approximation</span>, and it is <span class="hlt">accurate</span> for optically thin, thick, and intermediate systems. In the limit of a homogeneously distributed extinction coefficient, our method is very <span class="hlt">accurate</span> and exceptionally fast. We combine this fast method with a slower but more generally applicable method to describe realistic problems. We perform various test calculations, including a simplified protostellar collapse simulation. We also discuss possible future improvements.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/26597159','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/26597159"><span>The relationship between stochastic and deterministic quasi-steady state <span class="hlt">approximations</span>.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Kim, Jae Kyoung; Josić, Krešimir; Bennett, Matthew R</p> <p>2015-11-23</p> <p>The quasi steady-state <span class="hlt">approximation</span> (QSSA) is frequently used to reduce deterministic models of biochemical networks. The resulting equations provide a simplified description of the network in terms of non-elementary reaction functions (e.g. Hill functions). Such deterministic reductions are frequently a basis for heuristic stochastic models in which non-elementary reaction functions are used to define reaction propensities. Despite their popularity, it remains unclear when such stochastic reductions are valid. It is frequently assumed that the stochastic reduction can be trusted whenever its deterministic counterpart is <span class="hlt">accurate</span>. However, a number of recent examples show that this is not necessarily the case. Here we explain the origin of these discrepancies, and demonstrate a clear relationship between the accuracy of the deterministic and the stochastic QSSA for examples widely used in biological systems. With an analysis of a two-state promoter model, and numerical simulations for a variety of other models, we find that the stochastic QSSA is <span class="hlt">accurate</span> whenever its deterministic counterpart provides an <span class="hlt">accurate</span> <span class="hlt">approximation</span> over a range of initial conditions which cover the likely fluctuations from the quasi steady-state (QSS). We conjecture that this relationship provides a simple and computationally inexpensive way to test the accuracy of reduced stochastic models using deterministic simulations. The stochastic QSSA is one of the most popular multi-scale stochastic simulation methods. While the use of QSSA, and the resulting non-elementary functions has been justified in the deterministic case, it is not clear when their stochastic counterparts are <span class="hlt">accurate</span>. In this study, we show how the accuracy of the stochastic QSSA can be tested using their deterministic counterparts providing a concrete method to test when non-elementary rate functions can be used in stochastic simulations.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/biblio/21067438-bloch-approximation-periodically-perforated-media','SCIGOV-STC'); return false;" href="https://www.osti.gov/biblio/21067438-bloch-approximation-periodically-perforated-media"><span>The Bloch <span class="hlt">Approximation</span> in Periodically Perforated Media</span></a></p> <p><a target="_blank" href="http://www.osti.gov/search">DOE Office of Scientific and Technical Information (OSTI.GOV)</a></p> <p>Conca, C.; Gomez, D., E-mail: gomezdel@unican.es; Lobo, M.</p> <p>2005-06-15</p> <p>We consider a periodically heterogeneous and perforated medium filling an open domain {omega} of R{sup N}. Assuming that the size of the periodicity of the structure and of the holes is O({epsilon}),we study the asymptotic behavior, as {epsilon} {sup {yields}} 0, of the <span class="hlt">solution</span> of an elliptic boundary value problem with strongly oscillating coefficients posed in {omega}{sup {epsilon}}({omega}{sup {epsilon}} being {omega} minus the holes) with a Neumann condition on the boundary of the holes. We use Bloch wave decomposition to introduce an <span class="hlt">approximation</span> of the <span class="hlt">solution</span> in the energy norm which can be computed from the homogenized <span class="hlt">solution</span> and themore » first Bloch eigenfunction. We first consider the case where {omega}is R{sup N} and then localize the problem for abounded domain {omega}, considering a homogeneous Dirichlet condition on the boundary of {omega}.« less</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2009TMP...159..853N','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2009TMP...159..853N"><span>Padé <span class="hlt">approximations</span> for Painlevé I and II transcendents</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Novokshenov, V. Yu.</p> <p>2009-06-01</p> <p>We use a version of the Fair-Luke algorithm to find the Padé <span class="hlt">approximate</span> <span class="hlt">solutions</span> of the Painlevé I and II equations. We find the distributions of poles for the well-known Ablowitz-Segur and Hastings-McLeod <span class="hlt">solutions</span> of the Painlevé II equation. We show that the Boutroux tritronquée <span class="hlt">solution</span> of the Painleé I equation has poles only in the critical sector of the complex plane. The algorithm allows checking other analytic properties of the Painlevé transcendents, such as the asymptotic behavior at infinity in the complex plane.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19790049964&hterms=Poisson&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D70%26Ntt%3DPoisson','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19790049964&hterms=Poisson&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D70%26Ntt%3DPoisson"><span>The <span class="hlt">accurate</span> <span class="hlt">solution</span> of Poisson's equation by expansion in Chebyshev polynomials</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Haidvogel, D. B.; Zang, T.</p> <p>1979-01-01</p> <p>A Chebyshev expansion technique is applied to Poisson's equation on a square with homogeneous Dirichlet boundary conditions. The spectral equations are solved in two ways - by alternating direction and by matrix diagonalization methods. <span class="hlt">Solutions</span> are sought to both oscillatory and mildly singular problems. The accuracy and efficiency of the Chebyshev approach compare favorably with those of standard second- and fourth-order finite-difference methods.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/28654275','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/28654275"><span><span class="hlt">Accurate</span> Quasiparticle Spectra from the T-Matrix Self-Energy and the Particle-Particle Random Phase <span class="hlt">Approximation</span>.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Zhang, Du; Su, Neil Qiang; Yang, Weitao</p> <p>2017-07-20</p> <p>The GW self-energy, especially G 0 W 0 based on the particle-hole random phase <span class="hlt">approximation</span> (phRPA), is widely used to study quasiparticle (QP) energies. Motivated by the desirable features of the particle-particle (pp) RPA compared to the conventional phRPA, we explore the pp counterpart of GW, that is, the T-matrix self-energy, formulated with the eigenvectors and eigenvalues of the ppRPA matrix. We demonstrate the accuracy of the T-matrix method for molecular QP energies, highlighting the importance of the pp channel for calculating QP spectra.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017EntIS..11..434Z','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017EntIS..11..434Z"><span>Identification of <span class="hlt">approximately</span> duplicate material records in ERP systems</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Zong, Wei; Wu, Feng; Chu, Lap-Keung; Sculli, Domenic</p> <p>2017-03-01</p> <p>The quality of master data is crucial for the <span class="hlt">accurate</span> functioning of the various modules of an enterprise resource planning (ERP) system. This study addresses specific data problems arising from the generation of <span class="hlt">approximately</span> duplicate material records in ERP databases. Such problems are mainly due to the firm's lack of unique and global identifiers for the material records, and to the arbitrary assignment of alternative names for the same material by various users. Traditional duplicate detection methods are ineffective in identifying such <span class="hlt">approximately</span> duplicate material records because these methods typically rely on string comparisons of each field. To address this problem, a machine learning-based framework is developed to recognise semantic similarity between strings and to further identify and reunify <span class="hlt">approximately</span> duplicate material records - a process referred to as de-duplication in this article. First, the keywords of the material records are extracted to form vectors of discriminating words. Second, a machine learning method using a probabilistic neural network is applied to determine the semantic similarity between these material records. The approach was evaluated using data from a real case study. The test results indicate that the proposed method outperforms traditional algorithms in identifying <span class="hlt">approximately</span> duplicate material records.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/pages/biblio/1407456-incomplete-sparse-approximate-inverses-parallel-preconditioning','SCIGOV-DOEP'); return false;" href="https://www.osti.gov/pages/biblio/1407456-incomplete-sparse-approximate-inverses-parallel-preconditioning"><span>Incomplete Sparse <span class="hlt">Approximate</span> Inverses for Parallel Preconditioning</span></a></p> <p><a target="_blank" href="http://www.osti.gov/pages">DOE PAGES</a></p> <p>Anzt, Hartwig; Huckle, Thomas K.; Bräckle, Jürgen; ...</p> <p>2017-10-28</p> <p>In this study, we propose a new preconditioning method that can be seen as a generalization of block-Jacobi methods, or as a simplification of the sparse <span class="hlt">approximate</span> inverse (SAI) preconditioners. The “Incomplete Sparse <span class="hlt">Approximate</span> Inverses” (ISAI) is in particular efficient in the <span class="hlt">solution</span> of sparse triangular linear systems of equations. Those arise, for example, in the context of incomplete factorization preconditioning. ISAI preconditioners can be generated via an algorithm providing fine-grained parallelism, which makes them attractive for hardware with a high concurrency level. Finally, in a study covering a large number of matrices, we identify the ISAI preconditioner as anmore » attractive alternative to exact triangular solves in the context of incomplete factorization preconditioning.« less</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/1999A%26A...351..827E','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/1999A%26A...351..827E"><span>Investigations of the Local supercluster velocity field. II. A study using Tolman-Bondi <span class="hlt">solution</span> and galaxies with <span class="hlt">accurate</span> distances from the Cepheid PL-relation</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Ekholm, T.; Lanoix, P.; Teerikorpi, P.; Paturel, G.; Fouqué, P.</p> <p>1999-11-01</p> <p>A sample of 32 galaxies with <span class="hlt">accurate</span> distance moduli from the Cepheid PL-relation (Lanoix \\cite{Lanoix99}) has been used to study the dynamical behaviour of the Local (Virgo) supercluster. We used analytical Tolman-Bondi (TB) <span class="hlt">solutions</span> for a spherically symmetric density excess embedded in the Einstein-deSitter universe (q_0=0.5). Using 12 galaxies within Theta =30degr from the centre we found a mass estimate of 1.62M_virial for the Virgo cluster. This agrees with the finding of Teerikorpi et al. (\\cite{Teerikorpi92}) that TB-estimate may be larger than virial mass estimate from Tully & Shaya (\\cite{Tully84}). Our conclusions do not critically depend on our primary choice of the global H_0=57 km s-1 Mpc{-1} established from SNe Ia (Lanoix \\cite{Lanoix99}). The remaining galaxies outside Virgo region do not disagree with this value. Finally, we also found a TB-<span class="hlt">solution</span> with the H_0 and q_0 cited yielding exactly one virial mass for the Virgo cluster.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2018SuMi..118..308A','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2018SuMi..118..308A"><span>A new <span class="hlt">approximation</span> of Fermi-Dirac integrals of order 1/2 for degenerate semiconductor devices</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>AlQurashi, Ahmed; Selvakumar, C. R.</p> <p>2018-06-01</p> <p>There had been tremendous growth in the field of Integrated circuits (ICs) in the past fifty years. Scaling laws mandated both lateral and vertical dimensions to be reduced and a steady increase in doping densities. Most of the modern semiconductor devices have invariably heavily doped regions where Fermi-Dirac Integrals are required. Several attempts have been devoted to developing analytical <span class="hlt">approximations</span> for Fermi-Dirac Integrals since numerical computations of Fermi-Dirac Integrals are difficult to use in semiconductor devices, although there are several highly <span class="hlt">accurate</span> tabulated functions available. Most of these analytical expressions are not sufficiently suitable to be employed in semiconductor device applications due to their poor accuracy, the requirement of complicated calculations, and difficulties in differentiating and integrating. A new <span class="hlt">approximation</span> has been developed for the Fermi-Dirac integrals of the order 1/2 by using Prony's method and discussed in this paper. The <span class="hlt">approximation</span> is <span class="hlt">accurate</span> enough (Mean Absolute Error (MAE) = 0.38%) and easy enough to be used in semiconductor device equations. The new <span class="hlt">approximation</span> of Fermi-Dirac Integrals is applied to a more generalized Einstein Relation which is an important relation in semiconductor devices.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2006CG.....32..912P','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2006CG.....32..912P"><span>Parallel adaptive discontinuous Galerkin <span class="hlt">approximation</span> for thin layer avalanche modeling</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Patra, A. K.; Nichita, C. C.; Bauer, A. C.; Pitman, E. B.; Bursik, M.; Sheridan, M. F.</p> <p>2006-08-01</p> <p>This paper describes the development of highly <span class="hlt">accurate</span> adaptive discontinuous Galerkin schemes for the <span class="hlt">solution</span> of the equations arising from a thin layer type model of debris flows. Such flows have wide applicability in the analysis of avalanches induced by many natural calamities, e.g. volcanoes, earthquakes, etc. These schemes are coupled with special parallel <span class="hlt">solution</span> methodologies to produce a simulation tool capable of very high-order numerical accuracy. The methodology successfully replicates cold rock avalanches at Mount Rainier, Washington and hot volcanic particulate flows at Colima Volcano, Mexico.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/21038975','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/21038975"><span>Rational <span class="hlt">approximations</span> to rational models: alternative algorithms for category learning.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Sanborn, Adam N; Griffiths, Thomas L; Navarro, Daniel J</p> <p>2010-10-01</p> <p>Rational models of cognition typically consider the abstract computational problems posed by the environment, assuming that people are capable of optimally solving those problems. This differs from more traditional formal models of cognition, which focus on the psychological processes responsible for behavior. A basic challenge for rational models is thus explaining how optimal <span class="hlt">solutions</span> can be <span class="hlt">approximated</span> by psychological processes. We outline a general strategy for answering this question, namely to explore the psychological plausibility of <span class="hlt">approximation</span> algorithms developed in computer science and statistics. In particular, we argue that Monte Carlo methods provide a source of rational process models that connect optimal <span class="hlt">solutions</span> to psychological processes. We support this argument through a detailed example, applying this approach to Anderson's (1990, 1991) rational model of categorization (RMC), which involves a particularly challenging computational problem. Drawing on a connection between the RMC and ideas from nonparametric Bayesian statistics, we propose 2 alternative algorithms for <span class="hlt">approximate</span> inference in this model. The algorithms we consider include Gibbs sampling, a procedure appropriate when all stimuli are presented simultaneously, and particle filters, which sequentially <span class="hlt">approximate</span> the posterior distribution with a small number of samples that are updated as new data become available. Applying these algorithms to several existing datasets shows that a particle filter with a single particle provides a good description of human inferences.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/16460146','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/16460146"><span>The finite state projection algorithm for the <span class="hlt">solution</span> of the chemical master equation.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Munsky, Brian; Khammash, Mustafa</p> <p>2006-01-28</p> <p>This article introduces the finite state projection (FSP) method for use in the stochastic analysis of chemically reacting systems. One can describe the chemical populations of such systems with probability density vectors that evolve according to a set of linear ordinary differential equations known as the chemical master equation (CME). Unlike Monte Carlo methods such as the stochastic simulation algorithm (SSA) or tau leaping, the FSP directly solves or <span class="hlt">approximates</span> the <span class="hlt">solution</span> of the CME. If the CME describes a system that has a finite number of distinct population vectors, the FSP method provides an exact analytical <span class="hlt">solution</span>. When an infinite or extremely large number of population variations is possible, the state space can be truncated, and the FSP method provides a certificate of accuracy for how closely the truncated space <span class="hlt">approximation</span> matches the true <span class="hlt">solution</span>. The proposed FSP algorithm systematically increases the projection space in order to meet prespecified tolerance in the total probability density error. For any system in which a sufficiently <span class="hlt">accurate</span> FSP exists, the FSP algorithm is shown to converge in a finite number of steps. The FSP is utilized to solve two examples taken from the field of systems biology, and comparisons are made between the FSP, the SSA, and tau leaping algorithms. In both examples, the FSP outperforms the SSA in terms of accuracy as well as computational efficiency. Furthermore, due to very small molecular counts in these particular examples, the FSP also performs far more effectively than tau leaping methods.</p> </li> </ol> <div class="pull-right"> <ul class="pagination"> <li><a href="#" onclick='return showDiv("page_1");'>«</a></li> <li><a href="#" onclick='return showDiv("page_19");'>19</a></li> <li><a href="#" onclick='return showDiv("page_20");'>20</a></li> <li class="active"><span>21</span></li> <li><a href="#" onclick='return showDiv("page_22");'>22</a></li> <li><a href="#" onclick='return showDiv("page_23");'>23</a></li> <li><a href="#" onclick='return showDiv("page_25");'>»</a></li> </ul> </div> </div><!-- col-sm-12 --> </div><!-- row --> </div><!-- page_21 --> <div id="page_22" class="hiddenDiv"> <div class="row"> <div class="col-sm-12"> <div class="pull-right"> <ul class="pagination"> <li><a href="#" onclick='return showDiv("page_1");'>«</a></li> <li><a href="#" onclick='return showDiv("page_20");'>20</a></li> <li><a href="#" onclick='return showDiv("page_21");'>21</a></li> <li class="active"><span>22</span></li> <li><a href="#" onclick='return showDiv("page_23");'>23</a></li> <li><a href="#" onclick='return showDiv("page_24");'>24</a></li> <li><a href="#" onclick='return showDiv("page_25");'>»</a></li> </ul> </div> </div> </div> <div class="row"> <div class="col-sm-12"> <ol class="result-class" start="421"> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/20100039468','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/20100039468"><span>Application of <span class="hlt">Approximate</span> Unsteady Aerodynamics for Flutter Analysis</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Pak, Chan-gi; Li, Wesley W.</p> <p>2010-01-01</p> <p>A technique for <span class="hlt">approximating</span> the modal aerodynamic influence coefficient (AIC) matrices by using basis functions has been developed. A process for using the resulting <span class="hlt">approximated</span> modal AIC matrix in aeroelastic analysis has also been developed. The method requires the unsteady aerodynamics in frequency domain, and this methodology can be applied to the unsteady subsonic, transonic, and supersonic aerodynamics. The flutter <span class="hlt">solution</span> can be found by the classic methods, such as rational function <span class="hlt">approximation</span>, k, p-k, p, root locus et cetera. The unsteady aeroelastic analysis using unsteady subsonic aerodynamic <span class="hlt">approximation</span> is demonstrated herein. The technique presented is shown to offer consistent flutter speed prediction on an aerostructures test wing (ATW) 2 and a hybrid wing body (HWB) type of vehicle configuration with negligible loss in precision. This method computes AICs that are functions of the changing parameters being studied and are generated within minutes of CPU time instead of hours. These results may have practical application in parametric flutter analyses as well as more efficient multidisciplinary design and optimization studies.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/biblio/22403372-approximating-high-dimensional-dynamics-barycentric-coordinates-linear-programming','SCIGOV-STC'); return false;" href="https://www.osti.gov/biblio/22403372-approximating-high-dimensional-dynamics-barycentric-coordinates-linear-programming"><span><span class="hlt">Approximating</span> high-dimensional dynamics by barycentric coordinates with linear programming</span></a></p> <p><a target="_blank" href="http://www.osti.gov/search">DOE Office of Scientific and Technical Information (OSTI.GOV)</a></p> <p>Hirata, Yoshito, E-mail: yoshito@sat.t.u-tokyo.ac.jp; Aihara, Kazuyuki; Suzuki, Hideyuki</p> <p></p> <p>The increasing development of novel methods and techniques facilitates the measurement of high-dimensional time series but challenges our ability for <span class="hlt">accurate</span> modeling and predictions. The use of a general mathematical model requires the inclusion of many parameters, which are difficult to be fitted for relatively short high-dimensional time series observed. Here, we propose a novel method to <span class="hlt">accurately</span> model a high-dimensional time series. Our method extends the barycentric coordinates to high-dimensional phase space by employing linear programming, and allowing the <span class="hlt">approximation</span> errors explicitly. The extension helps to produce free-running time-series predictions that preserve typical topological, dynamical, and/or geometric characteristics ofmore » the underlying attractors more <span class="hlt">accurately</span> than the radial basis function model that is widely used. The method can be broadly applied, from helping to improve weather forecasting, to creating electronic instruments that sound more natural, and to comprehensively understanding complex biological data.« less</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2018EPJWC.17508006B','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2018EPJWC.17508006B"><span>Real-time dynamics of matrix quantum mechanics beyond the classical <span class="hlt">approximation</span></span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Buividovich, Pavel; Hanada, Masanori; Schäfer, Andreas</p> <p>2018-03-01</p> <p>We describe a numerical method which allows to go beyond the classical <span class="hlt">approximation</span> for the real-time dynamics of many-body systems by <span class="hlt">approximating</span> the many-body Wigner function by the most general Gaussian function with time-dependent mean and dispersion. On a simple example of a classically chaotic system with two degrees of freedom we demonstrate that this Gaussian state <span class="hlt">approximation</span> is <span class="hlt">accurate</span> for significantly smaller field strengths and longer times than the classical one. Applying this <span class="hlt">approximation</span> to matrix quantum mechanics, we demonstrate that the quantum Lyapunov exponents are in general smaller than their classical counterparts, and even seem to vanish below some temperature. This behavior resembles the finite-temperature phase transition which was found for this system in Monte-Carlo simulations, and ensures that the system does not violate the Maldacena-Shenker-Stanford bound λL < 2πT, which inevitably happens for classical dynamics at sufficiently small temperatures.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19850022413','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19850022413"><span>On the convergence of difference <span class="hlt">approximations</span> to scalar conservation laws</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Osher, S.; Tadmor, E.</p> <p>1985-01-01</p> <p>A unified treatment of explicit in time, two level, second order resolution, total variation diminishing, <span class="hlt">approximations</span> to scalar conservation laws are presented. The schemes are assumed only to have conservation form and incremental form. A modified flux and a viscosity coefficient are introduced and results in terms of the latter are obtained. The existence of a cell entropy inequality is discussed and such an equality for all entropies is shown to imply that the scheme is an E scheme on monotone (actually more general) data, hence at most only first order <span class="hlt">accurate</span> in general. Convergence for total variation diminishing-second order resolution schemes <span class="hlt">approximating</span> convex or concave conservation laws is shown by enforcing a single discrete entropy inequality.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19880037825&hterms=levels+law&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D30%26Ntt%3DA%2Blevels%2Blaw','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19880037825&hterms=levels+law&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D30%26Ntt%3DA%2Blevels%2Blaw"><span>On the convergence of difference <span class="hlt">approximations</span> to scalar conservation laws</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Osher, Stanley; Tadmor, Eitan</p> <p>1988-01-01</p> <p>A unified treatment is given for time-explicit, two-level, second-order-resolution (SOR), total-variation-diminishing (TVD) <span class="hlt">approximations</span> to scalar conservation laws. The schemes are assumed only to have conservation form and incremental form. A modified flux and a viscosity coefficient are introduced to obtain results in terms of the latter. The existence of a cell entropy inequality is discussed, and such an equality for all entropies is shown to imply that the scheme is an E scheme on monotone (actually more general) data, hence at most only first-order <span class="hlt">accurate</span> in general. Convergence for TVD-SOR schemes <span class="hlt">approximating</span> convex or concave conservation laws is shown by enforcing a single discrete entropy inequality.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2018OPhy...16...22G','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2018OPhy...16...22G"><span>Constructing analytic <span class="hlt">solutions</span> on the Tricomi equation</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Ghiasi, Emran Khoshrouye; Saleh, Reza</p> <p>2018-04-01</p> <p>In this paper, homotopy analysis method (HAM) and variational iteration method (VIM) are utilized to derive the <span class="hlt">approximate</span> <span class="hlt">solutions</span> of the Tricomi equation. Afterwards, the HAM is optimized to accelerate the convergence of the series <span class="hlt">solution</span> by minimizing its square residual error at any order of the <span class="hlt">approximation</span>. It is found that effect of the optimal values of auxiliary parameter on the convergence of the series <span class="hlt">solution</span> is not negligible. Furthermore, the present results are found to agree well with those obtained through a closed-form equation available in the literature. To conclude, it is seen that the two are effective to achieve the <span class="hlt">solution</span> of the partial differential equations.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/pages/biblio/1379538-low-rank-approximation-g0w0-calculations','SCIGOV-DOEP'); return false;" href="https://www.osti.gov/pages/biblio/1379538-low-rank-approximation-g0w0-calculations"><span>Low rank <span class="hlt">approximation</span> in G 0W 0 calculations</span></a></p> <p><a target="_blank" href="http://www.osti.gov/pages">DOE PAGES</a></p> <p>Shao, MeiYue; Lin, Lin; Yang, Chao; ...</p> <p>2016-06-04</p> <p>The single particle energies obtained in a Kohn-Sham density functional theory (DFT) calculation are generally known to be poor <span class="hlt">approximations</span> to electron excitation energies that are measured in tr ansport, tunneling and spectroscopic experiments such as photo-emission spectroscopy. The correction to these energies can be obtained from the poles of a single particle Green’s function derived from a many-body perturbation theory. From a computational perspective, the accuracy and efficiency of such an approach depends on how a self energy term that properly accounts for dynamic screening of electrons is <span class="hlt">approximated</span>. The G 0W 0 <span class="hlt">approximation</span> is a widely used techniquemore » in which the self energy is expressed as the convolution of a noninteracting Green’s function (G 0) and a screened Coulomb interaction (W 0) in the frequency domain. The computational cost associated with such a convolution is high due to the high complexity of evaluating W 0 at multiple frequencies. In this paper, we discuss how the cost of G 0W 0 calculation can be reduced by constructing a low rank <span class="hlt">approximation</span> to the frequency dependent part of W 0 . In particular, we examine the effect of such a low rank <span class="hlt">approximation</span> on the accuracy of the G 0W 0 <span class="hlt">approximation</span>. We also discuss how the numerical convolution of G 0 and W 0 can be evaluated efficiently and <span class="hlt">accurately</span> by using a contour deformation technique with an appropriate choice of the contour.« less</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/23205976','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/23205976"><span>An asymptotically consistent <span class="hlt">approximant</span> method with application to soft- and hard-sphere fluids.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Barlow, N S; Schultz, A J; Weinstein, S J; Kofke, D A</p> <p>2012-11-28</p> <p>A modified Padé <span class="hlt">approximant</span> is used to construct an equation of state, which has the same large-density asymptotic behavior as the model fluid being described, while still retaining the low-density behavior of the virial equation of state (virial series). Within this framework, all sequences of rational functions that are analytic in the physical domain converge to the correct behavior at the same rate, eliminating the ambiguity of choosing the correct form of Padé <span class="hlt">approximant</span>. The method is applied to fluids composed of "soft" spherical particles with separation distance r interacting through an inverse-power pair potential, φ = ε(σ∕r)(n), where ε and σ are model parameters and n is the "hardness" of the spheres. For n < 9, the <span class="hlt">approximants</span> provide a significant improvement over the 8-term virial series, when compared against molecular simulation data. For n ≥ 9, both the <span class="hlt">approximants</span> and the 8-term virial series give an <span class="hlt">accurate</span> description of the fluid behavior, when compared with simulation data. When taking the limit as n → ∞, an equation of state for hard spheres is obtained, which is closer to simulation data than the 10-term virial series for hard spheres, and is comparable in accuracy to other recently proposed equations of state. By applying a least square fit to the <span class="hlt">approximants</span>, we obtain a general and <span class="hlt">accurate</span> soft-sphere equation of state as a function of n, valid over the full range of density in the fluid phase.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/21954203','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/21954203"><span><span class="hlt">Approximate</span> dynamic programming for optimal stationary control with control-dependent noise.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Jiang, Yu; Jiang, Zhong-Ping</p> <p>2011-12-01</p> <p>This brief studies the stochastic optimal control problem via reinforcement learning and <span class="hlt">approximate</span>/adaptive dynamic programming (ADP). A policy iteration algorithm is derived in the presence of both additive and multiplicative noise using Itô calculus. The expectation of the <span class="hlt">approximated</span> cost matrix is guaranteed to converge to the <span class="hlt">solution</span> of some algebraic Riccati equation that gives rise to the optimal cost value. Moreover, the covariance of the <span class="hlt">approximated</span> cost matrix can be reduced by increasing the length of time interval between two consecutive iterations. Finally, a numerical example is given to illustrate the efficiency of the proposed ADP methodology.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19900042273&hterms=millwater&qs=N%3D0%26Ntk%3DAuthor-Name%26Ntx%3Dmode%2Bmatchall%26Ntt%3Dmillwater','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19900042273&hterms=millwater&qs=N%3D0%26Ntk%3DAuthor-Name%26Ntx%3Dmode%2Bmatchall%26Ntt%3Dmillwater"><span>Application of the probabilistic <span class="hlt">approximate</span> analysis method to a turbopump blade analysis. [for Space Shuttle Main Engine</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Thacker, B. H.; Mcclung, R. C.; Millwater, H. R.</p> <p>1990-01-01</p> <p>An eigenvalue analysis of a typical space propulsion system turbopump blade is presented using an <span class="hlt">approximate</span> probabilistic analysis methodology. The methodology was developed originally to investigate the feasibility of computing probabilistic structural response using closed-form <span class="hlt">approximate</span> models. This paper extends the methodology to structures for which simple closed-form <span class="hlt">solutions</span> do not exist. The finite element method will be used for this demonstration, but the concepts apply to any numerical method. The results agree with detailed analysis results and indicate the usefulness of using a probabilistic <span class="hlt">approximate</span> analysis in determining efficient <span class="hlt">solution</span> strategies.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017JMP....58j2104F','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017JMP....58j2104F"><span>Some remarks concerning the centrifugal term <span class="hlt">approximation</span></span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Ferreira, F. J. S.; Bezerra, V. B.</p> <p>2017-10-01</p> <p>We generalize the Pekeris <span class="hlt">approximation</span> for the centrifugal term potential, l/(l +1 ) r2 , and use this to obtain the <span class="hlt">solutions</span> of the radial Schrödinger equation for the arbitrary angular quantum number, l, of the Hulthén potential. We also obtain the expressions for the bound state energies corresponding to this potential and calculate their values for different states and compare with other results presented in the literature. We also consider some models of physical potentials, namely, the Eckart potential, the Poschl-Teller potentials, the Rosen-Morse potential, the Woods-Saxon potential, and the Manning-Rosen potential. Thus, following straightforward the example corresponding to the Hulthén potential, we show what the radial <span class="hlt">solutions</span> and the energy spectra for these potentials are.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://pubs.er.usgs.gov/publication/70025921','USGSPUBS'); return false;" href="https://pubs.er.usgs.gov/publication/70025921"><span>A finite-volume ELLAM for three-dimensional <span class="hlt">solute</span>-transport modeling</span></a></p> <p><a target="_blank" href="http://pubs.er.usgs.gov/pubs/index.jsp?view=adv">USGS Publications Warehouse</a></p> <p>Russell, T.F.; Heberton, C.I.; Konikow, Leonard F.; Hornberger, G.Z.</p> <p>2003-01-01</p> <p>A three-dimensional finite-volume ELLAM method has been developed, tested, and successfully implemented as part of the U.S. Geological Survey (USGS) MODFLOW-2000 ground water modeling package. It is included as a solver option for the Ground Water Transport process. The FVELLAM uses space-time finite volumes oriented along the streamlines of the flow field to solve an integral form of the <span class="hlt">solute</span>-transport equation, thus combining local and global mass conservation with the advantages of Eulerian-Lagrangian characteristic methods. The USGS FVELLAM code simulates <span class="hlt">solute</span> transport in flowing ground water for a single dissolved <span class="hlt">solute</span> constituent and represents the processes of advective transport, hydrodynamic dispersion, mixing from fluid sources, retardation, and decay. Implicit time discretization of the dispersive and source/sink terms is combined with a Lagrangian treatment of advection, in which forward tracking moves mass to the new time level, distributing mass among destination cells using <span class="hlt">approximate</span> indicator functions. This allows the use of large transport time increments (large Courant numbers) with <span class="hlt">accurate</span> results, even for advection-dominated systems (large Peclet numbers). Four test cases, including comparisons with analytical <span class="hlt">solutions</span> and benchmarking against other numerical codes, are presented that indicate that the FVELLAM can usually yield excellent results, even if relatively few transport time steps are used, although the quality of the results is problem-dependent.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=2669769','PMC'); return false;" href="https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=2669769"><span>An efficient and <span class="hlt">accurate</span> molecular alignment and docking technique using ab initio quality scoring</span></a></p> <p><a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pmc">PubMed Central</a></p> <p>Füsti-Molnár, László; Merz, Kenneth M.</p> <p>2008-01-01</p> <p>An <span class="hlt">accurate</span> and efficient molecular alignment technique is presented based on first principle electronic structure calculations. This new scheme maximizes quantum similarity matrices in the relative orientation of the molecules and uses Fourier transform techniques for two purposes. First, building up the numerical representation of true ab initio electronic densities and their Coulomb potentials is accelerated by the previously described Fourier transform Coulomb method. Second, the Fourier convolution technique is applied for accelerating optimizations in the translational coordinates. In order to avoid any interpolation error, the necessary analytical formulas are derived for the transformation of the ab initio wavefunctions in rotational coordinates. The results of our first implementation for a small test set are analyzed in detail and compared with published results of the literature. A new way of refinement of existing shape based alignments is also proposed by using Fourier convolutions of ab initio or other <span class="hlt">approximate</span> electron densities. This new alignment technique is generally applicable for overlap, Coulomb, kinetic energy, etc., quantum similarity measures and can be extended to a genuine docking <span class="hlt">solution</span> with ab initio scoring. PMID:18624561</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/20100039588','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/20100039588"><span>Basis Function <span class="hlt">Approximation</span> of Transonic Aerodynamic Influence Coefficient Matrix</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Li, Wesley Waisang; Pak, Chan-Gi</p> <p>2010-01-01</p> <p>A technique for <span class="hlt">approximating</span> the modal aerodynamic influence coefficients [AIC] matrices by using basis functions has been developed and validated. An application of the resulting <span class="hlt">approximated</span> modal AIC matrix for a flutter analysis in transonic speed regime has been demonstrated. This methodology can be applied to the unsteady subsonic, transonic and supersonic aerodynamics. The method requires the unsteady aerodynamics in frequency-domain. The flutter <span class="hlt">solution</span> can be found by the classic methods, such as rational function <span class="hlt">approximation</span>, k, p-k, p, root-locus et cetera. The unsteady aeroelastic analysis for design optimization using unsteady transonic aerodynamic <span class="hlt">approximation</span> is being demonstrated using the ZAERO(TradeMark) flutter solver (ZONA Technology Incorporated, Scottsdale, Arizona). The technique presented has been shown to offer consistent flutter speed prediction on an aerostructures test wing [ATW] 2 configuration with negligible loss in precision in transonic speed regime. These results may have practical significance in the analysis of aircraft aeroelastic calculation and could lead to a more efficient design optimization cycle</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/20110024032','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/20110024032"><span>Basis Function <span class="hlt">Approximation</span> of Transonic Aerodynamic Influence Coefficient Matrix</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Li, Wesley W.; Pak, Chan-gi</p> <p>2011-01-01</p> <p>A technique for <span class="hlt">approximating</span> the modal aerodynamic influence coefficients matrices by using basis functions has been developed and validated. An application of the resulting <span class="hlt">approximated</span> modal aerodynamic influence coefficients matrix for a flutter analysis in transonic speed regime has been demonstrated. This methodology can be applied to the unsteady subsonic, transonic, and supersonic aerodynamics. The method requires the unsteady aerodynamics in frequency-domain. The flutter <span class="hlt">solution</span> can be found by the classic methods, such as rational function <span class="hlt">approximation</span>, k, p-k, p, root-locus et cetera. The unsteady aeroelastic analysis for design optimization using unsteady transonic aerodynamic <span class="hlt">approximation</span> is being demonstrated using the ZAERO flutter solver (ZONA Technology Incorporated, Scottsdale, Arizona). The technique presented has been shown to offer consistent flutter speed prediction on an aerostructures test wing 2 configuration with negligible loss in precision in transonic speed regime. These results may have practical significance in the analysis of aircraft aeroelastic calculation and could lead to a more efficient design optimization cycle.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017JPhA...50i3001S','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017JPhA...50i3001S"><span><span class="hlt">Approximation</span> and inference methods for stochastic biochemical kinetics—a tutorial review</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Schnoerr, David; Sanguinetti, Guido; Grima, Ramon</p> <p>2017-03-01</p> <p>Stochastic fluctuations of molecule numbers are ubiquitous in biological systems. Important examples include gene expression and enzymatic processes in living cells. Such systems are typically modelled as chemical reaction networks whose dynamics are governed by the chemical master equation. Despite its simple structure, no analytic <span class="hlt">solutions</span> to the chemical master equation are known for most systems. Moreover, stochastic simulations are computationally expensive, making systematic analysis and statistical inference a challenging task. Consequently, significant effort has been spent in recent decades on the development of efficient <span class="hlt">approximation</span> and inference methods. This article gives an introduction to basic modelling concepts as well as an overview of state of the art methods. First, we motivate and introduce deterministic and stochastic methods for modelling chemical networks, and give an overview of simulation and exact <span class="hlt">solution</span> methods. Next, we discuss several <span class="hlt">approximation</span> methods, including the chemical Langevin equation, the system size expansion, moment closure <span class="hlt">approximations</span>, time-scale separation <span class="hlt">approximations</span> and hybrid methods. We discuss their various properties and review recent advances and remaining challenges for these methods. We present a comparison of several of these methods by means of a numerical case study and highlight some of their respective advantages and disadvantages. Finally, we discuss the problem of inference from experimental data in the Bayesian framework and review recent methods developed the literature. In summary, this review gives a self-contained introduction to modelling, <span class="hlt">approximations</span> and inference methods for stochastic chemical kinetics.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/26574370','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/26574370"><span>Molecular Excitation Energies from Time-Dependent Density Functional Theory Employing Random-Phase <span class="hlt">Approximation</span> Hessians with Exact Exchange.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Heßelmann, Andreas</p> <p>2015-04-14</p> <p>Molecular excitation energies have been calculated with time-dependent density-functional theory (TDDFT) using random-phase <span class="hlt">approximation</span> Hessians augmented with exact exchange contributions in various orders. It has been observed that this approach yields fairly <span class="hlt">accurate</span> local valence excitations if combined with <span class="hlt">accurate</span> asymptotically corrected exchange-correlation potentials used in the ground-state Kohn-Sham calculations. The inclusion of long-range particle-particle with hole-hole interactions in the kernel leads to errors of 0.14 eV only for the lowest excitations of a selection of three alkene, three carbonyl, and five azabenzene molecules, thus surpassing the accuracy of a number of common TDDFT and even some wave function correlation methods. In the case of long-range charge-transfer excitations, the method typically underestimates <span class="hlt">accurate</span> reference excitation energies by 8% on average, which is better than with standard hybrid-GGA functionals but worse compared to range-separated functional <span class="hlt">approximations</span>.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/biblio/6234737-analytic-approximation-random-muffin-tin-alloys','SCIGOV-STC'); return false;" href="https://www.osti.gov/biblio/6234737-analytic-approximation-random-muffin-tin-alloys"><span>Analytic <span class="hlt">approximation</span> for random muffin-tin alloys</span></a></p> <p><a target="_blank" href="http://www.osti.gov/search">DOE Office of Scientific and Technical Information (OSTI.GOV)</a></p> <p>Mills, R.; Gray, L.J.; Kaplan, T.</p> <p>1983-03-15</p> <p>The methods introduced in a previous paper under the name of ''traveling-cluster <span class="hlt">approximation</span>'' (TCA) are applied, in a multiple-scattering approach, to the case of a random muffin-tin substitutional alloy. This permits the iterative part of a self-consistent calculation to be carried out entirely in terms of on-the-energy-shell scattering amplitudes. Off-shell components of the mean resolvent, needed for the calculation of spectral functions, are obtained by standard methods involving single-site scattering wave functions. The single-site TCA is just the usual coherent-potential <span class="hlt">approximation</span>, expressed in a form particularly suited for iteration. A fixed-point theorem is proved for the general t-matrix TCA, ensuringmore » convergence upon iteration to a unique self-consistent <span class="hlt">solution</span> with the physically essential Herglotz properties.« less</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/16241374','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/16241374"><span>Piecewise linear emulator of the nonlinear Schrödinger equation and the resulting analytic <span class="hlt">solutions</span> for Bose-Einstein condensates.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Theodorakis, Stavros</p> <p>2003-06-01</p> <p>We emulate the cubic term Psi(3) in the nonlinear Schrödinger equation by a piecewise linear term, thus reducing the problem to a set of uncoupled linear inhomogeneous differential equations. The resulting analytic expressions constitute an excellent <span class="hlt">approximation</span> to the exact <span class="hlt">solutions</span>, as is explicitly shown in the case of the kink, the vortex, and a delta function trap. Such a piecewise linear emulation can be used for any differential equation where the only nonlinearity is a Psi(3) one. In particular, it can be used for the nonlinear Schrödinger equation in the presence of harmonic traps, giving analytic Bose-Einstein condensate <span class="hlt">solutions</span> that reproduce very <span class="hlt">accurately</span> the numerically calculated ones in one, two, and three dimensions.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2016JCoPh.319..163D','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2016JCoPh.319..163D"><span>A simple robust and <span class="hlt">accurate</span> a posteriori sub-cell finite volume limiter for the discontinuous Galerkin method on unstructured meshes</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Dumbser, Michael; Loubère, Raphaël</p> <p>2016-08-01</p> <p>In this paper we propose a simple, robust and <span class="hlt">accurate</span> nonlinear a posteriori stabilization of the Discontinuous Galerkin (DG) finite element method for the <span class="hlt">solution</span> of nonlinear hyperbolic PDE systems on unstructured triangular and tetrahedral meshes in two and three space dimensions. This novel a posteriori limiter, which has been recently proposed for the simple Cartesian grid case in [62], is able to resolve discontinuities at a sub-grid scale and is substantially extended here to general unstructured simplex meshes in 2D and 3D. It can be summarized as follows: At the beginning of each time step, an <span class="hlt">approximation</span> of the local minimum and maximum of the discrete <span class="hlt">solution</span> is computed for each cell, taking into account also the vertex neighbors of an element. Then, an unlimited discontinuous Galerkin scheme of <span class="hlt">approximation</span> degree N is run for one time step to produce a so-called candidate <span class="hlt">solution</span>. Subsequently, an a posteriori detection step checks the unlimited candidate <span class="hlt">solution</span> at time t n + 1 for positivity, absence of floating point errors and whether the discrete <span class="hlt">solution</span> has remained within or at least very close to the bounds given by the local minimum and maximum computed in the first step. Elements that do not satisfy all the previously mentioned detection criteria are flagged as troubled cells. For these troubled cells, the candidate <span class="hlt">solution</span> is discarded as inappropriate and consequently needs to be recomputed. Within these troubled cells the old discrete <span class="hlt">solution</span> at the previous time tn is scattered onto small sub-cells (Ns = 2 N + 1 sub-cells per element edge), in order to obtain a set of sub-cell averages at time tn. Then, a more robust second order TVD finite volume scheme is applied to update the sub-cell averages within the troubled DG cells from time tn to time t n + 1. The new sub-grid data at time t n + 1 are finally gathered back into a valid cell-centered DG polynomial of degree N by using a classical conservative and higher order</p> </li> </ol> <div class="pull-right"> <ul class="pagination"> <li><a href="#" onclick='return showDiv("page_1");'>«</a></li> <li><a href="#" onclick='return showDiv("page_20");'>20</a></li> <li><a href="#" onclick='return showDiv("page_21");'>21</a></li> <li class="active"><span>22</span></li> <li><a href="#" onclick='return showDiv("page_23");'>23</a></li> <li><a href="#" onclick='return showDiv("page_24");'>24</a></li> <li><a href="#" onclick='return showDiv("page_25");'>»</a></li> </ul> </div> </div><!-- col-sm-12 --> </div><!-- row --> </div><!-- page_22 --> <div id="page_23" class="hiddenDiv"> <div class="row"> <div class="col-sm-12"> <div class="pull-right"> <ul class="pagination"> <li><a href="#" onclick='return showDiv("page_1");'>«</a></li> <li><a href="#" onclick='return showDiv("page_21");'>21</a></li> <li><a href="#" onclick='return showDiv("page_22");'>22</a></li> <li class="active"><span>23</span></li> <li><a href="#" onclick='return showDiv("page_24");'>24</a></li> <li><a href="#" onclick='return showDiv("page_25");'>25</a></li> <li><a href="#" onclick='return showDiv("page_25");'>»</a></li> </ul> </div> </div> </div> <div class="row"> <div class="col-sm-12"> <ol class="result-class" start="441"> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/biblio/22679883-effect-limber-flat-sky-approximations-galaxy-weak-lensing','SCIGOV-STC'); return false;" href="https://www.osti.gov/biblio/22679883-effect-limber-flat-sky-approximations-galaxy-weak-lensing"><span>The effect of Limber and flat-sky <span class="hlt">approximations</span> on galaxy weak lensing</span></a></p> <p><a target="_blank" href="http://www.osti.gov/search">DOE Office of Scientific and Technical Information (OSTI.GOV)</a></p> <p>Lemos, Pablo; Challinor, Anthony; Efstathiou, George, E-mail: pl411@cam.ac.uk, E-mail: a.d.challinor@ast.cam.ac.uk, E-mail: gpe@ast.cam.ac.uk</p> <p></p> <p>We review the effect of the commonly-used Limber and flat-sky <span class="hlt">approximations</span> on the calculation of shear power spectra and correlation functions for galaxy weak lensing. These <span class="hlt">approximations</span> are <span class="hlt">accurate</span> at small scales, but it has been claimed recently that their impact on low multipoles could lead to an increase in the amplitude of the mass fluctuations inferred from surveys such as CFHTLenS, reducing the tension between galaxy weak lensing and the amplitude determined by Planck from observations of the cosmic microwave background. Here, we explore the impact of these <span class="hlt">approximations</span> on cosmological parameters derived from weak lensing surveys, using themore » CFHTLenS data as a test case. We conclude that the use of small-angle <span class="hlt">approximations</span> for cosmological parameter estimation is negligible for current data, and does not contribute to the tension between current weak lensing surveys and Planck.« less</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2003JChPh.11911526F','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2003JChPh.11911526F"><span>Representation of the exact relativistic electronic Hamiltonian within the regular <span class="hlt">approximation</span></span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Filatov, Michael; Cremer, Dieter</p> <p>2003-12-01</p> <p>The exact relativistic Hamiltonian for electronic states is expanded in terms of energy-independent linear operators within the regular <span class="hlt">approximation</span>. An effective relativistic Hamiltonian has been obtained, which yields in lowest order directly the infinite-order regular <span class="hlt">approximation</span> (IORA) rather than the zeroth-order regular <span class="hlt">approximation</span> method. Further perturbational expansion of the exact relativistic electronic energy utilizing the effective Hamiltonian leads to new methods based on ordinary (IORAn) or double [IORAn(2)] perturbation theory (n: order of expansion), which provide improved energies in atomic calculations. Energies calculated with IORA4 and IORA3(2) are <span class="hlt">accurate</span> up to c-20. Furthermore, IORA is improved by using the IORA wave function to calculate the Rayleigh quotient, which, if minimized, leads to the exact relativistic energy. The outstanding performance of this new IORA method coined scaled IORA is documented in atomic and molecular calculations.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2002JMPSo..50.2107S','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2002JMPSo..50.2107S"><span>A numerical <span class="hlt">approximation</span> to the elastic properties of sphere-reinforced composites</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Segurado, J.; Llorca, J.</p> <p>2002-10-01</p> <p>Three-dimensional cubic unit cells containing 30 non-overlapping identical spheres randomly distributed were generated using a new, modified random sequential adsortion algorithm suitable for particle volume fractions of up to 50%. The elastic constants of the ensemble of spheres embedded in a continuous and isotropic elastic matrix were computed through the finite element analysis of the three-dimensional periodic unit cells, whose size was chosen as a compromise between the minimum size required to obtain <span class="hlt">accurate</span> results in the statistical sense and the maximum one imposed by the computational cost. Three types of materials were studied: rigid spheres and spherical voids in an elastic matrix and a typical composite made up of glass spheres in an epoxy resin. The moduli obtained for different unit cells showed very little scatter, and the average values obtained from the analysis of four unit cells could be considered very close to the "exact" <span class="hlt">solution</span> to the problem, in agreement with the results of Drugan and Willis (J. Mech. Phys. Solids 44 (1996) 497) referring to the size of the representative volume element for elastic composites. They were used to assess the accuracy of three classical analytical models: the Mori-Tanaka mean-field analysis, the generalized self-consistent method, and Torquato's third-order <span class="hlt">approximation</span>.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/23554899','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/23554899"><span>DendroBLAST: <span class="hlt">approximate</span> phylogenetic trees in the absence of multiple sequence alignments.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Kelly, Steven; Maini, Philip K</p> <p>2013-01-01</p> <p>The rapidly growing availability of genome information has created considerable demand for both fast and <span class="hlt">accurate</span> phylogenetic inference algorithms. We present a novel method called DendroBLAST for reconstructing phylogenetic dendrograms/trees from protein sequences using BLAST. This method differs from other methods by incorporating a simple model of sequence evolution to test the effect of introducing sequence changes on the reliability of the bipartitions in the inferred tree. Using realistic simulated sequence data we demonstrate that this method produces phylogenetic trees that are more <span class="hlt">accurate</span> than other commonly-used distance based methods though not as <span class="hlt">accurate</span> as maximum likelihood methods from good quality multiple sequence alignments. In addition to tests on simulated data, we use DendroBLAST to generate input trees for a supertree reconstruction of the phylogeny of the Archaea. This independent analysis produces an <span class="hlt">approximate</span> phylogeny of the Archaea that has both high precision and recall when compared to previously published analysis of the same dataset using conventional methods. Taken together these results demonstrate that <span class="hlt">approximate</span> phylogenetic trees can be produced in the absence of multiple sequence alignments, and we propose that these trees will provide a platform for improving and informing downstream bioinformatic analysis. A web implementation of the DendroBLAST method is freely available for use at http://www.dendroblast.com/.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://eric.ed.gov/?q=Pendulum&pg=3&id=EJ901519','ERIC'); return false;" href="https://eric.ed.gov/?q=Pendulum&pg=3&id=EJ901519"><span>Improvements in the <span class="hlt">Approximate</span> Formulae for the Period of the Simple Pendulum</span></a></p> <p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p> <p>Turkyilmazoglu, M.</p> <p>2010-01-01</p> <p>This paper is concerned with improvements in some exact formulae for the period of the simple pendulum problem. Two recently presented formulae are re-examined and refined rationally, yielding more <span class="hlt">accurate</span> <span class="hlt">approximate</span> periods. Based on the improved expressions here, a particular new formula is proposed for the period. It is shown that the derived…</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2018PhRvF...3e3401L','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2018PhRvF...3e3401L"><span>Variational method enabling simplified <span class="hlt">solutions</span> to the linearized Boltzmann equation for oscillatory gas flows</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Ladiges, Daniel R.; Sader, John E.</p> <p>2018-05-01</p> <p>Nanomechanical resonators and sensors, operated in ambient conditions, often generate low-Mach-number oscillating rarefied gas flows. Cercignani [C. Cercignani, J. Stat. Phys. 1, 297 (1969), 10.1007/BF01007482] proposed a variational principle for the linearized Boltzmann equation, which can be used to derive <span class="hlt">approximate</span> analytical <span class="hlt">solutions</span> of steady (time-independent) flows. Here we extend and generalize this principle to unsteady oscillatory rarefied flows and thus accommodate resonating nanomechanical devices. This includes a mathematical approach that facilitates its general use and allows for systematic improvements in accuracy. This formulation is demonstrated for two canonical flow problems: oscillatory Couette flow and Stokes' second problem. <span class="hlt">Approximate</span> analytical formulas giving the bulk velocity and shear stress, valid for arbitrary oscillation frequency, are obtained for Couette flow. For Stokes' second problem, a simple system of ordinary differential equations is derived which may be solved to obtain the desired flow fields. Using this framework, a simple and <span class="hlt">accurate</span> formula is provided for the shear stress at the oscillating boundary, again for arbitrary frequency, which may prove useful in application. These <span class="hlt">solutions</span> are easily implemented on any symbolic or numerical package, such as Mathematica or matlab, facilitating the characterization of flows produced by nanomechanical devices and providing insight into the underlying flow physics.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/12765364','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/12765364"><span>Helmholtz and parabolic equation <span class="hlt">solutions</span> to a benchmark problem in ocean acoustics.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Larsson, Elisabeth; Abrahamsson, Leif</p> <p>2003-05-01</p> <p>The Helmholtz equation (HE) describes wave propagation in applications such as acoustics and electromagnetics. For realistic problems, solving the HE is often too expensive. Instead, <span class="hlt">approximations</span> like the parabolic wave equation (PE) are used. For low-frequency shallow-water environments, one persistent problem is to assess the accuracy of the PE model. In this work, a recently developed HE solver that can handle a smoothly varying bathymetry, variable material properties, and layered materials, is used for an investigation of the errors in PE <span class="hlt">solutions</span>. In the HE solver, a preconditioned Krylov subspace method is applied to the discretized equations. The preconditioner combines domain decomposition and fast transform techniques. A benchmark problem with upslope-downslope propagation over a penetrable lossy seamount is solved. The numerical experiments show that, for the same bathymetry, a soft and slow bottom gives very similar HE and PE <span class="hlt">solutions</span>, whereas the PE model is far from <span class="hlt">accurate</span> for a hard and fast bottom. A first attempt to estimate the error is made by computing the relative deviation from the energy balance for the PE <span class="hlt">solution</span>. This measure gives an indication of the magnitude of the error, but cannot be used as a strict error bound.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://www.dtic.mil/docs/citations/ADA623313','DTIC-ST'); return false;" href="http://www.dtic.mil/docs/citations/ADA623313"><span>High-Order <span class="hlt">Accurate</span> <span class="hlt">Solutions</span> to the Helmholtz Equation in the Presence of Boundary Singularities</span></a></p> <p><a target="_blank" href="http://www.dtic.mil/">DTIC Science & Technology</a></p> <p></p> <p>2015-03-31</p> <p>FD scheme is only consistent for classical <span class="hlt">solutions</span> of the PDE . For this reason, we implement the method of singularity subtraction as a means for...regularity due to the boundary conditions. This is because the FD scheme is only consistent for classical <span class="hlt">solutions</span> of the PDE . For this reason, we...Introduction In the present work, we develop a high-order numerical method for solving linear elliptic PDEs with well-behaved variable coefficients on</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2016AIPC.1776i0049T','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2016AIPC.1776i0049T"><span><span class="hlt">Approximating</span> a nonlinear advanced-delayed equation from acoustics</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Teodoro, M. Filomena</p> <p>2016-10-01</p> <p>We <span class="hlt">approximate</span> the <span class="hlt">solution</span> of a particular non-linear mixed type functional differential equation from physiology, the mucosal wave model of the vocal oscillation during phonation. The mathematical equation models a superficial wave propagating through the tissues. The numerical scheme is adapted from the work presented in [1, 2, 3], using homotopy analysis method (HAM) to solve the non linear mixed type equation under study.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=2894168','PMC'); return false;" href="https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=2894168"><span><span class="hlt">Accurate</span> Computation of Electric Field Enhancement Factors for Metallic Nanoparticles Using the Discrete Dipole <span class="hlt">Approximation</span></span></a></p> <p><a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pmc">PubMed Central</a></p> <p></p> <p>2010-01-01</p> <p>We model the response of nanoscale Ag prolate spheroids to an external uniform static electric field using simulations based on the discrete dipole <span class="hlt">approximation</span>, in which the spheroid is represented as a collection of polarizable subunits. We compare the results of simulations that employ subunit polarizabilities derived from the Clausius–Mossotti relation with those of simulations that employ polarizabilities that include a local environmental correction for subunits near the spheroid’s surface [Rahmani et al. Opt Lett 27: 2118 (2002)]. The simulations that employ corrected polarizabilities give predictions in very good agreement with exact results obtained by solving Laplace’s equation. In contrast, simulations that employ uncorrected Clausius–Mossotti polarizabilities substantially underestimate the extent of the electric field “hot spot” near the spheroid’s sharp tip, and give predictions for the field enhancement factor near the tip that are 30 to 50% too small. PMID:20672062</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/20040112050','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/20040112050"><span>An Operator Method for Field Moments from the Extended Parabolic Wave Equation and Analytical <span class="hlt">Solutions</span> of the First and Second Moments for Atmospheric Electromagnetic Wave Propagation</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Manning, Robert M.</p> <p>2004-01-01</p> <p>The extended wide-angle parabolic wave equation applied to electromagnetic wave propagation in random media is considered. A general operator equation is derived which gives the statistical moments of an electric field of a propagating wave. This expression is used to obtain the first and second order moments of the wave field and <span class="hlt">solutions</span> are found that transcend those which incorporate the full paraxial <span class="hlt">approximation</span> at the outset. Although these equations can be applied to any propagation scenario that satisfies the conditions of application of the extended parabolic wave equation, the example of propagation through atmospheric turbulence is used. It is shown that in the case of atmospheric wave propagation and under the Markov <span class="hlt">approximation</span> (i.e., the delta-correlation of the fluctuations in the direction of propagation), the usual parabolic equation in the paraxial <span class="hlt">approximation</span> is <span class="hlt">accurate</span> even at millimeter wavelengths. The comprehensive operator <span class="hlt">solution</span> also allows one to obtain expressions for the longitudinal (generalized) second order moment. This is also considered and the <span class="hlt">solution</span> for the atmospheric case is obtained and discussed. The methodology developed here can be applied to any qualifying situation involving random propagation through turbid or plasma environments that can be represented by a spectral density of permittivity fluctuations.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2018JSMTE..05.3404K','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2018JSMTE..05.3404K"><span><span class="hlt">Approximate</span> method of variational Bayesian matrix factorization/completion with sparse prior</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Kawasumi, Ryota; Takeda, Koujin</p> <p>2018-05-01</p> <p>We derive the analytical expression of a matrix factorization/completion <span class="hlt">solution</span> by the variational Bayes method, under the assumption that the observed matrix is originally the product of low-rank, dense and sparse matrices with additive noise. We assume the prior of a sparse matrix is a Laplace distribution by taking matrix sparsity into consideration. Then we use several <span class="hlt">approximations</span> for the derivation of a matrix factorization/completion <span class="hlt">solution</span>. By our <span class="hlt">solution</span>, we also numerically evaluate the performance of a sparse matrix reconstruction in matrix factorization, and completion of a missing matrix element in matrix completion.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19950018054','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19950018054"><span>The Method of Space-time Conservation Element and <span class="hlt">Solution</span> Element: Development of a New Implicit Solver</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Chang, S. C.; Wang, X. Y.; Chow, C. Y.; Himansu, A.</p> <p>1995-01-01</p> <p>The method of space-time conservation element and <span class="hlt">solution</span> element is a nontraditional numerical method designed from a physicist's perspective, i.e., its development is based more on physics than numerics. It uses only the simplest <span class="hlt">approximation</span> techniques and yet is capable of generating nearly perfect <span class="hlt">solutions</span> for a 2-D shock reflection problem used by Helen Yee and others. In addition to providing an overall view of the new method, we introduce a new concept in the design of implicit schemes, and use it to construct a highly <span class="hlt">accurate</span> solver for a convection-diffusion equation. It is shown that, in the inviscid case, this new scheme becomes explicit and its amplification factors are identical to those of the Leapfrog scheme. On the other hand, in the pure diffusion case, its principal amplification factor becomes the amplification factor of the Crank-Nicolson scheme.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19930013475','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19930013475"><span>A time-<span class="hlt">accurate</span> high-resolution TVD scheme for solving the Navier-Stokes equations</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Kim, Hyun Dae; Liu, Nan-Suey</p> <p>1992-01-01</p> <p>A total variation diminishing (TVD) scheme has been developed and incorporated into an existing time-<span class="hlt">accurate</span> high-resolution Navier-Stokes code. The accuracy and the robustness of the resulting <span class="hlt">solution</span> procedure have been assessed by performing many calculations in four different areas: shock tube flows, regular shock reflection, supersonic boundary layer, and shock boundary layer interactions. These numerical results compare well with corresponding exact <span class="hlt">solutions</span> or experimental data.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/20080040183','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/20080040183"><span><span class="hlt">Approximation</span> of Failure Probability Using Conditional Sampling</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Giesy. Daniel P.; Crespo, Luis G.; Kenney, Sean P.</p> <p>2008-01-01</p> <p>In analyzing systems which depend on uncertain parameters, one technique is to partition the uncertain parameter domain into a failure set and its complement, and judge the quality of the system by estimating the probability of failure. If this is done by a sampling technique such as Monte Carlo and the probability of failure is small, <span class="hlt">accurate</span> <span class="hlt">approximation</span> can require so many sample points that the computational expense is prohibitive. Previous work of the authors has shown how to bound the failure event by sets of such simple geometry that their probabilities can be calculated analytically. In this paper, it is shown how to make use of these failure bounding sets and conditional sampling within them to substantially reduce the computational burden of <span class="hlt">approximating</span> failure probability. It is also shown how the use of these sampling techniques improves the confidence intervals for the failure probability estimate for a given number of sample points and how they reduce the number of sample point analyses needed to achieve a given level of confidence.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19950012948','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19950012948"><span>Comparison of dynamical <span class="hlt">approximation</span> schemes for non-linear gravitational clustering</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Melott, Adrian L.</p> <p>1994-01-01</p> <p>We have recently conducted a controlled comparison of a number of <span class="hlt">approximations</span> for gravitational clustering against the same n-body simulations. These include ordinary linear perturbation theory (Eulerian), the adhesion <span class="hlt">approximation</span>, the frozen-flow <span class="hlt">approximation</span>, the Zel'dovich <span class="hlt">approximation</span> (describable as first-order Lagrangian perturbation theory), and its second-order generalization. In the last two cases we also created new versions of <span class="hlt">approximation</span> by truncation, i.e., smoothing the initial conditions by various smoothing window shapes and varying their sizes. The primary tool for comparing simulations to <span class="hlt">approximation</span> schemes was crosscorrelation of the evolved mass density fields, testing the extent to which mass was moved to the right place. The Zel'dovich <span class="hlt">approximation</span>, with initial convolution with a Gaussian e(exp -k(exp 2)/k(exp 2, sub G)) where k(sub G) is adjusted to be just into the nonlinear regime of the evolved model (details in text) worked extremely well. Its second-order generalization worked slightly better. All other schemes, including those proposed as generalizations of the Zel'dovich <span class="hlt">approximation</span> created by adding forces, were in fact generally worse by this measure. By explicitly checking, we verified that the success of our best-choice was a result of the best treatment of the phases of nonlinear Fourier components. Of all schemes tested, the adhesion <span class="hlt">approximation</span> produced the most <span class="hlt">accurate</span> nonlinear power spectrum and density distribution, but its phase errors suggest mass condensations were moved to slightly the wrong location. Due to its better reproduction of the mass density distribution function and power spectrum, it might be preferred for some uses. We recommend either n-body simulations or our modified versions of the Zel'dovich <span class="hlt">approximation</span>, depending upon the purpose. The theoretical implication is that pancaking is implicit in all cosmological gravitational clustering, at least from Gaussian initial conditions, even</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/26054026','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/26054026"><span>Meta-regression <span class="hlt">approximations</span> to reduce publication selection bias.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Stanley, T D; Doucouliagos, Hristos</p> <p>2014-03-01</p> <p>Publication selection bias is a serious challenge to the integrity of all empirical sciences. We derive meta-regression <span class="hlt">approximations</span> to reduce this bias. Our approach employs Taylor polynomial <span class="hlt">approximations</span> to the conditional mean of a truncated distribution. A quadratic <span class="hlt">approximation</span> without a linear term, precision-effect estimate with standard error (PEESE), is shown to have the smallest bias and mean squared error in most cases and to outperform conventional meta-analysis estimators, often by a great deal. Monte Carlo simulations also demonstrate how a new hybrid estimator that conditionally combines PEESE and the Egger regression intercept can provide a practical <span class="hlt">solution</span> to publication selection bias. PEESE is easily expanded to accommodate systematic heterogeneity along with complex and differential publication selection bias that is related to moderator variables. By providing an intuitive reason for these <span class="hlt">approximations</span>, we can also explain why the Egger regression works so well and when it does not. These meta-regression methods are applied to several policy-relevant areas of research including antidepressant effectiveness, the value of a statistical life, the minimum wage, and nicotine replacement therapy. Copyright © 2013 John Wiley & Sons, Ltd.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/26298108','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/26298108"><span>Communication: Analytic continuation of the virial series through the critical point using parametric <span class="hlt">approximants</span>.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Barlow, Nathaniel S; Schultz, Andrew J; Weinstein, Steven J; Kofke, David A</p> <p>2015-08-21</p> <p>The mathematical structure imposed by the thermodynamic critical point motivates an <span class="hlt">approximant</span> that synthesizes two theoretically sound equations of state: the parametric and the virial. The former is constructed to describe the critical region, incorporating all scaling laws; the latter is an expansion about zero density, developed from molecular considerations. The <span class="hlt">approximant</span> is shown to yield an equation of state capable of <span class="hlt">accurately</span> describing properties over a large portion of the thermodynamic parameter space, far greater than that covered by each treatment alone.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017PhRvL.118p3203H','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017PhRvL.118p3203H"><span>Strong Field Theories beyond Dipole <span class="hlt">Approximations</span> in Nonrelativistic Regimes</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>He, Pei-Lun; Lao, Di; He, Feng</p> <p>2017-04-01</p> <p>The exact nondipole Volkov <span class="hlt">solutions</span> to the Schrödinger equation and Pauli equation are found, based on which a strong field theory beyond the dipole <span class="hlt">approximation</span> is built for describing the nondipole effects in nonrelativistic laser driven electron dynamics. This theory is applied to investigate momentum partition laws for multiphoton and tunneling ionization and explicitly shows that the complex interplay of a laser field and Coulomb action may reverse the expected photoelectron momentum along the laser propagation direction. The magnetic-spin coupling does not bring observable effects on the photoelectron momentum distribution and can be neglected. Compared to the strong field <span class="hlt">approximation</span> within the dipole <span class="hlt">approximation</span>, this theory works in a much wider range of laser parameters and lays a solid foundation for describing nonrelativistic electron dynamics in both short wavelength and midinfrared regimes where nondipole effects are unavoidable.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19960045317','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19960045317"><span>Highly <span class="hlt">Accurate</span> Beam Torsion <span class="hlt">Solutions</span> Using the p-Version Finite Element Method</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Smith, James P.</p> <p>1996-01-01</p> <p>A new treatment of the classical beam torsion boundary value problem is applied. Using the p-version finite element method with shape functions based on Legendre polynomials, torsion <span class="hlt">solutions</span> for generic cross-sections comprised of isotropic materials are developed. Element shape functions for quadrilateral and triangular elements are discussed, and numerical examples are provided.</p> </li> </ol> <div class="pull-right"> <ul class="pagination"> <li><a href="#" onclick='return showDiv("page_1");'>«</a></li> <li><a href="#" onclick='return showDiv("page_21");'>21</a></li> <li><a href="#" onclick='return showDiv("page_22");'>22</a></li> <li class="active"><span>23</span></li> <li><a href="#" onclick='return showDiv("page_24");'>24</a></li> <li><a href="#" onclick='return showDiv("page_25");'>25</a></li> <li><a href="#" onclick='return showDiv("page_25");'>»</a></li> </ul> </div> </div><!-- col-sm-12 --> </div><!-- row --> </div><!-- page_23 --> <div id="page_24" class="hiddenDiv"> <div class="row"> <div class="col-sm-12"> <div class="pull-right"> <ul class="pagination"> <li><a href="#" onclick='return showDiv("page_1");'>«</a></li> <li><a href="#" onclick='return showDiv("page_21");'>21</a></li> <li><a href="#" onclick='return showDiv("page_22");'>22</a></li> <li><a href="#" onclick='return showDiv("page_23");'>23</a></li> <li class="active"><span>24</span></li> <li><a href="#" onclick='return showDiv("page_25");'>25</a></li> <li><a href="#" onclick='return showDiv("page_25");'>»</a></li> </ul> </div> </div> </div> <div class="row"> <div class="col-sm-12"> <ol class="result-class" start="461"> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017JPhCS.949a2006A','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017JPhCS.949a2006A"><span>Analytical <span class="hlt">approximations</span> for the oscillators with anti-symmetric quadratic nonlinearity</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Alal Hosen, Md.; Chowdhury, M. S. H.; Yeakub Ali, Mohammad; Faris Ismail, Ahmad</p> <p>2017-12-01</p> <p>A second-order ordinary differential equation involving anti-symmetric quadratic nonlinearity changes sign. The behaviour of the oscillators with an anti-symmetric quadratic nonlinearity is assumed to oscillate different in the positive and negative directions. In this reason, Harmonic Balance Method (HBM) cannot be directly applied. The main purpose of the present paper is to propose an analytical <span class="hlt">approximation</span> technique based on the HBM for obtaining <span class="hlt">approximate</span> angular frequencies and the corresponding periodic <span class="hlt">solutions</span> of the oscillators with anti-symmetric quadratic nonlinearity. After applying HBM, a set of complicated nonlinear algebraic equations is found. Analytical approach is not always fruitful for solving such kinds of nonlinear algebraic equations. In this article, two small parameters are found, for which the power series <span class="hlt">solution</span> produces desired results. Moreover, the amplitude-frequency relationship has also been determined in a novel analytical way. The presented technique gives excellent results as compared with the corresponding numerical results and is better than the existing ones.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/25903880','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/25903880"><span>Subsystem density functional theory with meta-generalized gradient <span class="hlt">approximation</span> exchange-correlation functionals.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Śmiga, Szymon; Fabiano, Eduardo; Laricchia, Savio; Constantin, Lucian A; Della Sala, Fabio</p> <p>2015-04-21</p> <p>We analyze the methodology and the performance of subsystem density functional theory (DFT) with meta-generalized gradient <span class="hlt">approximation</span> (meta-GGA) exchange-correlation functionals for non-bonded molecular systems. Meta-GGA functionals depend on the Kohn-Sham kinetic energy density (KED), which is not known as an explicit functional of the density. Therefore, they cannot be directly applied in subsystem DFT calculations. We propose a Laplacian-level <span class="hlt">approximation</span> to the KED which overcomes this limitation and provides a simple and <span class="hlt">accurate</span> way to apply meta-GGA exchange-correlation functionals in subsystem DFT calculations. The so obtained density and energy errors, with respect to the corresponding supermolecular calculations, are comparable with conventional approaches, depending almost exclusively on the <span class="hlt">approximations</span> in the non-additive kinetic embedding term. An embedding energy error decomposition explains the accuracy of our method.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2016amos.confE..42P','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2016amos.confE..42P"><span>Confidence Region of Least Squares <span class="hlt">Solution</span> for Single-Arc Observations</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Principe, G.; Armellin, R.; Lewis, H.</p> <p>2016-09-01</p> <p>The total number of active satellites, rocket bodies, and debris larger than 10 cm is currently about 20,000. Considering all resident space objects larger than 1 cm this rises to an estimated minimum of 500,000 objects. Latest generation sensor networks will be able to detect small-size objects, producing millions of observations per day. Due to observability constraints it is likely that long gaps between observations will occur for small objects. This requires to determine the space object (SO) orbit and to <span class="hlt">accurately</span> describe the associated uncertainty when observations are acquired on a single arc. The aim of this work is to revisit the classical least squares method taking advantage of the high order Taylor expansions enabled by differential algebra. In particular, the high order expansion of the residuals with respect to the state is used to implement an arbitrary order least squares solver, avoiding the typical <span class="hlt">approximations</span> of differential correction methods. In addition, the same expansions are used to <span class="hlt">accurately</span> characterize the confidence region of the <span class="hlt">solution</span>, going beyond the classical Gaussian distributions. The properties and performances of the proposed method are discussed using optical observations of objects in LEO, HEO, and GEO.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19950005402','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19950005402"><span><span class="hlt">Accurate</span> interlaminar stress recovery from finite element analysis</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Tessler, Alexander; Riggs, H. Ronald</p> <p>1994-01-01</p> <p>The accuracy and robustness of a two-dimensional smoothing methodology is examined for the problem of recovering <span class="hlt">accurate</span> interlaminar shear stress distributions in laminated composite and sandwich plates. The smoothing methodology is based on a variational formulation which combines discrete least-squares and penalty-constraint functionals in a single variational form. The smoothing analysis utilizes optimal strains computed at discrete locations in a finite element analysis. These discrete strain data are smoothed with a smoothing element discretization, producing superior accuracy strains and their first gradients. The approach enables the resulting smooth strain field to be practically C1-continuous throughout the domain of smoothing, exhibiting superconvergent properties of the smoothed quantity. The continuous strain gradients are also obtained directly from the <span class="hlt">solution</span>. The recovered strain gradients are subsequently employed in the integration o equilibrium equations to obtain <span class="hlt">accurate</span> interlaminar shear stresses. The problem is a simply-supported rectangular plate under a doubly sinusoidal load. The problem has an exact analytic <span class="hlt">solution</span> which serves as a measure of goodness of the recovered interlaminar shear stresses. The method has the versatility of being applicable to the analysis of rather general and complex structures built of distinct components and materials, such as found in aircraft design. For these types of structures, the smoothing is achieved with 'patches', each patch covering the domain in which the smoothed quantity is physically continuous.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/25158101','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/25158101"><span>Comparison of non-ideal <span class="hlt">solution</span> theories for multi-<span class="hlt">solute</span> <span class="hlt">solutions</span> in cryobiology and tabulation of required coefficients.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Zielinski, Michal W; McGann, Locksley E; Nychka, John A; Elliott, Janet A W</p> <p>2014-10-01</p> <p>Thermodynamic <span class="hlt">solution</span> theories allow the prediction of chemical potentials in <span class="hlt">solutions</span> of known composition. In cryobiology, such models are a critical component of many mathematical models that are used to simulate the biophysical processes occurring in cells and tissues during cryopreservation. A number of <span class="hlt">solution</span> theories, both thermodynamically ideal and non-ideal, have been proposed for use with cryobiological <span class="hlt">solutions</span>. In this work, we have evaluated two non-ideal <span class="hlt">solution</span> theories for predicting water chemical potential (i.e. osmolality) in multi-<span class="hlt">solute</span> <span class="hlt">solutions</span> relevant to cryobiology: the Elliott et al. form of the multi-<span class="hlt">solute</span> osmotic virial equation, and the Kleinhans and Mazur freezing point summation model. These two <span class="hlt">solution</span> theories require fitting to only single-<span class="hlt">solute</span> data, although they can make predictions in multi-<span class="hlt">solute</span> <span class="hlt">solutions</span>. The predictions of these non-ideal <span class="hlt">solution</span> theories were compared to predictions made using ideal dilute assumptions and to available literature multi-<span class="hlt">solute</span> experimental osmometric data. A single, consistent set of literature single-<span class="hlt">solute</span> <span class="hlt">solution</span> data was used to fit for the required <span class="hlt">solute</span>-specific coefficients for each of the non-ideal models. Our results indicate that the two non-ideal <span class="hlt">solution</span> theories have similar overall performance, and both give more <span class="hlt">accurate</span> predictions than ideal models. These results can be used to select between the non-ideal models for a specific multi-<span class="hlt">solute</span> <span class="hlt">solution</span>, and the updated coefficients provided in this work can be used to make the desired predictions. Copyright © 2014 The Authors. Published by Elsevier Inc. All rights reserved.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017CG....109...51M','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017CG....109...51M"><span>Big geo data surface <span class="hlt">approximation</span> using radial basis functions: A comparative study</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Majdisova, Zuzana; Skala, Vaclav</p> <p>2017-12-01</p> <p><span class="hlt">Approximation</span> of scattered data is often a task in many engineering problems. The Radial Basis Function (RBF) <span class="hlt">approximation</span> is appropriate for big scattered datasets in n-dimensional space. It is a non-separable <span class="hlt">approximation</span>, as it is based on the distance between two points. This method leads to the <span class="hlt">solution</span> of an overdetermined linear system of equations. In this paper the RBF <span class="hlt">approximation</span> methods are briefly described, a new approach to the RBF <span class="hlt">approximation</span> of big datasets is presented, and a comparison for different Compactly Supported RBFs (CS-RBFs) is made with respect to the accuracy of the computation. The proposed approach uses symmetry of a matrix, partitioning the matrix into blocks and data structures for storage of the sparse matrix. The experiments are performed for synthetic and real datasets.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/24288668','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/24288668"><span>Selecting summary statistics in <span class="hlt">approximate</span> Bayesian computation for calibrating stochastic models.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Burr, Tom; Skurikhin, Alexei</p> <p>2013-01-01</p> <p><span class="hlt">Approximate</span> Bayesian computation (ABC) is an approach for using measurement data to calibrate stochastic computer models, which are common in biology applications. ABC is becoming the "go-to" option when the data and/or parameter dimension is large because it relies on user-chosen summary statistics rather than the full data and is therefore computationally feasible. One technical challenge with ABC is that the quality of the <span class="hlt">approximation</span> to the posterior distribution of model parameters depends on the user-chosen summary statistics. In this paper, the user requirement to choose effective summary statistics in order to <span class="hlt">accurately</span> estimate the posterior distribution of model parameters is investigated and illustrated by example, using a model and corresponding real data of mitochondrial DNA population dynamics. We show that for some choices of summary statistics, the posterior distribution of model parameters is closely <span class="hlt">approximated</span> and for other choices of summary statistics, the posterior distribution is not closely <span class="hlt">approximated</span>. A strategy to choose effective summary statistics is suggested in cases where the stochastic computer model can be run at many trial parameter settings, as in the example.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19820015320','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19820015320"><span>An <span class="hlt">accurate</span> method for evaluating the kernel of the integral equation relating lift to downwash in unsteady potential flow</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Desmarais, R. N.</p> <p>1982-01-01</p> <p>The method is capable of generating <span class="hlt">approximations</span> of arbitrary accuracy. It is based on <span class="hlt">approximating</span> the algebraic part of the nonelementary integrals in the kernel by exponential functions and then integrating termwise. The exponent spacing in the <span class="hlt">approximation</span> is a geometric sequence. The coefficients and exponent multiplier of the exponential <span class="hlt">approximation</span> are computed by least squares so the method is completely automated. Exponential <span class="hlt">approximates</span> generated in this manner are two orders of magnitude more <span class="hlt">accurate</span> than the exponential <span class="hlt">approximation</span> that is currently most often used for this purpose. The method can be used to generate <span class="hlt">approximations</span> to attain any desired trade-off between accuracy and computing cost.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2010PhDT.......213A','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2010PhDT.......213A"><span>Quantum Monte Carlo: Faster, More Reliable, And More <span class="hlt">Accurate</span></span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Anderson, Amos Gerald</p> <p>2010-06-01</p> <p>The Schrodinger Equation has been available for about 83 years, but today, we still strain to apply it <span class="hlt">accurately</span> to molecules of interest. The difficulty is not theoretical in nature, but practical, since we're held back by lack of sufficient computing power. Consequently, effort is applied to find acceptable <span class="hlt">approximations</span> to facilitate real time <span class="hlt">solutions</span>. In the meantime, computer technology has begun rapidly advancing and changing the way we think about efficient algorithms. For those who can reorganize their formulas to take advantage of these changes and thereby lift some <span class="hlt">approximations</span>, incredible new opportunities await. Over the last decade, we've seen the emergence of a new kind of computer processor, the graphics card. Designed to accelerate computer games by optimizing quantity instead of quality in processor, they have become of sufficient quality to be useful to some scientists. In this thesis, we explore the first known use of a graphics card to computational chemistry by rewriting our Quantum Monte Carlo software into the requisite "data parallel" formalism. We find that notwithstanding precision considerations, we are able to speed up our software by about a factor of 6. The success of a Quantum Monte Carlo calculation depends on more than just processing power. It also requires the scientist to carefully design the trial wavefunction used to guide simulated electrons. We have studied the use of Generalized Valence Bond wavefunctions to simply, and yet effectively, captured the essential static correlation in atoms and molecules. Furthermore, we have developed significantly improved two particle correlation functions, designed with both flexibility and simplicity considerations, representing an effective and reliable way to add the necessary dynamic correlation. Lastly, we present our method for stabilizing the statistical nature of the calculation, by manipulating configuration weights, thus facilitating efficient and robust calculations. Our</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/23503227','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/23503227"><span>Validity of the paraxial <span class="hlt">approximation</span> for electron acceleration with radially polarized laser beams.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Marceau, Vincent; Varin, Charles; Piché, Michel</p> <p>2013-03-15</p> <p>In the study of laser-driven electron acceleration, it has become customary to work within the framework of paraxial wave optics. Using an exact <span class="hlt">solution</span> to the Helmholtz equation as well as its paraxial counterpart, we perform numerical simulations of electron acceleration with a high-power TM(01) beam. For beam waist sizes at which the paraxial <span class="hlt">approximation</span> was previously recognized valid, we highlight significant differences in the angular divergence and energy distribution of the electron bunches produced by the exact and the paraxial <span class="hlt">solutions</span>. Our results demonstrate that extra care has to be taken when working under the paraxial <span class="hlt">approximation</span> in the context of electron acceleration with radially polarized laser beams.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/23288415','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/23288415"><span><span class="hlt">Accurate</span> measurement of volume and shape of resting and activated blood platelets from light scattering.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Moskalensky, Alexander E; Yurkin, Maxim A; Konokhova, Anastasiya I; Strokotov, Dmitry I; Nekrasov, Vyacheslav M; Chernyshev, Andrei V; Tsvetovskaya, Galina A; Chikova, Elena D; Maltsev, Valeri P</p> <p>2013-01-01</p> <p>We introduce a novel approach for determination of volume and shape of individual blood platelets modeled as an oblate spheroid from angle-resolved light scattering with flow-cytometric technique. The light-scattering profiles (LSPs) of individual platelets were measured with the scanning flow cytometer and the platelet characteristics were determined from the <span class="hlt">solution</span> of the inverse light-scattering problem using the precomputed database of theoretical LSPs. We revealed a phenomenon of parameter compensation, which is partly explained in the framework of anomalous diffraction <span class="hlt">approximation</span>. To overcome this problem, additional a priori information on the platelet refractive index was used. It allowed us to determine the size of each platelet with subdiffraction precision and independent of the particular value of the platelet aspect ratio. The shape (spheroidal aspect ratio) distributions of platelets showed substantial differences between native and activated by 10 μM adenosine diphosphate samples. We expect that the new approach may find use in hematological analyzers for <span class="hlt">accurate</span> measurement of platelet volume distribution and for determination of the platelet activation efficiency.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/biblio/6637686-approximate-stoichiometry-rich-hydrocarbon-mixtures','SCIGOV-STC'); return false;" href="https://www.osti.gov/biblio/6637686-approximate-stoichiometry-rich-hydrocarbon-mixtures"><span><span class="hlt">Approximate</span> stoichiometry for rich hydrocarbon mixtures</span></a></p> <p><a target="_blank" href="http://www.osti.gov/search">DOE Office of Scientific and Technical Information (OSTI.GOV)</a></p> <p>Beans, E.W.</p> <p>1993-03-01</p> <p>The stoichiometry of lean mixtures can readily and <span class="hlt">accurately</span> be determined from the assumption that all the carbon oxidizes to carbon dioxide and all the hydrogen oxidizes to water. This assumption is valid up to an equivalence ratio ([sigma]) of 0.8 and can be used with little error up to [sigma] = 1. The composition of the products of a hydrocarbon burnt in air under the foregoing assumption can be obtained from simple carbon, hydrogen, oxygen and nitrogen balances. Given the composition, one can determine the energy released and/or the adiabatic flame temperature. For rich mixtures, the foregoing assumption, ofmore » course, is not valid. Hence, there is no easy way to determine the stoichiometry of the products of a rich mixture. The objective of this note is to present an equation' which will allow one to readily determine the composition of the products of rich hydrocarbon mixtures. The equation is based on equilibrium composition calculations and some assumptions regarding the characteristics of hydrocarbons. The equation gives <span class="hlt">approximate</span> results. However, the results are sufficiently <span class="hlt">accurate</span> for many situations. If more accuracy is wanted, one should use an equilibrium combustion program like the one by Gordon and McBride.« less</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19980151078','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19980151078"><span>Neural Network and Regression <span class="hlt">Approximations</span> in High Speed Civil Transport Aircraft Design Optimization</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Patniak, Surya N.; Guptill, James D.; Hopkins, Dale A.; Lavelle, Thomas M.</p> <p>1998-01-01</p> <p>Nonlinear mathematical-programming-based design optimization can be an elegant method. However, the calculations required to generate the merit function, constraints, and their gradients, which are frequently required, can make the process computational intensive. The computational burden can be greatly reduced by using <span class="hlt">approximating</span> analyzers derived from an original analyzer utilizing neural networks and linear regression methods. The experience gained from using both of these <span class="hlt">approximation</span> methods in the design optimization of a high speed civil transport aircraft is the subject of this paper. The Langley Research Center's Flight Optimization System was selected for the aircraft analysis. This software was exercised to generate a set of training data with which a neural network and a regression method were trained, thereby producing the two <span class="hlt">approximating</span> analyzers. The derived analyzers were coupled to the Lewis Research Center's CometBoards test bed to provide the optimization capability. With the combined software, both <span class="hlt">approximation</span> methods were examined for use in aircraft design optimization, and both performed satisfactorily. The CPU time for <span class="hlt">solution</span> of the problem, which had been measured in hours, was reduced to minutes with the neural network <span class="hlt">approximation</span> and to seconds with the regression method. Instability encountered in the aircraft analysis software at certain design points was also eliminated. On the other hand, there were costs and difficulties associated with training the <span class="hlt">approximating</span> analyzers. The CPU time required to generate the input-output pairs and to train the <span class="hlt">approximating</span> analyzers was seven times that required for <span class="hlt">solution</span> of the problem.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017AGUFMDI42A..06Q','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017AGUFMDI42A..06Q"><span>Well-Balanced Second-Order <span class="hlt">Approximation</span> of the Shallow Water Equations With Friction via Continuous Galerkin Finite Elements</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Quezada de Luna, M.; Farthing, M.; Guermond, J. L.; Kees, C. E.; Popov, B.</p> <p>2017-12-01</p> <p>The Shallow Water Equations (SWEs) are popular for modeling non-dispersive incompressible water waves where the horizontal wavelength is much larger than the vertical scales. They can be derived from the incompressible Navier-Stokes equations assuming a constant vertical velocity. The SWEs are important in Geophysical Fluid Dynamics for modeling surface gravity waves in shallow regimes; e.g., in the deep ocean. Some common geophysical applications are the evolution of tsunamis, river flooding and dam breaks, storm surge simulations, atmospheric flows and others. This work is concerned with the <span class="hlt">approximation</span> of the time-dependent Shallow Water Equations with friction using explicit time stepping and continuous finite elements. The objective is to construct a method that is at least second-order <span class="hlt">accurate</span> in space and third or higher-order <span class="hlt">accurate</span> in time, positivity preserving, well-balanced with respect to rest states, well-balanced with respect to steady sliding <span class="hlt">solutions</span> on inclined planes and robust with respect to dry states. Methods fulfilling the desired goals are common within the finite volume literature. However, to the best of our knowledge, schemes with the above properties are not well developed in the context of continuous finite elements. We start this work based on a finite element method that is second-order <span class="hlt">accurate</span> in space, positivity preserving and well-balanced with respect to rest states. We extend it by: modifying the artificial viscosity (via the entropy viscosity method) to deal with issues of loss of accuracy around local extrema, considering a singular Manning friction term handled via an explicit discretization under the usual CFL condition, considering a water height regularization that depends on the mesh size and is consistent with the polynomial <span class="hlt">approximation</span>, reducing dispersive errors introduced by lumping the mass matrix and others. After presenting the details of the method we show numerical tests that demonstrate the well</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2018JAG...152..100L','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2018JAG...152..100L"><span>Arrival-time picking method based on <span class="hlt">approximate</span> negentropy for microseismic data</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Li, Yue; Ni, Zhuo; Tian, Yanan</p> <p>2018-05-01</p> <p><span class="hlt">Accurate</span> and dependable picking of the first arrival time for microseismic data is an important part in microseismic monitoring, which directly affects analysis results of post-processing. This paper presents a new method based on <span class="hlt">approximate</span> negentropy (AN) theory for microseismic arrival time picking in condition of much lower signal-to-noise ratio (SNR). According to the differences in information characteristics between microseismic data and random noise, an appropriate <span class="hlt">approximation</span> of negentropy function is selected to minimize the effect of SNR. At the same time, a weighted function of the differences between maximum and minimum value of AN spectrum curve is designed to obtain a proper threshold function. In this way, the region of signal and noise is distinguished to pick the first arrival time <span class="hlt">accurately</span>. To demonstrate the effectiveness of AN method, we make many experiments on a series of synthetic data with different SNR from -1 dB to -12 dB and compare it with previously published Akaike information criterion (AIC) and short/long time average ratio (STA/LTA) methods. Experimental results indicate that these three methods can achieve well picking effect when SNR is from -1 dB to -8 dB. However, when SNR is as low as -8 dB to -12 dB, the proposed AN method yields more <span class="hlt">accurate</span> and stable picking result than AIC and STA/LTA methods. Furthermore, the application results of real three-component microseismic data also show that the new method is superior to the other two methods in accuracy and stability.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19910045449&hterms=feedforward+control&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D50%26Ntt%3Dfeedforward%2Bcontrol','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19910045449&hterms=feedforward+control&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D50%26Ntt%3Dfeedforward%2Bcontrol"><span>Neural networks for function <span class="hlt">approximation</span> in nonlinear control</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Linse, Dennis J.; Stengel, Robert F.</p> <p>1990-01-01</p> <p>Two neural network architectures are compared with a classical spline interpolation technique for the <span class="hlt">approximation</span> of functions useful in a nonlinear control system. A standard back-propagation feedforward neural network and a cerebellar model articulation controller (CMAC) neural network are presented, and their results are compared with a B-spline interpolation procedure that is updated using recursive least-squares parameter identification. Each method is able to <span class="hlt">accurately</span> represent a one-dimensional test function. Tradeoffs between size requirements, speed of operation, and speed of learning indicate that neural networks may be practical for identification and adaptation in a nonlinear control environment.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/biblio/1326627-approximate-maximum-likelihood-localization-solver','SCIGOV-STC'); return false;" href="https://www.osti.gov/biblio/1326627-approximate-maximum-likelihood-localization-solver"><span>A 3D <span class="hlt">approximate</span> maximum likelihood localization solver</span></a></p> <p><a target="_blank" href="http://www.osti.gov/search">DOE Office of Scientific and Technical Information (OSTI.GOV)</a></p> <p></p> <p>2016-09-23</p> <p>A robust three-dimensional solver was needed to <span class="hlt">accurately</span> and efficiently estimate the time sequence of locations of fish tagged with acoustic transmitters and vocalizing marine mammals to describe in sufficient detail the information needed to assess the function of dam-passage design alternatives and support Marine Renewable Energy. An <span class="hlt">approximate</span> maximum likelihood solver was developed using measurements of time difference of arrival from all hydrophones in receiving arrays on which a transmission was detected. Field experiments demonstrated that the developed solver performed significantly better in tracking efficiency and accuracy than other solvers described in the literature.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/20040110838','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/20040110838"><span>A Subsonic Aircraft Design Optimization With Neural Network and Regression <span class="hlt">Approximators</span></span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Patnaik, Surya N.; Coroneos, Rula M.; Guptill, James D.; Hopkins, Dale A.; Haller, William J.</p> <p>2004-01-01</p> <p>The Flight-Optimization-System (FLOPS) code encountered difficulty in analyzing a subsonic aircraft. The limitation made the design optimization problematic. The deficiencies have been alleviated through use of neural network and regression <span class="hlt">approximations</span>. The insight gained from using the <span class="hlt">approximators</span> is discussed in this paper. The FLOPS code is reviewed. Analysis models are developed and validated for each <span class="hlt">approximator</span>. The regression method appears to hug the data points, while the neural network <span class="hlt">approximation</span> follows a mean path. For an analysis cycle, the <span class="hlt">approximate</span> model required milliseconds of central processing unit (CPU) time versus seconds by the FLOPS code. Performance of the <span class="hlt">approximators</span> was satisfactory for aircraft analysis. A design optimization capability has been created by coupling the derived analyzers to the optimization test bed CometBoards. The <span class="hlt">approximators</span> were efficient reanalysis tools in the aircraft design optimization. Instability encountered in the FLOPS analyzer was eliminated. The convergence characteristics were improved for the design optimization. The CPU time required to calculate the optimum <span class="hlt">solution</span>, measured in hours with the FLOPS code was reduced to minutes with the neural network <span class="hlt">approximation</span> and to seconds with the regression method. Generation of the <span class="hlt">approximators</span> required the manipulation of a very large quantity of data. Design sensitivity with respect to the bounds of aircraft constraints is easily generated.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2010AdWR...33..813B','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2010AdWR...33..813B"><span>Hydraulic modeling of riverbank filtration systems with curved boundaries using analytic elements and series <span class="hlt">solutions</span></span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Bakker, Mark</p> <p>2010-08-01</p> <p>A new analytic <span class="hlt">solution</span> approach is presented for the modeling of steady flow to pumping wells near rivers in strip aquifers; all boundaries of the river and strip aquifer may be curved. The river penetrates the aquifer only partially and has a leaky stream bed. The water level in the river may vary spatially. Flow in the aquifer below the river is semi-confined while flow in the aquifer adjacent to the river is confined or unconfined and may be subject to areal recharge. Analytic <span class="hlt">solutions</span> are obtained through superposition of analytic elements and Fourier series. Boundary conditions are specified at collocation points along the boundaries. The number of collocation points is larger than the number of coefficients in the Fourier series and a <span class="hlt">solution</span> is obtained in the least squares sense. The <span class="hlt">solution</span> is analytic while boundary conditions are met <span class="hlt">approximately</span>. Very <span class="hlt">accurate</span> <span class="hlt">solutions</span> are obtained when enough terms are used in the series. Several examples are presented for domains with straight and curved boundaries, including a well pumping near a meandering river with a varying water level. The area of the river bottom where water infiltrates into the aquifer is delineated and the fraction of river water in the well water is computed for several cases.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/biblio/6056245-traveling-cluster-approximation-uncorrelated-amorphous-systems','SCIGOV-STC'); return false;" href="https://www.osti.gov/biblio/6056245-traveling-cluster-approximation-uncorrelated-amorphous-systems"><span>Traveling-cluster <span class="hlt">approximation</span> for uncorrelated amorphous systems</span></a></p> <p><a target="_blank" href="http://www.osti.gov/search">DOE Office of Scientific and Technical Information (OSTI.GOV)</a></p> <p>Sen, A.K.; Mills, R.; Kaplan, T.</p> <p>1984-11-15</p> <p>We have developed a formalism for including cluster effects in the one-electron Green's function for a positionally disordered (liquid or amorphous) system without any correlation among the scattering sites. This method is an extension of the technique known as the traveling-cluster <span class="hlt">approximation</span> (TCA) originally obtained and applied to a substitutional alloy by Mills and Ratanavararaksa. We have also proved the appropriate fixed-point theorem, which guarantees, for a bounded local potential, that the self-consistent equations always converge upon iteration to a unique, Herglotz <span class="hlt">solution</span>. To our knowledge, this is the only analytic theory for considering cluster effects. Furthermore, we have performedmore » some computer calculations in the pair TCA, for the model case of delta-function potentials on a one-dimensional random chain. These results have been compared with ''exact calculations'' (which, in principle, take into account all cluster effects) and with the coherent-potential <span class="hlt">approximation</span> (CPA), which is the single-site TCA. The density of states for the pair TCA clearly shows some improvement over the CPA and yet, apparently, the pair <span class="hlt">approximation</span> distorts some of the features of the exact results.« less</p> </li> </ol> <div class="pull-right"> <ul class="pagination"> <li><a href="#" onclick='return showDiv("page_1");'>«</a></li> <li><a href="#" onclick='return showDiv("page_21");'>21</a></li> <li><a href="#" onclick='return showDiv("page_22");'>22</a></li> <li><a href="#" onclick='return showDiv("page_23");'>23</a></li> <li class="active"><span>24</span></li> <li><a href="#" onclick='return showDiv("page_25");'>25</a></li> <li><a href="#" onclick='return showDiv("page_25");'>»</a></li> </ul> </div> </div><!-- col-sm-12 --> </div><!-- row --> </div><!-- page_24 --> <div id="page_25" class="hiddenDiv"> <div class="row"> <div class="col-sm-12"> <div class="pull-right"> <ul class="pagination"> <li><a href="#" onclick='return showDiv("page_1");'>«</a></li> <li><a href="#" onclick='return showDiv("page_21");'>21</a></li> <li><a href="#" onclick='return showDiv("page_22");'>22</a></li> <li><a href="#" onclick='return showDiv("page_23");'>23</a></li> <li><a href="#" onclick='return showDiv("page_24");'>24</a></li> <li class="active"><span>25</span></li> <li><a href="#" onclick='return showDiv("page_25");'>»</a></li> </ul> </div> </div> </div> <div class="row"> <div class="col-sm-12"> <ol class="result-class" start="481"> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2018MCM....54..243X','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2018MCM....54..243X"><span>An <span class="hlt">Approximate</span> <span class="hlt">Solution</span> to the Plastic Indentation of Circular Sandwich Panels</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Xie, Z.</p> <p>2018-05-01</p> <p>The plastic indentation response of circular sandwich panels loaded by the flat end of a cylinder is investigated employing a velocity field model. Using the principles of virtual velocities and minimum work, an expression for the indenter load in relation to the indenter displacement and displacement field of the deformed face sheet is derived. The analytical <span class="hlt">solutions</span> obtained are in good agreement with those found by simulations using the ABAQUS code. The radial tensile strain of the deformed face sheet and the ratio of energy absorption rate of the core to that of the face sheet are discussed.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=3976839','PMC'); return false;" href="https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=3976839"><span>An <span class="hlt">Approximation</span> <span class="hlt">Solution</span> to Refinery Crude Oil Scheduling Problem with Demand Uncertainty Using Joint Constrained Programming</span></a></p> <p><a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pmc">PubMed Central</a></p> <p>Duan, Qianqian; Yang, Genke; Xu, Guanglin; Pan, Changchun</p> <p>2014-01-01</p> <p>This paper is devoted to develop an <span class="hlt">approximation</span> method for scheduling refinery crude oil operations by taking into consideration the demand uncertainty. In the stochastic model the demand uncertainty is modeled as random variables which follow a joint multivariate distribution with a specific correlation structure. Compared to deterministic models in existing works, the stochastic model can be more practical for optimizing crude oil operations. Using joint chance constraints, the demand uncertainty is treated by specifying proximity level on the satisfaction of product demands. However, the joint chance constraints usually hold strong nonlinearity and consequently, it is still hard to handle it directly. In this paper, an <span class="hlt">approximation</span> method combines a relax-and-tight technique to <span class="hlt">approximately</span> transform the joint chance constraints to a serial of parameterized linear constraints so that the complicated problem can be attacked iteratively. The basic idea behind this approach is to <span class="hlt">approximate</span>, as much as possible, nonlinear constraints by a lot of easily handled linear constraints which will lead to a well balance between the problem complexity and tractability. Case studies are conducted to demonstrate the proposed methods. Results show that the operation cost can be reduced effectively compared with the case without considering the demand correlation. PMID:24757433</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/24757433','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/24757433"><span>An <span class="hlt">approximation</span> <span class="hlt">solution</span> to refinery crude oil scheduling problem with demand uncertainty using joint constrained programming.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Duan, Qianqian; Yang, Genke; Xu, Guanglin; Pan, Changchun</p> <p>2014-01-01</p> <p>This paper is devoted to develop an <span class="hlt">approximation</span> method for scheduling refinery crude oil operations by taking into consideration the demand uncertainty. In the stochastic model the demand uncertainty is modeled as random variables which follow a joint multivariate distribution with a specific correlation structure. Compared to deterministic models in existing works, the stochastic model can be more practical for optimizing crude oil operations. Using joint chance constraints, the demand uncertainty is treated by specifying proximity level on the satisfaction of product demands. However, the joint chance constraints usually hold strong nonlinearity and consequently, it is still hard to handle it directly. In this paper, an <span class="hlt">approximation</span> method combines a relax-and-tight technique to <span class="hlt">approximately</span> transform the joint chance constraints to a serial of parameterized linear constraints so that the complicated problem can be attacked iteratively. The basic idea behind this approach is to <span class="hlt">approximate</span>, as much as possible, nonlinear constraints by a lot of easily handled linear constraints which will lead to a well balance between the problem complexity and tractability. Case studies are conducted to demonstrate the proposed methods. Results show that the operation cost can be reduced effectively compared with the case without considering the demand correlation.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=4112111','PMC'); return false;" href="https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=4112111"><span>Two <span class="hlt">approximations</span> for the geometric model of signal amplification in an electron-multiplying charge-coupled device detector</span></a></p> <p><a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pmc">PubMed Central</a></p> <p>Chao, Jerry; Ram, Sripad; Ward, E. Sally; Ober, Raimund J.</p> <p>2014-01-01</p> <p>The extraction of information from images acquired under low light conditions represents a common task in diverse disciplines. In single molecule microscopy, for example, techniques for superresolution image reconstruction depend on the <span class="hlt">accurate</span> estimation of the locations of individual particles from generally low light images. In order to estimate a quantity of interest with high accuracy, however, an appropriate model for the image data is needed. To this end, we previously introduced a data model for an image that is acquired using the electron-multiplying charge-coupled device (EMCCD) detector, a technology of choice for low light imaging due to its ability to amplify weak signals significantly above its readout noise floor. Specifically, we proposed the use of a geometrically multiplied branching process to model the EMCCD detector’s stochastic signal amplification. Geometric multiplication, however, can be computationally expensive and challenging to work with analytically. We therefore describe here two <span class="hlt">approximations</span> for geometric multiplication that can be used instead. The high gain <span class="hlt">approximation</span> is appropriate when a high level of signal amplification is used, a scenario which corresponds to the typical usage of an EMCCD detector. It is an <span class="hlt">accurate</span> <span class="hlt">approximation</span> that is computationally more efficient, and can be used to perform maximum likelihood estimation on EMCCD image data. In contrast, the Gaussian <span class="hlt">approximation</span> is applicable at all levels of signal amplification, but is only <span class="hlt">accurate</span> when the initial signal to be amplified is relatively large. As we demonstrate, it can importantly facilitate the analysis of an information-theoretic quantity called the noise coefficient. PMID:25075263</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19820021683','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19820021683"><span>Burgers <span class="hlt">approximation</span> for two-dimensional flow past an ellipse</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Dorrepaal, J. M.</p> <p>1982-01-01</p> <p>A linearization of the Navier-Stokes equation due to Burgers in which vorticity is transported by the velocity field corresponding to continuous potential flow is examined. The governing equations are solved exactly for the two dimensional steady flow past an ellipse of arbitrary aspect ratio. The requirement of no slip along the surface of the ellipse results in an infinite algebraic system of linear equations for coefficients appearing in the <span class="hlt">solution</span>. The system is truncated at a point which gives reliable results for Reynolds numbers R in the range 0 R 5. Predictions of the Burgers <span class="hlt">approximation</span> regarding separation, drag and boundary layer behavior are investigated. In particular, Burgers linearization gives drag coefficients which are closer to observed experimental values than those obtained from Oseen's <span class="hlt">approximation</span>. In the special case of flow past a circular cylinder, Burgers <span class="hlt">approximation</span> predicts a boundary layer whose thickness is roughly proportional to R-1/2.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19900003958','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19900003958"><span>Corrections to the thin wall <span class="hlt">approximation</span> in general relativity</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Garfinkle, David; Gregory, Ruth</p> <p>1989-01-01</p> <p>The question is considered whether the thin wall formalism of Israel applies to the gravitating domain walls of a lambda phi(exp 4) theory. The coupled Einstein-scalar equations that describe the thick gravitating wall are expanded in powers of the thickness of the wall. The <span class="hlt">solutions</span> of the zeroth order equations reproduce the results of the usual Israel thin wall <span class="hlt">approximation</span> for domain walls. The <span class="hlt">solutions</span> of the first order equations provide corrections to the expressions for the stress-energy of the wall and to the Israel thin wall equations. The modified thin wall equations are then used to treat the motion of spherical and planar domain walls.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://www.dtic.mil/docs/citations/ADA136324','DTIC-ST'); return false;" href="http://www.dtic.mil/docs/citations/ADA136324"><span>Structure and Stability of Finite Dimensional <span class="hlt">Approximations</span> for Functional Differential Equations.</span></a></p> <p><a target="_blank" href="http://www.dtic.mil/">DTIC Science & Technology</a></p> <p></p> <p>1983-10-01</p> <p><span class="hlt">approximating</span> the <span class="hlt">solution</span> of the algebraic Riccati equation associated with a retarded system. However, there remained one open problem in the...theory much more elegant and efficient (see e.g. BERNIER- MANITIUS ( 3 ], MANITIUS (14], DELFOUR-MANITIUS (71). They have led to a number of new results...characteristic function of the interval I. It is well known that equation (2.1) admits a unique <span class="hlt">solution</span>2 n 12 x() e L 2o-h,-;iUn I W [0,_: 3 n ] for every</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2006JPhCS..46..200C','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2006JPhCS..46..200C"><span>Petascale self-consistent electromagnetic computations using scalable and <span class="hlt">accurate</span> algorithms for complex structures</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Cary, John R.; Abell, D.; Amundson, J.; Bruhwiler, D. L.; Busby, R.; Carlsson, J. A.; Dimitrov, D. A.; Kashdan, E.; Messmer, P.; Nieter, C.; Smithe, D. N.; Spentzouris, P.; Stoltz, P.; Trines, R. M.; Wang, H.; Werner, G. R.</p> <p>2006-09-01</p> <p>As the size and cost of particle accelerators escalate, high-performance computing plays an increasingly important role; optimization through <span class="hlt">accurate</span>, detailed computermodeling increases performance and reduces costs. But consequently, computer simulations face enormous challenges. Early <span class="hlt">approximation</span> methods, such as expansions in distance from the design orbit, were unable to supply detailed <span class="hlt">accurate</span> results, such as in the computation of wake fields in complex cavities. Since the advent of message-passing supercomputers with thousands of processors, earlier <span class="hlt">approximations</span> are no longer necessary, and it is now possible to compute wake fields, the effects of dampers, and self-consistent dynamics in cavities <span class="hlt">accurately</span>. In this environment, the focus has shifted towards the development and implementation of algorithms that scale to large numbers of processors. So-called charge-conserving algorithms evolve the electromagnetic fields without the need for any global solves (which are difficult to scale up to many processors). Using cut-cell (or embedded) boundaries, these algorithms can simulate the fields in complex accelerator cavities with curved walls. New implicit algorithms, which are stable for any time-step, conserve charge as well, allowing faster simulation of structures with details small compared to the characteristic wavelength. These algorithmic and computational advances have been implemented in the VORPAL7 Framework, a flexible, object-oriented, massively parallel computational application that allows run-time assembly of algorithms and objects, thus composing an application on the fly.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2016JCoPh.326..569M','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2016JCoPh.326..569M"><span>Semi-implicit iterative methods for low Mach number turbulent reacting flows: Operator splitting versus <span class="hlt">approximate</span> factorization</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>MacArt, Jonathan F.; Mueller, Michael E.</p> <p>2016-12-01</p> <p>Two formally second-order <span class="hlt">accurate</span>, semi-implicit, iterative methods for the <span class="hlt">solution</span> of scalar transport-reaction equations are developed for Direct Numerical Simulation (DNS) of low Mach number turbulent reacting flows. The first is a monolithic scheme based on a linearly implicit midpoint method utilizing an <span class="hlt">approximately</span> factorized exact Jacobian of the transport and reaction operators. The second is an operator splitting scheme based on the Strang splitting approach. The accuracy properties of these schemes, as well as their stability, cost, and the effect of chemical mechanism size on relative performance, are assessed in two one-dimensional test configurations comprising an unsteady premixed flame and an unsteady nonpremixed ignition, which have substantially different Damköhler numbers and relative stiffness of transport to chemistry. All schemes demonstrate their formal order of accuracy in the fully-coupled convergence tests. Compared to a (non-)factorized scheme with a diagonal <span class="hlt">approximation</span> to the chemical Jacobian, the monolithic, factorized scheme using the exact chemical Jacobian is shown to be both more stable and more economical. This is due to an improved convergence rate of the iterative procedure, and the difference between the two schemes in convergence rate grows as the time step increases. The stability properties of the Strang splitting scheme are demonstrated to outpace those of Lie splitting and monolithic schemes in simulations at high Damköhler number; however, in this regime, the monolithic scheme using the <span class="hlt">approximately</span> factorized exact Jacobian is found to be the most economical at practical CFL numbers. The performance of the schemes is further evaluated in a simulation of a three-dimensional, spatially evolving, turbulent nonpremixed planar jet flame.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2010JPhA...43X2001I','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2010JPhA...43X2001I"><span>FAST TRACK COMMUNICATION Time-dependent exact <span class="hlt">solutions</span> of the nonlinear Kompaneets equation</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Ibragimov, N. H.</p> <p>2010-12-01</p> <p>Time-dependent exact <span class="hlt">solutions</span> of the Kompaneets photon diffusion equation are obtained for several <span class="hlt">approximations</span> of this equation. One of the <span class="hlt">approximations</span> describes the case when the induced scattering is dominant. In this case, the Kompaneets equation has an additional symmetry which is used for constructing some exact <span class="hlt">solutions</span> as group invariant <span class="hlt">solutions</span>.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2018JEI....27a3011L','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2018JEI....27a3011L"><span>Local facet <span class="hlt">approximation</span> for image stitching</span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Li, Jing; Lai, Shiming; Liu, Yu; Wang, Zhengming; Zhang, Maojun</p> <p>2018-01-01</p> <p>Image stitching aims at eliminating multiview parallax and generating a seamless panorama given a set of input images. This paper proposes a local adaptive stitching method, which could achieve both <span class="hlt">accurate</span> and robust image alignments across the whole panorama. A transformation estimation model is introduced by <span class="hlt">approximating</span> the scene as a combination of neighboring facets. Then, the local adaptive stitching field is constructed using a series of linear systems of the facet parameters, which enables the parallax handling in three-dimensional space. We also provide a concise but effective global projectivity preserving technique that smoothly varies the transformations from local adaptive to global planar. The proposed model is capable of stitching both normal images and fisheye images. The efficiency of our method is quantitatively demonstrated in the comparative experiments on several challenging cases.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19810007660','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19810007660"><span>Perturbation <span class="hlt">solutions</span> of combustion instability problems</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Googerdy, A.; Peddieson, J., Jr.; Ventrice, M.</p> <p>1979-01-01</p> <p>A method involving <span class="hlt">approximate</span> modal analysis using the Galerkin method followed by an <span class="hlt">approximate</span> <span class="hlt">solution</span> of the resulting modal-amplitude equations by the two-variable perturbation method (method of multiple scales) is applied to two problems of pressure-sensitive nonlinear combustion instability in liquid-fuel rocket motors. One problem exhibits self-coupled instability while the other exhibits mode-coupled instability. In both cases it is possible to carry out the entire linear stability analysis and significant portions of the nonlinear stability analysis in closed form. In the problem of self-coupled instability the nonlinear stability boundary and <span class="hlt">approximate</span> forms of the limit-cycle amplitudes and growth and decay rates are determined in closed form while the exact limit-cycle amplitudes and growth and decay rates are found numerically. In the problem of mode-coupled instability the limit-cycle amplitudes are found in closed form while the growth and decay rates are found numerically. The behavior of the <span class="hlt">solutions</span> found by the perturbation method are in agreement with <span class="hlt">solutions</span> obtained using complex numerical methods.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2012EPJB...85..323S','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2012EPJB...85..323S"><span>Quantitative molecular orbital energies within a G0W0 <span class="hlt">approximation</span></span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Sharifzadeh, S.; Tamblyn, I.; Doak, P.; Darancet, P. T.; Neaton, J. B.</p> <p>2012-09-01</p> <p>Using many-body perturbation theory within a G 0 W 0 <span class="hlt">approximation</span>, with a plane wave basis set and using a starting point based on density functional theory within the generalized gradient <span class="hlt">approximation</span>, we explore routes for computing the ionization potential (IP), electron affinity (EA), and fundamental gap of three gas-phase molecules — benzene, thiophene, and (1,4) diamino-benzene — and compare with experiments. We examine the dependence of the IP and fundamental gap on the number of unoccupied states used to represent the dielectric function and the self energy, as well as the dielectric function plane-wave cutoff. We find that with an effective completion strategy for <span class="hlt">approximating</span> the unoccupied subspace, and a well converged dielectric function kinetic energy cutoff, the computed IPs and EAs are in excellent quantitative agreement with available experiment (within 0.2 eV), indicating that a one-shot G 0 W 0 approach can be very <span class="hlt">accurate</span> for calculating addition/removal energies of small organic molecules.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/20160013222','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/20160013222"><span>Optics of Water Microdroplets with Soot Inclusions: Exact Versus <span class="hlt">Approximate</span> Results</span></a></p> <p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p> <p>Liu, Li; Mishchenko, Michael I.</p> <p>2016-01-01</p> <p>We use the recently generalized version of the multi-sphere superposition T-matrix method (STMM) to compute the scattering and absorption properties of microscopic water droplets contaminated by black carbon. The soot material is assumed to be randomly distributed throughout the droplet interior in the form of numerous small spherical inclusions. Our numerically-exact STMM results are compared with <span class="hlt">approximate</span> ones obtained using the Maxwell-Garnett effective-medium <span class="hlt">approximation</span> (MGA) and the Monte Carlo ray-tracing <span class="hlt">approximation</span> (MCRTA). We show that the popular MGA can be used to calculate the droplet optical cross sections, single-scattering albedo, and asymmetry parameter provided that the soot inclusions are quasi-uniformly distributed throughout the droplet interior, but can fail in computations of the elements of the scattering matrix depending on the volume fraction of soot inclusions. The integral radiative characteristics computed with the MCRTA can deviate more significantly from their exact STMM counterparts, while <span class="hlt">accurate</span> MCRTA computations of the phase function require droplet size parameters substantially exceeding 60.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/biblio/21417246-poisson-nernst-planck-equations-simulating-biomolecular-diffusion-reaction-processes-finite-element-solutions','SCIGOV-STC'); return false;" href="https://www.osti.gov/biblio/21417246-poisson-nernst-planck-equations-simulating-biomolecular-diffusion-reaction-processes-finite-element-solutions"><span>Poisson-Nernst-Planck equations for simulating biomolecular diffusion-reaction processes I: Finite element <span class="hlt">solutions</span></span></a></p> <p><a target="_blank" href="http://www.osti.gov/search">DOE Office of Scientific and Technical Information (OSTI.GOV)</a></p> <p>Lu Benzhuo; Holst, Michael J.; Center for Theoretical Biological Physics, University of California San Diego, La Jolla, CA 92093</p> <p>2010-09-20</p> <p>In this paper we developed <span class="hlt">accurate</span> finite element methods for solving 3-D Poisson-Nernst-Planck (PNP) equations with singular permanent charges for simulating electrodiffusion in solvated biomolecular systems. The electrostatic Poisson equation was defined in the biomolecules and in the solvent, while the Nernst-Planck equation was defined only in the solvent. We applied a stable regularization scheme to remove the singular component of the electrostatic potential induced by the permanent charges inside biomolecules, and formulated regular, well-posed PNP equations. An inexact-Newton method was used to solve the coupled nonlinear elliptic equations for the steady problems; while an Adams-Bashforth-Crank-Nicolson method was devised formore » time integration for the unsteady electrodiffusion. We numerically investigated the conditioning of the stiffness matrices for the finite element <span class="hlt">approximations</span> of the two formulations of the Nernst-Planck equation, and theoretically proved that the transformed formulation is always associated with an ill-conditioned stiffness matrix. We also studied the electroneutrality of the <span class="hlt">solution</span> and its relation with the boundary conditions on the molecular surface, and concluded that a large net charge concentration is always present near the molecular surface due to the presence of multiple species of charged particles in the <span class="hlt">solution</span>. The numerical methods are shown to be <span class="hlt">accurate</span> and stable by various test problems, and are applicable to real large-scale biophysical electrodiffusion problems.« less</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=2922884','PMC'); return false;" href="https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=2922884"><span>Poisson-Nernst-Planck Equations for Simulating Biomolecular Diffusion-Reaction Processes I: Finite Element <span class="hlt">Solutions</span></span></a></p> <p><a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pmc">PubMed Central</a></p> <p>Lu, Benzhuo; Holst, Michael J.; McCammon, J. Andrew; Zhou, Y. C.</p> <p>2010-01-01</p> <p>In this paper we developed <span class="hlt">accurate</span> finite element methods for solving 3-D Poisson-Nernst-Planck (PNP) equations with singular permanent charges for electrodiffusion in solvated biomolecular systems. The electrostatic Poisson equation was defined in the biomolecules and in the solvent, while the Nernst-Planck equation was defined only in the solvent. We applied a stable regularization scheme to remove the singular component of the electrostatic potential induced by the permanent charges inside biomolecules, and formulated regular, well-posed PNP equations. An inexact-Newton method was used to solve the coupled nonlinear elliptic equations for the steady problems; while an Adams-Bashforth-Crank-Nicolson method was devised for time integration for the unsteady electrodiffusion. We numerically investigated the conditioning of the stiffness matrices for the finite element <span class="hlt">approximations</span> of the two formulations of the Nernst-Planck equation, and theoretically proved that the transformed formulation is always associated with an ill-conditioned stiffness matrix. We also studied the electroneutrality of the <span class="hlt">solution</span> and its relation with the boundary conditions on the molecular surface, and concluded that a large net charge concentration is always present near the molecular surface due to the presence of multiple species of charged particles in the <span class="hlt">solution</span>. The numerical methods are shown to be <span class="hlt">accurate</span> and stable by various test problems, and are applicable to real large-scale biophysical electrodiffusion problems. PMID:21709855</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/21709855','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/21709855"><span>Poisson-Nernst-Planck Equations for Simulating Biomolecular Diffusion-Reaction Processes I: Finite Element <span class="hlt">Solutions</span>.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Lu, Benzhuo; Holst, Michael J; McCammon, J Andrew; Zhou, Y C</p> <p>2010-09-20</p> <p>In this paper we developed <span class="hlt">accurate</span> finite element methods for solving 3-D Poisson-Nernst-Planck (PNP) equations with singular permanent charges for electrodiffusion in solvated biomolecular systems. The electrostatic Poisson equation was defined in the biomolecules and in the solvent, while the Nernst-Planck equation was defined only in the solvent. We applied a stable regularization scheme to remove the singular component of the electrostatic potential induced by the permanent charges inside biomolecules, and formulated regular, well-posed PNP equations. An inexact-Newton method was used to solve the coupled nonlinear elliptic equations for the steady problems; while an Adams-Bashforth-Crank-Nicolson method was devised for time integration for the unsteady electrodiffusion. We numerically investigated the conditioning of the stiffness matrices for the finite element <span class="hlt">approximations</span> of the two formulations of the Nernst-Planck equation, and theoretically proved that the transformed formulation is always associated with an ill-conditioned stiffness matrix. We also studied the electroneutrality of the <span class="hlt">solution</span> and its relation with the boundary conditions on the molecular surface, and concluded that a large net charge concentration is always present near the molecular surface due to the presence of multiple species of charged particles in the <span class="hlt">solution</span>. The numerical methods are shown to be <span class="hlt">accurate</span> and stable by various test problems, and are applicable to real large-scale biophysical electrodiffusion problems.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/24875786','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/24875786"><span>Optimal sparse <span class="hlt">approximation</span> with integrate and fire neurons.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Shapero, Samuel; Zhu, Mengchen; Hasler, Jennifer; Rozell, Christopher</p> <p>2014-08-01</p> <p>Sparse <span class="hlt">approximation</span> is a hypothesized coding strategy where a population of sensory neurons (e.g. V1) encodes a stimulus using as few active neurons as possible. We present the Spiking LCA (locally competitive algorithm), a rate encoded Spiking Neural Network (SNN) of integrate and fire neurons that calculate sparse <span class="hlt">approximations</span>. The Spiking LCA is designed to be equivalent to the nonspiking LCA, an analog dynamical system that converges on a ℓ(1)-norm sparse <span class="hlt">approximations</span> exponentially. We show that the firing rate of the Spiking LCA converges on the same <span class="hlt">solution</span> as the analog LCA, with an error inversely proportional to the sampling time. We simulate in NEURON a network of 128 neuron pairs that encode 8 × 8 pixel image patches, demonstrating that the network converges to nearly optimal encodings within 20 ms of biological time. We also show that when using more biophysically realistic parameters in the neurons, the gain function encourages additional ℓ(0)-norm sparsity in the encoding, relative both to ideal neurons and digital solvers.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2014JCAP...06..054F','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2014JCAP...06..054F"><span>Swiss-cheese models and the Dyer-Roeder <span class="hlt">approximation</span></span></a></p> <p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p> <p>Fleury, Pierre</p> <p>2014-06-01</p> <p>In view of interpreting the cosmological observations precisely, especially when they involve narrow light beams, it is crucial to understand how light propagates in our statistically homogeneous, clumpy, Universe. Among the various approaches to tackle this issue, Swiss-cheese models propose an inhomogeneous spacetime geometry which is an exact <span class="hlt">solution</span> of Einstein's equation, while the Dyer-Roeder <span class="hlt">approximation</span> deals with inhomogeneity in an effective way. In this article, we demonstrate that the distance-redshift relation of a certain class of Swiss-cheese models is the same as the one predicted by the Dyer-Roeder approach, at a well-controlled level of <span class="hlt">approximation</span>. Both methods are therefore equivalent when applied to the interpretation of, e.g., supernova obervations. The proof relies on completely analytical arguments, and is illustrated by numerical results.</p> </li> <li> <p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/26807042','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/26807042"><span>Verification of cardiac mechanics software: benchmark problems and <span class="hlt">solutions</span> for testing active and passive material behaviour.</span></a></p> <p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p> <p>Land, Sander; Gurev, Viatcheslav; Arens, Sander; Augustin, Christoph M; Baron, Lukas; Blake, Robert; Bradley, Chris; Castro, Sebastian; Crozier, Andrew; Favino, Marco; Fastl, Thomas E; Fritz, Thomas; Gao, Hao; Gizzi, Alessio; Griffith, Boyce E; Hurtado, Daniel E; Krause, Rolf; Luo, Xiaoyu; Nash, Martyn P; Pezzuto, Simone; Plank, Gernot; Rossi, Simone; Ruprecht, Daniel; Seemann, Gunnar; Smith, Nicolas P; Sundnes, Joakim; Rice, J Jeremy; Trayanova, Natalia; Wang, Dafang; Jenny Wang, Zhinuo; Niederer, Steven A</p> <p>2015-12-08</p> <p>Models of cardiac mechanics are increasingly used to investigate cardiac physiology. These models are characterized by a high level of complexity, including the particular anisotropic material properties of biological tissue and the actively contracting material. A large number of independent simulation codes have been developed, but a consistent way of verifying the accuracy and replicability of simulations is lacking. To aid in the verification of current and future cardiac mechanics solvers, this study provides three benchmark problems for cardiac mechanics. These benchmark problems test the ability to <span class="hlt">accurately</span> simulate pressure-type forces that depend on the deformed objects geometry, anisotropic and spatially varying material properties similar to those seen in the left ventricle and active contractile forces. The benchmark was solved by 11 different groups to generate consensus <span class="hlt">solutions</span>, with typical differences in higher-resolution <span class="hlt">solutions</span> at <span class="hlt">approximately</span> 0.5%, and consistent results between linear, quadratic and cubic finite elements as well as different approaches to simulating incompressible materials. Online tools and <span class="hlt">solutions</span> are made available to allow these tests to be effectively used in verification of future cardiac mechanics software.</p> </li> </ol> <div class="pull-right"> <ul class="pagination"> <li><a href="#" onclick='return showDiv("page_1");'>«</a></li> <li><a href="#" onclick='return showDiv("page_21");'>21</a></li> <li><a href="#" onclick='return showDiv("page_22");'>22</a></li> <li><a href="#" onclick='return showDiv("page_23");'>23</a></li> <li><a href="#" onclick='return showDiv("page_24");'>24</a></li> <li class="active"><span>25</span></li> <li><a href="#" onclick='return showDiv("page_25");'>»</a></li> </ul> </div> </div><!-- col-sm-12 --> </div><!-- row --> </div><!-- page_25 --> <div class="footer-extlink text-muted" style="margin-bottom:1rem; text-align:center;">Some links on this page may take you to non-federal websites. Their policies may differ from this site.</div> </div><!-- container --> <footer><a id="backToTop" href="#top"> </a><nav><a id="backToTop" href="#top"> </a><ul class="links"><a id="backToTop" href="#top"> </a><li><a id="backToTop" href="#top"></a><a href="/sitemap.html">Site Map</a></li> <li><a href="/members/index.html">Members Only</a></li> <li><a href="/website-policies.html">Website Policies</a></li> <li><a href="https://doe.responsibledisclosure.com/hc/en-us" target="_blank">Vulnerability Disclosure Program</a></li> <li><a href="/contact.html">Contact Us</a></li> </ul> <div class="small">Science.gov is maintained by the U.S. Department of Energy's <a href="https://www.osti.gov/" target="_blank">Office of Scientific and Technical Information</a>, in partnership with <a href="https://www.cendi.gov/" target="_blank">CENDI</a>.</div> </nav> </footer> <script type="text/javascript"><!-- // var lastDiv = ""; function showDiv(divName) { // hide last div if (lastDiv) { document.getElementById(lastDiv).className = "hiddenDiv"; } //if value of the box is not nothing and an object with that name exists, then change the class if (divName && document.getElementById(divName)) { document.getElementById(divName).className = "visibleDiv"; lastDiv = divName; } } //--> </script> <script> /** * Function that tracks a click on an outbound link in Google Analytics. * This function takes a valid URL string as an argument, and uses that URL string * as the event label. */ var trackOutboundLink = function(url,collectionCode) { try { h = window.open(url); setTimeout(function() { ga('send', 'event', 'topic-page-click-through', collectionCode, url); }, 1000); } catch(err){} }; </script> <!-- Google Analytics --> <script> (function(i,s,o,g,r,a,m){i['GoogleAnalyticsObject']=r;i[r]=i[r]||function(){ (i[r].q=i[r].q||[]).push(arguments)},i[r].l=1*new Date();a=s.createElement(o), m=s.getElementsByTagName(o)[0];a.async=1;a.src=g;m.parentNode.insertBefore(a,m) })(window,document,'script','//www.google-analytics.com/analytics.js','ga'); ga('create', 'UA-1122789-34', 'auto'); ga('send', 'pageview'); </script> <!-- End Google Analytics --> <script> showDiv('page_1') </script> </body> </html>