Efficient algorithms for function approximation with piecewise linear sigmoidal networks.
Hush, D R; Horne, B
1998-01-01
This paper presents a computationally efficient algorithm for function approximation with piecewise linear sigmoidal nodes. A one hidden layer network is constructed one node at a time using the well-known method of fitting the residual. The task of fitting an individual node is accomplished using a new algorithm that searches for the best fit by solving a sequence of quadratic programming problems. This approach offers significant advantages over derivative-based search algorithms (e.g., backpropagation and its extensions). Unique characteristics of this algorithm include: finite step convergence, a simple stopping criterion, solutions that are independent of initial conditions, good scaling properties and a robust numerical implementation. Empirical results are included to illustrate these characteristics.
Phase patterns in finite oscillator networks with insights from the piecewise linear approximation
NASA Astrophysics Data System (ADS)
Goldstein, Daniel
2015-03-01
Recent experiments on spatially extend arrays of droplets containing Belousov-Zhabotinsky reactants have shown a rich variety of spatio-temporal patterns. Motivated by this experimental set up, we study a simple model of chemical oscillators in the highly nonlinear excitable regime in order to gain insight into the mechanism giving rise to the observed multistable attractors. When coupled, these two attractors have different preferred phase synchronizations, leading to complex behavior. We study rings of coupled oscillators and observe a rich array of oscillating patterns. We combine Turing analysis and a piecewise linear approximation to better understand the observed patterns.
NASA Astrophysics Data System (ADS)
Ibáñez, Javier; Hernández, Vicente
2011-03-01
Differential Matrix Riccati Equations (DMREs) appear in several branches of science such as applied physics and engineering. For example, these equations play a fundamental role in control theory, optimal control, filtering and estimation, decoupling and order reduction, etc. In this paper a new method based on a theorem proved in this paper is described for solving DMREs by a piecewise-linearized approach. This method is applied for developing two block-oriented algorithms based on diagonal Padé approximants. MATLAB versions of the above algorithms are developed, comparing, under equal conditions, accuracy and computational costs with other piecewise-linearized algorithms implemented by the authors. Experimental results show the advantages of solving stiff or non-stiff DMREs by the implemented algorithms.
Optimal piecewise locally linear modeling
NASA Astrophysics Data System (ADS)
Harris, Chris J.; Hong, Xia; Feng, M.
1999-03-01
Associative memory networks such as Radial Basis Functions, Neurofuzzy and Fuzzy Logic used for modelling nonlinear processes suffer from the curse of dimensionality (COD), in that as the input dimension increases the parameterization, computation cost, training data requirements, etc. increase exponentially. Here a new algorithm is introduced for the construction of a Delaunay input space partitioned optimal piecewise locally linear models to overcome the COD as well as generate locally linear models directly amenable to linear control and estimation algorithms. The training of the model is configured as a new mixture of experts network with a new fast decision rule derived using convex set theory. A very fast simulated reannealing (VFSR) algorithm is utilized to search a global optimal solution of the Delaunay input space partition. A benchmark non-linear time series is used to demonstrate the new approach.
Nonlinear Modeling by Assembling Piecewise Linear Models
NASA Technical Reports Server (NTRS)
Yao, Weigang; Liou, Meng-Sing
2013-01-01
To preserve nonlinearity of a full order system over a parameters range of interest, we propose a simple modeling approach by assembling a set of piecewise local solutions, including the first-order Taylor series terms expanded about some sampling states. The work by Rewienski and White inspired our use of piecewise linear local solutions. The assembly of these local approximations is accomplished by assigning nonlinear weights, through radial basis functions in this study. The efficacy of the proposed procedure is validated for a two-dimensional airfoil moving at different Mach numbers and pitching motions, under which the flow exhibits prominent nonlinear behaviors. All results confirm that our nonlinear model is accurate and stable for predicting not only aerodynamic forces but also detailed flowfields. Moreover, the model is robustness-accurate for inputs considerably different from the base trajectory in form and magnitude. This modeling preserves nonlinearity of the problems considered in a rather simple and accurate manner.
Tamborrino, Massimiliano
2016-06-01
The first passage time density of a diffusion process to a time varying threshold is of primary interest in different fields. Here, we consider a Brownian motion in presence of an exponentially decaying threshold to model the neuronal spiking activity. Since analytical expressions of the first passage time density are not available, we propose to approximate the curved boundary by means of a continuous two-piecewise linear threshold. Explicit expressions for the first passage time density towards the new boundary are provided. First, we introduce different approximating linear thresholds. Then, we describe how to choose the optimal one minimizing the distance to the curved boundary, and hence the error in the corresponding passage time density. Theoretical means, variances and coefficients of variation given by our method are compared with empirical quantities from simulated data. Moreover, a further comparison with firing statistics derived under the assumption of a small amplitude of the time-dependent change in the threshold, is also carried out. Finally, maximum likelihood and moment estimators of the parameters of the model are derived and applied on simulated data. PMID:27106189
Tamborrino, Massimiliano
2016-06-01
The first passage time density of a diffusion process to a time varying threshold is of primary interest in different fields. Here, we consider a Brownian motion in presence of an exponentially decaying threshold to model the neuronal spiking activity. Since analytical expressions of the first passage time density are not available, we propose to approximate the curved boundary by means of a continuous two-piecewise linear threshold. Explicit expressions for the first passage time density towards the new boundary are provided. First, we introduce different approximating linear thresholds. Then, we describe how to choose the optimal one minimizing the distance to the curved boundary, and hence the error in the corresponding passage time density. Theoretical means, variances and coefficients of variation given by our method are compared with empirical quantities from simulated data. Moreover, a further comparison with firing statistics derived under the assumption of a small amplitude of the time-dependent change in the threshold, is also carried out. Finally, maximum likelihood and moment estimators of the parameters of the model are derived and applied on simulated data.
Mutual inductance between piecewise-linear loops
NASA Astrophysics Data System (ADS)
Cristina Barroso, Ana; Silva, J. P.
2013-11-01
We consider a current-carrying wire loop made out of linear segments of arbitrary sizes and directions in three-dimensional space. We develop expressions to calculate its vector potential and magnetic field at all points in space. We then calculate the mutual inductance between two such (non-intersecting) piecewise-linear loops. As simple applications, we consider in detail the mutual inductance between two square wires of equal length that either lie in the same plane or lie in parallel horizontal planes with their centers on the same vertical axis. Our expressions can also be used to obtain approximations to the mutual inductance between wires of arbitrary three-dimensional shapes.
Transforming the canonical piecewise-linear model into a smooth-piecewise representation.
Jimenez-Fernandez, Victor M; Jimenez-Fernandez, Maribel; Vazquez-Leal, Hector; Muñoz-Aguirre, Evodio; Cerecedo-Nuñez, Hector H; Filobello-Niño, Uriel A; Castro-Gonzalez, Francisco J
2016-01-01
A smoothed representation (based on natural exponential and logarithmic functions) for the canonical piecewise-linear model, is presented. The result is a completely differentiable formulation that exhibits interesting properties, like preserving the parameters of the original piecewise-linear model in such a way that they can be directly inherited to the smooth model in order to determine their parameters, the capability of controlling not only the smoothness grade, but also the approximation accuracy at specific breakpoint locations, a lower or equal overshooting for high order derivatives in comparison with other approaches, and the additional advantage of being expressed in a reduced mathematical form with only two types of inverse functions (logarithmic and exponential). By numerical simulation examples, this proposal is verified and well-illustrated. PMID:27652185
Slope Estimation in Noisy Piecewise Linear Functions✩
Ingle, Atul; Bucklew, James; Sethares, William; Varghese, Tomy
2014-01-01
This paper discusses the development of a slope estimation algorithm called MAPSlope for piecewise linear data that is corrupted by Gaussian noise. The number and locations of slope change points (also known as breakpoints) are assumed to be unknown a priori though it is assumed that the possible range of slope values lies within known bounds. A stochastic hidden Markov model that is general enough to encompass real world sources of piecewise linear data is used to model the transitions between slope values and the problem of slope estimation is addressed using a Bayesian maximum a posteriori approach. The set of possible slope values is discretized, enabling the design of a dynamic programming algorithm for posterior density maximization. Numerical simulations are used to justify choice of a reasonable number of quantization levels and also to analyze mean squared error performance of the proposed algorithm. An alternating maximization algorithm is proposed for estimation of unknown model parameters and a convergence result for the method is provided. Finally, results using data from political science, finance and medical imaging applications are presented to demonstrate the practical utility of this procedure. PMID:25419020
Piecewise linear and Boolean models of chemical reaction networks.
Veliz-Cuba, Alan; Kumar, Ajit; Josić, Krešimir
2014-12-01
Models of biochemical networks are frequently complex and high-dimensional. Reduction methods that preserve important dynamical properties are therefore essential for their study. Interactions in biochemical networks are frequently modeled using Hill functions ([Formula: see text]). Reduced ODEs and Boolean approximations of such model networks have been studied extensively when the exponent [Formula: see text] is large. However, while the case of small constant [Formula: see text] appears in practice, it is not well understood. We provide a mathematical analysis of this limit and show that a reduction to a set of piecewise linear ODEs and Boolean networks can be mathematically justified. The piecewise linear systems have closed-form solutions that closely track those of the fully nonlinear model. The simpler, Boolean network can be used to study the qualitative behavior of the original system. We justify the reduction using geometric singular perturbation theory and compact convergence, and illustrate the results in network models of a toggle switch and an oscillator.
Piecewise linear dimension reduction for nonnegative data
NASA Astrophysics Data System (ADS)
Shen, Bin; Wang, Qifan; Allebach, Jan P.
2015-03-01
In past decade, the increasing popularity of imaging devices, especially smart phones, has led to a great increase in the amount of visual data. The rapidly increasing large scale data pose challenges to the storage and computational resources, and make many computer vision and pattern recognition tasks prohibitively expensive. Dimension reduction techniques explore hidden structures of the original high dimensional data and learn new low dimensional representation to alleviate the challenges. Popular dimension reduction techniques, such as PCA and NMF, do an efficient linear mapping to low dimensional space, while nonlinear techniques overcomes the limitation of linearity at the cost of expensive computational cost (e.g. computing the pairwise distance to find the geodesic distance). In this paper, a piecewise linear dimension reduction technique with global consistency and smoothness constraint is proposed to overcome the restriction of linearity at relatively low cost. Extensive experimental results show that the proposed methods outperform the linear method in the scenario of clustering both consistently and significantly.
A prototype piecewise-linear dynamic attenuator
NASA Astrophysics Data System (ADS)
Hsieh, Scott S.; Peng, Mark V.; May, Christopher A.; Shunhavanich, Picha; Fleischmann, Dominik; Pelc, Norbert J.
2016-07-01
The piecewise-linear dynamic attenuator has been proposed as a mechanism in CT scanning for personalizing the x-ray illumination on a patient- and application-specific basis. Previous simulations have shown benefits in image quality, scatter, and dose objectives. We report on the first prototype implementation. This prototype is reduced in scale and speed and is integrated into a tabletop CT system with a smaller field of view (25 cm) and longer scan time (42 s) compared to a clinical system. Stainless steel wedges were machined and affixed to linear actuators, which were in turn held secure by a frame built using rapid prototyping technologies. The actuators were computer-controlled, with characteristic noise of about 100 microns. Simulations suggest that in a clinical setting, the impact of actuator noise could lead to artifacts of only 1 HU. Ring artifacts were minimized by careful design of the wedges. A water beam hardening correction was applied and the scan was collimated to reduce scatter. We scanned a 16 cm water cylinder phantom as well as an anthropomorphic pediatric phantom. The artifacts present in reconstructed images are comparable to artifacts normally seen with this tabletop system. Compared to a flat-field reference scan, increased detectability at reduced dose is shown and streaking is reduced. Artifacts are modest in our images and further refinement is possible. Issues of mechanical speed and stability in the challenging clinical CT environment will be addressed in a future design.
A prototype piecewise-linear dynamic attenuator.
Hsieh, Scott S; Peng, Mark V; May, Christopher A; Shunhavanich, Picha; Fleischmann, Dominik; Pelc, Norbert J
2016-07-01
The piecewise-linear dynamic attenuator has been proposed as a mechanism in CT scanning for personalizing the x-ray illumination on a patient- and application-specific basis. Previous simulations have shown benefits in image quality, scatter, and dose objectives. We report on the first prototype implementation. This prototype is reduced in scale and speed and is integrated into a tabletop CT system with a smaller field of view (25 cm) and longer scan time (42 s) compared to a clinical system. Stainless steel wedges were machined and affixed to linear actuators, which were in turn held secure by a frame built using rapid prototyping technologies. The actuators were computer-controlled, with characteristic noise of about 100 microns. Simulations suggest that in a clinical setting, the impact of actuator noise could lead to artifacts of only 1 HU. Ring artifacts were minimized by careful design of the wedges. A water beam hardening correction was applied and the scan was collimated to reduce scatter. We scanned a 16 cm water cylinder phantom as well as an anthropomorphic pediatric phantom. The artifacts present in reconstructed images are comparable to artifacts normally seen with this tabletop system. Compared to a flat-field reference scan, increased detectability at reduced dose is shown and streaking is reduced. Artifacts are modest in our images and further refinement is possible. Issues of mechanical speed and stability in the challenging clinical CT environment will be addressed in a future design. PMID:27284705
Pitch contour stylization using an optimal piecewise polynomial approximation
Ghosh, Prasanta Kumar; Narayanan, Shrikanth S.
2014-01-01
We propose a dynamic programming (DP) based piecewise polynomial approximation of discrete data such that the L2 norm of the approximation error is minimized. We apply this technique for the stylization of speech pitch contour. Objective evaluation verifies that the DP based technique indeed yields minimum mean square error (MSE) compared to other approximation methods. Subjective evaluation reveals that the quality of the synthesized speech using stylized pitch contour obtained by the DP method is almost identical to that of the original speech. PMID:24453471
Convergence of multipoint Pade approximants of piecewise analytic functions
Buslaev, Viktor I
2013-02-28
The behaviour as n{yields}{infinity} of multipoint Pade approximants to a function which is (piecewise) holomorphic on a union of finitely many continua is investigated. The convergence of multipoint Pade approximants is proved for a function which extends holomorphically from these continua to a union of domains whose boundaries have a certain symmetry property. An analogue of Stahl's theorem is established for two-point Pade approximants to a pair of functions, either of which is a multivalued analytic function with finitely many branch points. Bibliography: 11 titles.
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.
Selmic, R R; Lewis, F L
2002-01-01
One of the most important properties of neural nets (NNs) for control purposes is the universal approximation property. Unfortunately,, this property is generally proven for continuous functions. In most real industrial control systems there are nonsmooth functions (e.g., piecewise continuous) for which approximation results in the literature are sparse. Examples include friction, deadzone, backlash, and so on. It is found that attempts to approximate piecewise continuous functions using smooth activation functions require many NN nodes and many training iterations, and still do not yield very good results. Therefore, a novel neural-network structure is given for approximation of piecewise continuous functions of the sort that appear in friction, deadzone, backlash, and other motion control actuator nonlinearities. The novel NN consists of neurons having standard sigmoid activation functions, plus some additional neurons having a special class of nonsmooth activation functions termed "jump approximation basis function." Two types of nonsmooth jump approximation basis functions are determined- a polynomial-like basis and a sigmoid-like basis. This modified NN with additional neurons having "jump approximation" activation functions can approximate any piecewise continuous function with discontinuities at a finite number of known points. Applications of the new NN structure are made to rigid-link robotic systems with friction nonlinearities. Friction is a nonlinear effect that can limit the performance of industrial control systems; it occurs in all mechanical systems and therefore is unavoidable in control systems. It can cause tracking errors, limit cycles, and other undesirable effects. Often, inexact friction compensation is used with standard adaptive techniques that require models that are linear in the unknown parameters. It is shown here how a certain class of augmented NN, capable of approximating piecewise continuous functions, can be used for friction
Progressive Image Coding by Hierarchical Linear Approximation.
ERIC Educational Resources Information Center
Wu, Xiaolin; Fang, Yonggang
1994-01-01
Proposes a scheme of hierarchical piecewise linear approximation as an adaptive image pyramid. A progressive image coder comes naturally from the proposed image pyramid. The new pyramid is semantically more powerful than regular tessellation but syntactically simpler than free segmentation. This compromise between adaptability and complexity…
Virtual Estimator for Piecewise Linear Systems Based on Observability Analysis
Morales-Morales, Cornelio; Adam-Medina, Manuel; Cervantes, Ilse; Vela-Valdés and, Luis G.; García Beltrán, Carlos Daniel
2013-01-01
This article proposes a virtual sensor for piecewise linear systems based on observability analysis that is in function of a commutation law related with the system's outpu. This virtual sensor is also known as a state estimator. Besides, it presents a detector of active mode when the commutation sequences of each linear subsystem are arbitrary and unknown. For the previous, this article proposes a set of virtual estimators that discern the commutation paths of the system and allow estimating their output. In this work a methodology in order to test the observability for piecewise linear systems with discrete time is proposed. An academic example is presented to show the obtained results. PMID:23447007
Bailey, T S; Adams, M L; Yang, B; Zika, M R
2005-07-15
We develop a piecewise linear (PWL) Galerkin finite element spatial discretization for the multi-dimensional radiation diffusion equation. It uses piecewise linear weight and basis functions in the finite element approximation, and it can be applied on arbitrary polygonal (2D) or polyhedral (3D) grids. We show that this new PWL method gives solutions comparable to those from Palmer's finite-volume method. However, since the PWL method produces a symmetric positive definite coefficient matrix, it should be substantially more computationally efficient than Palmer's method, which produces an asymmetric matrix. We conclude that the Galerkin PWL method is an attractive option for solving diffusion equations on unstructured grids.
The Piecewise Linear Reactive Flow Rate Model
Vitello, P; Souers, P C
2005-07-22
Conclusions are: (1) Early calibrations of the Piece Wise Linear reactive flow model have shown that it allows for very accurate agreement with data for a broad range of detonation wave strengths. (2) The ability to vary the rate at specific pressures has shown that corner turning involves competition between the strong wave that travels roughly in a straight line and growth at low pressure of a new wave that turns corners sharply. (3) The inclusion of a low pressure de-sensitization rate is essential to preserving the dead zone at large times as is observed.
Analytical results for resonance and runup in piecewise linear bathymetries
NASA Astrophysics Data System (ADS)
Fuentes, Mauricio; Riquelme, Sebastián; Ruiz, Javier; Campos, Jaime
2015-04-01
A general method of solution for the runup evolution and some analytical results concerning a more general bathymetry than a canonical sloping beach model are presented. We studied theoretically the water wave elevation and runup generated on a continuous piecewise linear bathymetry, by solving analytically the linear shallow water wave equations in the 1+1 dimensional case. Non-horizontal linear segments are assumed and we develop an specific matrix propagator scheme, similar to the ones used in the propagation of elastic seismic wave fields in layered media, to obtain an exact integral form for the runup. A general closed expression for the maximum runup was computed analytically via the Cauchy's residue Theorem for an incident solitary wave and isosceles leading-depression N-wave in the case of n+1 linear segments. It is already known that maximum run-up strongly depends only on the closest slope to the shore, although this has not been mathematically demonstrated yet for arbitraries bathymetries. Analytical and numerical verifications were done to check the validity of the asymptotic maximum runup and we provided the mathematical and bathymetrical conditions that must be satisfied by the model to obtain correct asymptotic solutions. We applied our model to study the runup evolution on a more realistic bathymetry than a canonical sloping beach model. The seabed in a Chilean subduction zone was approximated - from the trench to the shore - by two linear segments adjusting the continental slope and shelf. Assuming an incident solitary wave, the two linear segment bathymetry generates a larger runup than the simple sloping beach model. We also discussed about the differences in the runup evolution computed numerically from incident leading-depression and -elevation isosceles N-waves. In the latter case, the water elevation at the shore shows a symmetrical behavior in terms of theirs waveforms. Finally, we applied our solution to study the resonance effects due to
The feasibility of a piecewise-linear dynamic bowtie filter
Hsieh, Scott S.; Pelc, Norbert J.
2013-03-15
Purpose: The prepatient attenuator (or 'bowtie filter') in CT is used to modulate the flux as a function of fan angle of the x-ray beam incident on the patient. Traditional, static bowtie filters are tailored only for very generic scans and for the average patient. The authors propose a design for a dynamic bowtie that can produce a time-dependent piecewise-linear attenuation profile. This dynamic bowtie may reduce dynamic range, dose or scatter, but in this work they focus on its ability to reduce dynamic range, which may be particularly important for systems employing photon-counting detectors. Methods: The dynamic bowtie is composed of a set of triangular wedges. Each wedge is independently moved in order to produce a time-dependent piecewise-linear attenuation profile. Simulations of the bowtie are conducted to estimate the dynamic range reduction in six clinical datasets. The control of the dynamic bowtie is determined by solving a convex optimization problem, and the dose is estimated using Monte Carlo techniques. Beam hardening artifacts are also simulated. Results: The dynamic range is reduced by factors ranging from 2.4 to 27 depending on the part of the body studied. With a dynamic range minimization objective, the dose to the patient can be reduced from 6% to 33% while maintaining peak image noise. Further reduction in dose may be possible with a specific dose reduction objective. Beam hardening artifacts are suppressed with a two-pass algorithm. Conclusions: A dynamic bowtie producing a time-dependent, piecewise-linear attenuation profile is possible and can be used to modulate the flux of the scanner to the imaging task. Initial simulations show a large reduction in dynamic range. Several other applications are possible.
Neighboring extremal guidance for systems with a piecewise linear control
NASA Technical Reports Server (NTRS)
Hull, David G.; Helfrich, Clifford E.
1991-01-01
The neighboring extremal feedback control law is developed for systems with a piecewise linear control for the case where the optimal control is obtained by nonlinear programming techniques. To develop the control perturbation for a given deviation from the nominal path, the second variation is minimized subject to the constraint that the final conditions be satisfied. This process leads to a feedback relationship between the control perturbation and the measured deviation from the nominal state. A simple example, the lunar launch problem, is used to demonstrate the validity of the guidance law. For model errors on the order of 5 percent, the results indicate that 5 percent errors occur in the final conditions.
Piecewise linear manifolds: Einstein metrics and Ricci flows
NASA Astrophysics Data System (ADS)
Schrader, Robert
2016-05-01
This article provides an attempt to extend concepts from the theory of Riemannian manifolds to piecewise linear (p.l.) spaces. In particular we propose an analogue of the Ricci tensor, which we give the name of an Einstein vector field. On a given set of p.l. spaces we define and discuss (normalized) Einstein flows. p.l. Einstein metrics are defined and examples are provided. Criteria for flows to approach Einstein metrics are formulated. Second variations of the total scalar curvature at a specific Einstein space are calculated. Dedicated to Ludwig Faddeev on the occasion of his 80th birthday.
Understanding cardiac alternans: A piecewise linear modeling framework
NASA Astrophysics Data System (ADS)
Thul, R.; Coombes, S.
2010-12-01
Cardiac alternans is a beat-to-beat alternation in action potential duration (APD) and intracellular calcium (Ca2+) cycling seen in cardiac myocytes under rapid pacing that is believed to be a precursor to fibrillation. The cellular mechanisms of these rhythms and the coupling between cellular Ca2+ and voltage dynamics have been extensively studied leading to the development of a class of physiologically detailed models. These have been shown numerically to reproduce many of the features of myocyte response to pacing, including alternans, and have been analyzed mathematically using various approximation techniques that allow for the formulation of a low dimensional map to describe the evolution of APDs. The seminal work by Shiferaw and Karma is of particular interest in this regard [Shiferaw, Y. and Karma, A., "Turing instability mediated by voltage and calcium diffusion in paced cardiac cells," Proc. Natl. Acad. Sci. U.S.A. 103, 5670-5675 (2006)]. Here, we establish that the key dynamical behaviors of the Shiferaw-Karma model are arranged around a set of switches. These are shown to be the main elements for organizing the nonlinear behavior of the model. Exploiting this observation, we show that a piecewise linear caricature of the Shiferaw-Karma model, with a set of appropriate switching manifolds, can be constructed that preserves the physiological interpretation of the original model while being amenable to a systematic mathematical analysis. In illustration of this point, we formulate the dynamics of Ca2+ cycling (in response to pacing) and compute the properties of periodic orbits in terms of a stroboscopic map that can be constructed without approximation. Using this, we show that alternans emerge via a period-doubling instability and track this bifurcation in terms of physiologically important parameters. We also show that when coupled to a spatially extended model for Ca2+ transport, the model supports spatially varying patterns of alternans. We analyze
Bailey, Teresa S. Adams, Marvin L. Yang, Brian Zika, Michael R.
2008-04-01
We develop a piecewise linear (PWL) Galerkin finite element spatial discretization for the multi-dimensional radiation diffusion equation. It uses recently introduced piecewise linear weight and basis functions in the finite element approximation and it can be applied on arbitrary polygonal (2D) or polyhedral (3D) grids. We first demonstrate some analytical properties of the PWL method and perform a simple mode analysis to compare the PWL method with Palmer's vertex-centered finite-volume method and with a bilinear continuous finite element method. We then show that this new PWL method gives solutions comparable to those from Palmer's. However, since the PWL method produces a symmetric positive-definite coefficient matrix, it should be substantially more computationally efficient than Palmer's method, which produces an asymmetric matrix. We conclude that the Galerkin PWL method is an attractive option for solving diffusion equations on unstructured grids.
Lee, Lee-Min; Jean, Fu-Rong
2016-08-01
The hidden Markov models have been widely applied to systems with sequential data. However, the conditional independence of the state outputs will limit the output of a hidden Markov model to be a piecewise constant random sequence, which is not a good approximation for many real processes. In this paper, a high-order hidden Markov model for piecewise linear processes is proposed to better approximate the behavior of a real process. A parameter estimation method based on the expectation-maximization algorithm was derived for the proposed model. Experiments on speech recognition of noisy Mandarin digits were conducted to examine the effectiveness of the proposed method. Experimental results show that the proposed method can reduce the recognition error rate compared to a baseline hidden Markov model. PMID:27586781
Chaotic dynamics and diffusion in a piecewise linear equation
Shahrear, Pabel; Glass, Leon; Edwards, Rod
2015-03-15
Genetic interactions are often modeled by logical networks in which time is discrete and all gene activity states update simultaneously. However, there is no synchronizing clock in organisms. An alternative model assumes that the logical network is preserved and plays a key role in driving the dynamics in piecewise nonlinear differential equations. We examine dynamics in a particular 4-dimensional equation of this class. In the equation, two of the variables form a negative feedback loop that drives a second negative feedback loop. By modifying the original equations by eliminating exponential decay, we generate a modified system that is amenable to detailed analysis. In the modified system, we can determine in detail the Poincaré (return) map on a cross section to the flow. By analyzing the eigenvalues of the map for the different trajectories, we are able to show that except for a set of measure 0, the flow must necessarily have an eigenvalue greater than 1 and hence there is sensitive dependence on initial conditions. Further, there is an irregular oscillation whose amplitude is described by a diffusive process that is well-modeled by the Irwin-Hall distribution. There is a large class of other piecewise-linear networks that might be analyzed using similar methods. The analysis gives insight into possible origins of chaotic dynamics in periodically forced dynamical systems.
Chaotic dynamics and diffusion in a piecewise linear equation
NASA Astrophysics Data System (ADS)
Shahrear, Pabel; Glass, Leon; Edwards, Rod
2015-03-01
Genetic interactions are often modeled by logical networks in which time is discrete and all gene activity states update simultaneously. However, there is no synchronizing clock in organisms. An alternative model assumes that the logical network is preserved and plays a key role in driving the dynamics in piecewise nonlinear differential equations. We examine dynamics in a particular 4-dimensional equation of this class. In the equation, two of the variables form a negative feedback loop that drives a second negative feedback loop. By modifying the original equations by eliminating exponential decay, we generate a modified system that is amenable to detailed analysis. In the modified system, we can determine in detail the Poincaré (return) map on a cross section to the flow. By analyzing the eigenvalues of the map for the different trajectories, we are able to show that except for a set of measure 0, the flow must necessarily have an eigenvalue greater than 1 and hence there is sensitive dependence on initial conditions. Further, there is an irregular oscillation whose amplitude is described by a diffusive process that is well-modeled by the Irwin-Hall distribution. There is a large class of other piecewise-linear networks that might be analyzed using similar methods. The analysis gives insight into possible origins of chaotic dynamics in periodically forced dynamical systems.
Elasticity in Amorphous Solids: Nonlinear or Piecewise Linear?
Dubey, Awadhesh K; Procaccia, Itamar; Shor, Carmel A B Z; Singh, Murari
2016-02-26
Quasistatic strain-controlled measurements of stress versus strain curves in macroscopic amorphous solids result in a nonlinear-looking curve that ends up either in mechanical collapse or in a steady state with fluctuations around a mean stress that remains constant with increasing strain. It is therefore very tempting to fit a nonlinear expansion of the stress in powers of the strain. We argue here that at low temperatures the meaning of such an expansion needs to be reconsidered. We point out the enormous difference between quenched and annealed averages of the stress versus strain curves and propose that a useful description of the mechanical response is given by a stress (or strain) -dependent shear modulus for which a theoretical evaluation exists. The elastic response is piecewise linear rather than nonlinear.
An improved piecewise linear chaotic map based image encryption algorithm.
Hu, Yuping; Zhu, Congxu; Wang, Zhijian
2014-01-01
An image encryption algorithm based on improved piecewise linear chaotic map (MPWLCM) model was proposed. The algorithm uses the MPWLCM to permute and diffuse plain image simultaneously. Due to the sensitivity to initial key values, system parameters, and ergodicity in chaotic system, two pseudorandom sequences are designed and used in the processes of permutation and diffusion. The order of processing pixels is not in accordance with the index of pixels, but it is from beginning or end alternately. The cipher feedback was introduced in diffusion process. Test results and security analysis show that not only the scheme can achieve good encryption results but also its key space is large enough to resist against brute attack.
A Neurodynamic Approach for Real-Time Scheduling via Maximizing Piecewise Linear Utility.
Guo, Zhishan; Baruah, Sanjoy K
2016-02-01
In this paper, we study a set of real-time scheduling problems whose objectives can be expressed as piecewise linear utility functions. This model has very wide applications in scheduling-related problems, such as mixed criticality, response time minimization, and tardiness analysis. Approximation schemes and matrix vectorization techniques are applied to transform scheduling problems into linear constraint optimization with a piecewise linear and concave objective; thus, a neural network-based optimization method can be adopted to solve such scheduling problems efficiently. This neural network model has a parallel structure, and can also be implemented on circuits, on which the converging time can be significantly limited to meet real-time requirements. Examples are provided to illustrate how to solve the optimization problem and to form a schedule. An approximation ratio bound of 0.5 is further provided. Experimental studies on a large number of randomly generated sets suggest that our algorithm is optimal when the set is nonoverloaded, and outperforms existing typical scheduling strategies when there is overload. Moreover, the number of steps for finding an approximate solution remains at the same level when the size of the problem (number of jobs within a set) increases. PMID:26336153
Optimal Piecewise Linear Basis Functions in Two Dimensions
Brooks III, E D; Szoke, A
2009-01-26
We use a variational approach to optimize the center point coefficients associated with the piecewise linear basis functions introduced by Stone and Adams [1], for polygonal zones in two Cartesian dimensions. Our strategy provides optimal center point coefficients, as a function of the location of the center point, by minimizing the error induced when the basis function interpolation is used for the solution of the time independent diffusion equation within the polygonal zone. By using optimal center point coefficients, one expects to minimize the errors that occur when these basis functions are used to discretize diffusion equations, or transport equations in optically thick zones (where they approach the solution of the diffusion equation). Our optimal center point coefficients satisfy the requirements placed upon the basis functions for any location of the center point. We also find that the location of the center point can be optimized, but this requires numerical calculations. Curiously, the optimum center point location is independent of the values of the dependent variable on the corners only for quadrilaterals.
Robust estimation of motion blur kernel using a piecewise-linear model.
Sungchan Oh; Gyeonghwan Kim
2014-03-01
Blur kernel estimation is a crucial step in the deblurring process for images. Estimation of the kernel, especially in the presence of noise, is easily perturbed, and the quality of the resulting deblurred images is hence degraded. Since every motion blur in a single exposure image can be represented by 2D parametric curves, we adopt a piecewise-linear model to approximate the curves for the reliable blur kernel estimation. The model is found to be an effective tradeoff between flexibility and robustness as it takes advantage of two extremes: (1) the generic model, represented by a discrete 2D function, which has a high degree of freedom (DOF) for the maximum flexibility but suffers from noise and (2) the linear model, which enhances robustness and simplicity but has limited expressiveness due to its low DOF. We evaluate several deblurring methods based on not only the generic model, but also the piecewise-linear model as an alternative. After analyzing the experiment results using real-world images with significant levels of noise and a benchmark data set, we conclude that the proposed model is not only robust with respect to noise, but also flexible in dealing with various types of blur.
Inelastic strain analogy for piecewise linear computation of creep residues in built-up structures
NASA Technical Reports Server (NTRS)
Jenkins, Jerald M.
1987-01-01
An analogy between inelastic strains caused by temperature and those caused by creep is presented in terms of isotropic elasticity. It is shown how the theoretical aspects can be blended with existing finite-element computer programs to exact a piecewise linear solution. The creep effect is determined by using the thermal stress computational approach, if appropriate alterations are made to the thermal expansion of the individual elements. The overall transient solution is achieved by consecutive piecewise linear iterations. The total residue caused by creep is obtained by accumulating creep residues for each iteration and then resubmitting the total residues for each element as an equivalent input. A typical creep law is tested for incremental time convergence. The results indicate that the approach is practical, with a valid indication of the extent of creep after approximately 20 hr of incremental time. The general analogy between body forces and inelastic strain gradients is discussed with respect to how an inelastic problem can be worked as an elastic problem.
Dose reduction using a dynamic, piecewise-linear attenuator
Hsieh, Scott S.; Fleischmann, Dominik; Pelc, Norbert J.
2014-02-15
Purpose: The authors recently proposed a dynamic, prepatient x-ray attenuator capable of producing a piecewise-linear attenuation profile customized to each patient and viewing angle. This attenuator was intended to reduce scatter-to-primary ratio (SPR), dynamic range, and dose by redistributing flux. In this work the authors tested the ability of the attenuator to reduce dose and SPR in simulations. Methods: The authors selected four clinical applications, including routine full field-of-view scans of the thorax and abdomen, and targeted reconstruction tasks for an abdominal aortic aneurysm and the pancreas. Raw data were estimated by forward projection of the image volume datasets. The dynamic attenuator was controlled to reduce dose while maintaining peak variance by solving a convex optimization problem, assuminga priori knowledge of the patient anatomy. In targeted reconstruction tasks, the noise in specific regions was given increased weighting. A system with a standard attenuator (or “bowtie filter”) was used as a reference, and used either convex optimized tube current modulation (TCM) or a standard TCM heuristic. The noise of the scan was determined analytically while the dose was estimated using Monte Carlo simulations. Scatter was also estimated using Monte Carlo simulations. The sensitivity of the dynamic attenuator to patient centering was also examined by shifting the abdomen in 2 cm intervals. Results: Compared to a reference system with optimized TCM, use of the dynamic attenuator reduced dose by about 30% in routine scans and 50% in targeted scans. Compared to the TCM heuristics which are typically used withouta priori knowledge, the dose reduction is about 50% for routine scans. The dynamic attenuator gives the ability to redistribute noise and variance and produces more uniform noise profiles than systems with a conventional bowtie filter. The SPR was also modestly reduced by 10% in the thorax and 24% in the abdomen. Imaging with the dynamic
NASA Astrophysics Data System (ADS)
Saito, Asaki; Yasutomi, Shin-ichi; Tamura, Jun-ichi; Ito, Shunji
2015-06-01
We introduce a true orbit generation method enabling exact simulations of dynamical systems defined by arbitrary-dimensional piecewise linear fractional maps, including piecewise linear maps, with rational coefficients. This method can generate sufficiently long true orbits which reproduce typical behaviors (inherent behaviors) of these systems, by properly selecting algebraic numbers in accordance with the dimension of the target system, and involving only integer arithmetic. By applying our method to three dynamical systems—that is, the baker's transformation, the map associated with a modified Jacobi-Perron algorithm, and an open flow system—we demonstrate that it can reproduce their typical behaviors that have been very difficult to reproduce with conventional simulation methods. In particular, for the first two maps, we show that we can generate true orbits displaying the same statistical properties as typical orbits, by estimating the marginal densities of their invariant measures. For the open flow system, we show that an obtained true orbit correctly converges to the stable period-1 orbit, which is inherently possessed by the system.
Saito, Asaki; Yasutomi, Shin-ichi; Tamura, Jun-ichi; Ito, Shunji
2015-06-01
We introduce a true orbit generation method enabling exact simulations of dynamical systems defined by arbitrary-dimensional piecewise linear fractional maps, including piecewise linear maps, with rational coefficients. This method can generate sufficiently long true orbits which reproduce typical behaviors (inherent behaviors) of these systems, by properly selecting algebraic numbers in accordance with the dimension of the target system, and involving only integer arithmetic. By applying our method to three dynamical systems-that is, the baker's transformation, the map associated with a modified Jacobi-Perron algorithm, and an open flow system-we demonstrate that it can reproduce their typical behaviors that have been very difficult to reproduce with conventional simulation methods. In particular, for the first two maps, we show that we can generate true orbits displaying the same statistical properties as typical orbits, by estimating the marginal densities of their invariant measures. For the open flow system, we show that an obtained true orbit correctly converges to the stable period-1 orbit, which is inherently possessed by the system.
On the Limit Cycles for a Class of Continuous Piecewise Linear Differential Systems with Three Zones
NASA Astrophysics Data System (ADS)
Lima, Maurício Firmino Silva; Pessoa, Claudio; Pereira, Weber F.
Lima and Llibre [2012] have studied a class of planar continuous piecewise linear vector fields with three zones. Using the Poincaré map, they proved that this class admits always a unique limit cycle, which is hyperbolic. The class studied in [Lima & Llibre, 2012] belongs to a larger set of planar continuous piecewise linear vector fields with three zones that can be separated into four other classes. Here, we consider some of these classes and we prove that some of them always admit a unique limit cycle, which is hyperbolic. However we find a class that does not have limit cycles.
Canards in a minimal piecewise-linear square-wave burster
NASA Astrophysics Data System (ADS)
Desroches, M.; Fernández-García, S.; Krupa, M.
2016-07-01
We construct a piecewise-linear (PWL) approximation of the Hindmarsh-Rose (HR) neuron model that is minimal, in the sense that the vector field has the least number of linearity zones, in order to reproduce all the dynamics present in the original HR model with classical parameter values. This includes square-wave bursting and also special trajectories called canards, which possess long repelling segments and organise the transitions between stable bursting patterns with n and n + 1 spikes, also referred to as spike-adding canard explosions. We propose a first approximation of the smooth HR model, using a continuous PWL system, and show that its fast subsystem cannot possess a homoclinic bifurcation, which is necessary to obtain proper square-wave bursting. We then relax the assumption of continuity of the vector field across all zones, and we show that we can obtain a homoclinic bifurcation in the fast subsystem. We use the recently developed canard theory for PWL systems in order to reproduce the spike-adding canard explosion feature of the HR model as studied, e.g., in Desroches et al., Chaos 23(4), 046106 (2013).
Canards in a minimal piecewise-linear square-wave burster.
Desroches, M; Fernández-García, S; Krupa, M
2016-07-01
We construct a piecewise-linear (PWL) approximation of the Hindmarsh-Rose (HR) neuron model that is minimal, in the sense that the vector field has the least number of linearity zones, in order to reproduce all the dynamics present in the original HR model with classical parameter values. This includes square-wave bursting and also special trajectories called canards, which possess long repelling segments and organise the transitions between stable bursting patterns with n and n + 1 spikes, also referred to as spike-adding canard explosions. We propose a first approximation of the smooth HR model, using a continuous PWL system, and show that its fast subsystem cannot possess a homoclinic bifurcation, which is necessary to obtain proper square-wave bursting. We then relax the assumption of continuity of the vector field across all zones, and we show that we can obtain a homoclinic bifurcation in the fast subsystem. We use the recently developed canard theory for PWL systems in order to reproduce the spike-adding canard explosion feature of the HR model as studied, e.g., in Desroches et al., Chaos 23(4), 046106 (2013). PMID:27475071
Canards in a minimal piecewise-linear square-wave burster.
Desroches, M; Fernández-García, S; Krupa, M
2016-07-01
We construct a piecewise-linear (PWL) approximation of the Hindmarsh-Rose (HR) neuron model that is minimal, in the sense that the vector field has the least number of linearity zones, in order to reproduce all the dynamics present in the original HR model with classical parameter values. This includes square-wave bursting and also special trajectories called canards, which possess long repelling segments and organise the transitions between stable bursting patterns with n and n + 1 spikes, also referred to as spike-adding canard explosions. We propose a first approximation of the smooth HR model, using a continuous PWL system, and show that its fast subsystem cannot possess a homoclinic bifurcation, which is necessary to obtain proper square-wave bursting. We then relax the assumption of continuity of the vector field across all zones, and we show that we can obtain a homoclinic bifurcation in the fast subsystem. We use the recently developed canard theory for PWL systems in order to reproduce the spike-adding canard explosion feature of the HR model as studied, e.g., in Desroches et al., Chaos 23(4), 046106 (2013).
Piecewise linear hypersurfaces using the marching cubes algorithm
NASA Astrophysics Data System (ADS)
Roberts, Jonathan C.; Hill, Steve
1999-03-01
Surface visualization is very important within scientific visualization. The surfaces depict a value of equal density (an isosurface) or display the surrounds of specified objects within the data. Likewise, in two dimensions contour plots may be used to display the information. Thus similarly, in four dimensions hypersurfaces may be formed around hyperobjects. These surfaces (or contours) are often formed from a set of connected triangles (or lines). These piecewise segments represent the simplest non-degenerate object of that dimension and are named simplices. In four dimensions a simplex is represented by a tetrahedron, which is also known as a 3- simplex. Thus, a continuous n dimensional surface may be represented by a lattice of connected n-1 dimensional simplices. This lattice of connected simplices may be calculated over a set of adjacent n dimensional cubes, via for example the Marching Cubes Algorithm. We propose that the methods of this local-cell tiling method may be usefully- applied to four dimensions and potentially to N-dimensions. Thus, we organize the large number of traversal cases and major cases; introduce the notion of a sub-case (that enables the large number of cases to be further reduced); and describe three methods for implementing the Marching Cubes lookup table in four-dimensions.
Mixed-mode oscillations in a stochastic, piecewise-linear system
NASA Astrophysics Data System (ADS)
Simpson, D. J. W.; Kuske, R.
2011-07-01
We analyse a piecewise-linear FitzHugh-Nagumo model. The system exhibits a canard near which both small amplitude and large amplitude periodic orbits exist. The addition of small noise induces mixed-mode oscillations (MMOs) in the vicinity of the canard point. We determine the effect of each model parameter on the stochastically driven MMOs. In particular we show that any parameter variation (such as a modification of the piecewise-linear function in the model) that leaves the ratio of noise amplitude to time-scale separation unchanged typically has little effect on the width of the interval of the primary bifurcation parameter over which MMOs occur. In that sense, the MMOs are robust. Furthermore, we show that the piecewise-linear model exhibits MMOs more readily than the classical FitzHugh-Nagumo model for which a cubic polynomial is the only nonlinearity. By studying a piecewise-linear model, we are able to explain results using analytical expressions and compare these with numerical investigations.
Piecewise-homogeneous model for electron side injection into linear plasma waves
NASA Astrophysics Data System (ADS)
Golovanov, A. A.; Kostyukov, I. Yu.
2016-09-01
An analytical piecewise-homogeneous model for electron side injection into linear plasma waves is developed. The dynamics of transverse betatron oscillations are studied. Based on the characteristics of the transversal motion the longitudinal motion of electrons is described. The electron parameters for which the electron trapping and subsequent acceleration are possible are estimated. The analytical results are verified by numerical simulations in the scope of the piecewise-homogeneous model. The results predicted by this model are also compared to the results given by a more realistic inhomogeneous model.
Sun Wei; Huang, Guo H.; Lv Ying; Li Gongchen
2012-06-15
Highlights: Black-Right-Pointing-Pointer Inexact piecewise-linearization-based fuzzy flexible programming is proposed. Black-Right-Pointing-Pointer It's the first application to waste management under multiple complexities. Black-Right-Pointing-Pointer It tackles nonlinear economies-of-scale effects in interval-parameter constraints. Black-Right-Pointing-Pointer It estimates costs more accurately than the linear-regression-based model. Black-Right-Pointing-Pointer Uncertainties are decreased and more satisfactory interval solutions are obtained. - Abstract: To tackle nonlinear economies-of-scale (EOS) effects in interval-parameter constraints for a representative waste management problem, an inexact piecewise-linearization-based fuzzy flexible programming (IPFP) model is developed. In IPFP, interval parameters for waste amounts and transportation/operation costs can be quantified; aspiration levels for net system costs, as well as tolerance intervals for both capacities of waste treatment facilities and waste generation rates can be reflected; and the nonlinear EOS effects transformed from objective function to constraints can be approximated. An interactive algorithm is proposed for solving the IPFP model, which in nature is an interval-parameter mixed-integer quadratically constrained programming model. To demonstrate the IPFP's advantages, two alternative models are developed to compare their performances. One is a conventional linear-regression-based inexact fuzzy programming model (IPFP2) and the other is an IPFP model with all right-hand-sides of fussy constraints being the corresponding interval numbers (IPFP3). The comparison results between IPFP and IPFP2 indicate that the optimized waste amounts would have the similar patterns in both models. However, when dealing with EOS effects in constraints, the IPFP2 may underestimate the net system costs while the IPFP can estimate the costs more accurately. The comparison results between IPFP and IPFP3 indicate
Akaishi, A; Shudo, A
2009-12-01
We investigate the stickiness of the two-dimensional piecewise linear map with a family of marginal unstable periodic orbits (FMUPOs), and show that a series of unstable periodic orbits accumulating to FMUPOs plays a significant role to give rise to the power law correlation of trajectories. We can explicitly specify the sticky zone in which unstable periodic orbits whose stability increases algebraically exist, and find that there exists a hierarchy in accumulating periodic orbits. In particular, the periodic orbits with linearly increasing stability play the role of fundamental cycles as in the hyperbolic systems, which allows us to apply the method of cycle expansion. We also study the recurrence time distribution, especially discussing the position and size of the recurrence region. Following the definition adopted in one-dimensional maps, we show that the recurrence time distribution has an exponential part in the short time regime and an asymptotic power law part. The analysis on the crossover time T(c)(*) between these two regimes implies T(c)(*) approximately -log[micro(R)] where micro(R) denotes the area of the recurrence region.
NASA Astrophysics Data System (ADS)
Vörös, Jozef
2016-07-01
The paper deals with the parameter identification of cascade nonlinear dynamic systems with noninvertible piecewise linear input nonlinearities and backlash output nonlinearities. Application of the key term separation principle provides special expressions for the corresponding nonlinear model description that are linear in parameters. A least squares based iterative technique allows estimation of all the model parameters based on measured input/output data. Simulation studies illustrate the feasibility of proposed identification method.
Jump bifurcations in some degenerate planar piecewise linear differential systems with three zones
NASA Astrophysics Data System (ADS)
Euzébio, Rodrigo; Pazim, Rubens; Ponce, Enrique
2016-06-01
We consider continuous piecewise-linear differential systems with three zones where the central one is degenerate, that is, the determinant of its linear part vanishes. By moving one parameter which is associated to the equilibrium position, we detect some new bifurcations exhibiting jump transitions both in the equilibrium location and in the appearance of limit cycles. In particular, we introduce the scabbard bifurcation, characterized by the birth of a limit cycle from a continuum of equilibrium points. Some of the studied bifurcations are detected, after an appropriate choice of parameters, in a piecewise linear Morris-Lecar model for the activity of a single neuron activity, which is usually considered as a reduction of the celebrated Hodgkin-Huxley equations.
A piecewise linear state variable technique for real time propulsion system simulation
NASA Technical Reports Server (NTRS)
Mihaloew, J. R.; Roth, S. P.
1982-01-01
The emphasis on increased aircraft and propulsion control system integration and piloted simulation has created a need for higher fidelity real time dynamic propulsion models. A real time propulsion system modeling technique which satisfies this need and which provides the capabilities needed to evaluate propulsion system performance and aircraft system interaction on manned flight simulators was developed and demonstrated using flight simulator facilities at NASA Ames. A piecewise linear state variable technique is used. This technique provides the system accuracy, stability and transient response required for integrated aircraft and propulsion control system studies. The real time dynamic model includes the detail and flexibility required for the evaluation of critical control parameters and propulsion component limits over a limited flight envelope. The model contains approximately 7.0 K bytes of in-line computational code and 14.7 K of block data. It has an 8.9 ms cycle time on a Xerox Sigma 9 computer. A Pegasus-Harrier propulsion system was used as a baseline for developing the mathematical modeling and simulation technique. A hydromechanical and water injection control system was also simulated. The model was programmed for interfacing with a Harrier aircraft simulation at NASA Ames. Descriptions of the real time methodology and model capabilities are presented.
A Piecewise Linear State Variable Technique for Real Time Propulsion System Simulation
NASA Technical Reports Server (NTRS)
Mihaloew, J. R.; Roth, S. P.
1982-01-01
The emphasis on increased aircraft and propulsion control system integration and piloted simulation has created a need for higher fidelity real time dynamic propulsion models. A real time propulsion system modeling technique which satisfies this need and which provides the capabilities needed to evaluate propulsion system performance and aircraft system interaction on manned flight simulators was developed and demonstrated using flight simulator facilities at NASA Ames. A piecewise linear state variable technique is used. This technique provides the system accuracy, stability and transient response required for integrated aircraft and propulsion control system studies. The real time dynamic model includes the detail and flexibility required for the evaluation of critical control parameters and propulsion component limits over a limited flight envelope. The model contains approximately 7.0 K bytes of in-line computational code and 14.7 K of block data. It has an 8.9 ms cycle time on a Xerox Sigma 9 computer. A Pegasus-Harrier propulsion system was used as a baseline for developing the mathematical modeling and simulation technique. A hydromechanical and water injection control system was also simulated. The model was programmed for interfacing with a Harrier aircraft simulation at NASA Ames. Descriptions of the real time methodology and model capabilities are presented.
Piecewise polynomial chaos expansion with an application to brake squeal of a linear brake system
NASA Astrophysics Data System (ADS)
Sarrouy, E.; Dessombz, O.; Sinou, J.-J.
2013-02-01
This paper proposes numerical developments based on polynomial chaos (PC) expansions to process stochastic eigenvalue problems efficiently. These developments are applied to the problem of linear stability calculations for a simplified brake system: the stability of a finite element model of a brake is investigated when its friction coefficient or the contact stiffness are modeled as random parameters. Getting rid of the statistical point of view of the PC method but keeping the principle of a polynomial decomposition of eigenvalues and eigenvectors, the stochastic space is decomposed into several elements to realize a low degree piecewise polynomial approximation of these quantities. An approach relying on continuation principles is compared to the classical dichotomy method to build the partition. Moreover, a criterion for testing accuracy of the decomposition over each cell of the partition without requiring evaluation of exact eigenmodes is proposed and implemented. Several random distributions are tested, including a uniform-like law for description of friction coefficient variation. Results are compared to Monte Carlo simulations so as to determine the method accuracy and efficiency. Some general rules relative to the influence of the friction coefficient or the contact stiffness are also inferred from these calculations.
NASA Astrophysics Data System (ADS)
Shimizu, Kuniyasu; Inaba, Naohiko
2016-03-01
This study investigates mixed-mode oscillations (MMOs) generated by weakly driven piecewise-linear Bonhoeffer-van der Pol and Fitzhugh-Nagumo dynamics. Such a simple piecewise-linear oscillator can generate extremely complex MMO bifurcations such as mixed-mode oscillation-incrementing bifurcations (MMOIBs) and intermittently chaotic MMOs. These remarkable bifurcations are confirmed using explicit solutions of the piecewise-linear differential equation. Moreover, Lorenz plots are introduced, which strongly suggest that MMOIBs occur successively many times, and show that each MMO sequence is surrounded by chaos.
Vazquez-Leal, H; Jimenez-Fernandez, V M; Benhammouda, B; Filobello-Nino, U; Sarmiento-Reyes, A; Ramirez-Pinero, A; Marin-Hernandez, A; Huerta-Chua, J
2014-01-01
We present a homotopy continuation method (HCM) for finding multiple operating points of nonlinear circuits composed of devices modelled by using piecewise linear (PWL) representations. We propose an adaptation of the modified spheres path tracking algorithm to trace the homotopy trajectories of PWL circuits. In order to assess the benefits of this proposal, four nonlinear circuits composed of piecewise linear modelled devices are analysed to determine their multiple operating points. The results show that HCM can find multiple solutions within a single homotopy trajectory. Furthermore, we take advantage of the fact that homotopy trajectories are PWL curves meant to replace the multidimensional interpolation and fine tuning stages of the path tracking algorithm with a simple and highly accurate procedure based on the parametric straight line equation.
Vazquez-Leal, H.; Jimenez-Fernandez, V. M.; Benhammouda, B.; Filobello-Nino, U.; Sarmiento-Reyes, A.; Ramirez-Pinero, A.; Marin-Hernandez, A.; Huerta-Chua, J.
2014-01-01
We present a homotopy continuation method (HCM) for finding multiple operating points of nonlinear circuits composed of devices modelled by using piecewise linear (PWL) representations. We propose an adaptation of the modified spheres path tracking algorithm to trace the homotopy trajectories of PWL circuits. In order to assess the benefits of this proposal, four nonlinear circuits composed of piecewise linear modelled devices are analysed to determine their multiple operating points. The results show that HCM can find multiple solutions within a single homotopy trajectory. Furthermore, we take advantage of the fact that homotopy trajectories are PWL curves meant to replace the multidimensional interpolation and fine tuning stages of the path tracking algorithm with a simple and highly accurate procedure based on the parametric straight line equation. PMID:25184157
Bailey, T S; Chang, J H; Warsa, J S; Adams, M L
2010-12-22
We present a new spatial discretization of the discrete-ordinates transport equation in two-dimensional Cartesian (X-Y) geometry for arbitrary polygonal meshes. The discretization is a discontinuous finite element method (DFEM) that utilizes piecewise bi-linear (PWBL) basis functions, which are formally introduced in this paper. We also present a series of numerical results on quadrilateral and polygonal grids and compare these results to a variety of other spatial discretizations that have been shown to be successful on these grid types. Finally, we note that the properties of the PWBL basis functions are such that the leading-order piecewise bi-linear discontinuous finite element (PWBLD) solution will satisfy a reasonably accurate diffusion discretization in the thick diffusion limit, making the PWBLD method a viable candidate for many different classes of transport problems.
Vazquez-Leal, H; Jimenez-Fernandez, V M; Benhammouda, B; Filobello-Nino, U; Sarmiento-Reyes, A; Ramirez-Pinero, A; Marin-Hernandez, A; Huerta-Chua, J
2014-01-01
We present a homotopy continuation method (HCM) for finding multiple operating points of nonlinear circuits composed of devices modelled by using piecewise linear (PWL) representations. We propose an adaptation of the modified spheres path tracking algorithm to trace the homotopy trajectories of PWL circuits. In order to assess the benefits of this proposal, four nonlinear circuits composed of piecewise linear modelled devices are analysed to determine their multiple operating points. The results show that HCM can find multiple solutions within a single homotopy trajectory. Furthermore, we take advantage of the fact that homotopy trajectories are PWL curves meant to replace the multidimensional interpolation and fine tuning stages of the path tracking algorithm with a simple and highly accurate procedure based on the parametric straight line equation. PMID:25184157
The runup of long waves around piecewise linear bathymetries
NASA Astrophysics Data System (ADS)
Kanoglu, Utku
The evolution of waves on beaches is the quintissential problem of coastal engineering. Most practical problems involve directional waveforms with complex spectral distributions. In the last ten years consensus has emerged that certain terminal effects such as coastal flooding and inundation are mainly affected by the infragravity waves, i.e. the long wave part of the incident spectrum. These waves can be described by the shallow-water wave equations, which are also the standard model for tsunamis or tidal waves. Interest in these equations has rekindled because comparisons with both large-scale laboratory data and field data have demonstrated a remarkable and surprising capability to model complex evolution phenomena, and in particular the maximum runup. The maximum runup is arguably the single most important parameter in the design of coastal structures such as seawalls and dikes and for evaluating the inundation potential of tsunamis. A general method for solving developing exact solutions of the shallow-water wave equations is developed for determining the amplification factor of incident long waves as a function of the incident wave characteristics and the topographic variation. This method is then applied to the different ocean topographies composed of linearly varying depth segments and of constant depth segments, also known as composite beaches. Asysmptotic expressions are derived for the runup of solitary waves, and a series of large-scale laboratory experiments was conducted and is described. The analytical results are found in good agreement with the laboratory data for the time histories of free surface elevations and the maximum runup heights. An important result is that the maximum runup on the continental slope and shelf case is governed by the onshore slope, i.e., the slope which includes the initial shoreline, at least for the long waves of tsunami scales. In the last part of the study the evolution of solitary waves around circular islands is
Sun, Wei; Huang, Guo H; Lv, Ying; Li, Gongchen
2012-06-01
To tackle nonlinear economies-of-scale (EOS) effects in interval-parameter constraints for a representative waste management problem, an inexact piecewise-linearization-based fuzzy flexible programming (IPFP) model is developed. In IPFP, interval parameters for waste amounts and transportation/operation costs can be quantified; aspiration levels for net system costs, as well as tolerance intervals for both capacities of waste treatment facilities and waste generation rates can be reflected; and the nonlinear EOS effects transformed from objective function to constraints can be approximated. An interactive algorithm is proposed for solving the IPFP model, which in nature is an interval-parameter mixed-integer quadratically constrained programming model. To demonstrate the IPFP's advantages, two alternative models are developed to compare their performances. One is a conventional linear-regression-based inexact fuzzy programming model (IPFP2) and the other is an IPFP model with all right-hand-sides of fussy constraints being the corresponding interval numbers (IPFP3). The comparison results between IPFP and IPFP2 indicate that the optimized waste amounts would have the similar patterns in both models. However, when dealing with EOS effects in constraints, the IPFP2 may underestimate the net system costs while the IPFP can estimate the costs more accurately. The comparison results between IPFP and IPFP3 indicate that their solutions would be significantly different. The decreased system uncertainties in IPFP's solutions demonstrate its effectiveness for providing more satisfactory interval solutions than IPFP3. Following its first application to waste management, the IPFP can be potentially applied to other environmental problems under multiple complexities. PMID:22370050
Sun, Wei; Huang, Guo H; Lv, Ying; Li, Gongchen
2012-06-01
To tackle nonlinear economies-of-scale (EOS) effects in interval-parameter constraints for a representative waste management problem, an inexact piecewise-linearization-based fuzzy flexible programming (IPFP) model is developed. In IPFP, interval parameters for waste amounts and transportation/operation costs can be quantified; aspiration levels for net system costs, as well as tolerance intervals for both capacities of waste treatment facilities and waste generation rates can be reflected; and the nonlinear EOS effects transformed from objective function to constraints can be approximated. An interactive algorithm is proposed for solving the IPFP model, which in nature is an interval-parameter mixed-integer quadratically constrained programming model. To demonstrate the IPFP's advantages, two alternative models are developed to compare their performances. One is a conventional linear-regression-based inexact fuzzy programming model (IPFP2) and the other is an IPFP model with all right-hand-sides of fussy constraints being the corresponding interval numbers (IPFP3). The comparison results between IPFP and IPFP2 indicate that the optimized waste amounts would have the similar patterns in both models. However, when dealing with EOS effects in constraints, the IPFP2 may underestimate the net system costs while the IPFP can estimate the costs more accurately. The comparison results between IPFP and IPFP3 indicate that their solutions would be significantly different. The decreased system uncertainties in IPFP's solutions demonstrate its effectiveness for providing more satisfactory interval solutions than IPFP3. Following its first application to waste management, the IPFP can be potentially applied to other environmental problems under multiple complexities.
Fast-rolling shutter compensation based on piecewise quadratic approximation of a camera trajectory
NASA Astrophysics Data System (ADS)
Lee, Yun Gu; Kai, Guo
2014-09-01
Rolling shutter effect commonly exists in a video camera or a mobile phone equipped with a complementary metal-oxide semiconductor sensor, caused by a row-by-row exposure mechanism. As video resolution in both spatial and temporal domains increases dramatically, removing rolling shutter effect fast and effectively becomes a challenging problem, especially for devices with limited hardware resources. We propose a fast method to compensate rolling shutter effect, which uses a piecewise quadratic function to approximate a camera trajectory. The duration of a quadratic function in each segment is equal to one frame (or half-frame), and each quadratic function is described by an initial velocity and a constant acceleration. The velocity and acceleration of each segment are estimated using only a few global (or semiglobal) motion vectors, which can be simply predicted from fast motion estimation algorithms. Then geometric image distortion at each scanline is inferred from the predicted camera trajectory for compensation. Experimental results on mobile phones with full-HD video demonstrate that our method can not only be implemented in real time, but also achieve satisfactory visual quality.
On the fold-Hopf bifurcation for continuous piecewise linear differential systems with symmetry.
Llibre, Jaume; Ponce, Enrique; Ros, Javier; Torres, Francisco
2010-09-01
In this paper a partial unfolding for an analog to the fold-Hopf bifurcation in three-dimensional symmetric piecewise linear differential systems is obtained. A particular biparametric family of such systems is studied starting from a very degenerate configuration of nonhyperbolic periodic orbits and looking for the possible bifurcation of limit cycles. It is proved that four limit cycles can coexist after perturbation of the original configuration, and other two limit cycles are conjectured. It is shown that the described bifurcation scenario appears for appropriate values of parameters in the celebrated Chua's oscillator.
NASA Astrophysics Data System (ADS)
Jovanović, Jelena; Denić, Dragan
2016-02-01
A cost-effective method for resolution increase of a two-stage piecewise linear analog-to-digital converter used for sensor linearization is proposed in this paper. In both conversion stages flash analog-to-digital converters are employed. Resolution increase by one bit per conversion stage is performed by introducing one additional comparator in front of each of two flash analog-to-digital converters, while the converters' resolutions remain the same. As a result, the number of employed comparators, as well as the circuit complexity and the power consumption originating from employed comparators are for almost 50 % lower in comparison to the same parameters referring to the linearization circuit of the conventional design and of the same resolution. Since the number of employed comparators is significantly reduced according to the proposed method, special modifications of the linearization circuit are needed in order to properly adjust reference voltages of employed comparators.
CS2: a Piecewise-Linear Model for Large Strain Consolidation
NASA Astrophysics Data System (ADS)
Fox, Patrick J.; Berles, James D.
1997-07-01
This paper presents a piecewise-linear finite-difference model for one-dimensional large strain consolidation called CS2. CS2 is developed using a fixed Eulerian co-ordinate system and constitutive relationships which are defined by discrete data points. The model is dimensionless such that solutions are independent of the initial height of the compressible layer and the absolute magnitude of the hydraulic conductivity of the soil. The capability of CS2 is illustrated using four example problems involving small strain, large strain, self-weight, and non-linear constitutive relationships. In each case, the performance of the model is comparable to other available analytical and numerical solutions. Using CS2, correction factors are developed for the conventional Terzaghi theory which account for the effect of vertical strain on computed values by elapsed time and maximum excess pore pressure during consolidation.
Spectral analysis and an area-preserving extension of a piecewise linear intermittent map
NASA Astrophysics Data System (ADS)
Miyaguchi, Tomoshige; Aizawa, Yoji
2007-06-01
We investigate the spectral properties of a one-dimensional piecewise linear intermittent map, which has not only a marginal fixed point but also a singular structure suppressing injections of the orbits into neighborhoods of the marginal fixed point. We explicitly derive generalized eigenvalues and eigenfunctions of the Frobenius-Perron operator of the map for classes of observables and piecewise constant initial densities, and it is found that the Frobenius-Perron operator has two simple real eigenvalues 1 and λdɛ(-1,0) and a continuous spectrum on the real line [0,1]. From these spectral properties, we also found that this system exhibits a power law decay of correlations. This analytical result is found to be in a good agreement with numerical simulations. Moreover, the system can be extended to an area-preserving invertible map defined on the unit square. This extended system is similar to the baker transformation, but does not satisfy hyperbolicity. A relation between this area-preserving map and a billiard system is also discussed.
NASA Astrophysics Data System (ADS)
Hsieh, Scott S.; Pelc, Norbert J.
2014-06-01
Photon counting x-ray detectors (PCXDs) offer several advantages compared to standard energy-integrating x-ray detectors, but also face significant challenges. One key challenge is the high count rates required in CT. At high count rates, PCXDs exhibit count rate loss and show reduced detective quantum efficiency in signal-rich (or high flux) measurements. In order to reduce count rate requirements, a dynamic beam-shaping filter can be used to redistribute flux incident on the patient. We study the piecewise-linear attenuator in conjunction with PCXDs without energy discrimination capabilities. We examined three detector models: the classic nonparalyzable and paralyzable detector models, and a ‘hybrid’ detector model which is a weighted average of the two which approximates an existing, real detector (Taguchi et al 2011 Med. Phys. 38 1089-102 ). We derive analytic expressions for the variance of the CT measurements for these detectors. These expressions are used with raw data estimated from DICOM image files of an abdomen and a thorax to estimate variance in reconstructed images for both the dynamic attenuator and a static beam-shaping (‘bowtie’) filter. By redistributing flux, the dynamic attenuator reduces dose by 40% without increasing peak variance for the ideal detector. For non-ideal PCXDs, the impact of count rate loss is also reduced. The nonparalyzable detector shows little impact from count rate loss, but with the paralyzable model, count rate loss leads to noise streaks that can be controlled with the dynamic attenuator. With the hybrid model, the characteristic count rates required before noise streaks dominate the reconstruction are reduced by a factor of 2 to 3. We conclude that the piecewise-linear attenuator can reduce the count rate requirements of the PCXD in addition to improving dose efficiency. The magnitude of this reduction depends on the detector, with paralyzable detectors showing much greater benefit than nonparalyzable detectors.
Hsieh, Scott S.; Pelc, Norbert J.
2014-01-01
Photon counting x-ray detectors (PCXDs) offer several advantages compared to standard, energy-integrating x-ray detectors but also face significant challenges. One key challenge is the high count rates required in CT. At high count rates, PCXDs exhibit count rate loss and show reduced detective quantum efficiency in signal-rich (or high flux) measurements. In order to reduce count rate requirements, a dynamic beam-shaping filter can be used to redistribute flux incident on the patient. We study the piecewise-linear attenuator in conjunction with PCXDs without energy discrimination capabilities. We examined three detector models: the classic nonparalyzable and paralyzable detector models, and a “hybrid” detector model which is a weighted average of the two which approximates an existing, real detector (Taguchi et al, Med Phys 2011). We derive analytic expressions for the variance of the CT measurements for these detectors. These expressions are used with raw data estimated from DICOM image files of an abdomen and a thorax to estimate variance in reconstructed images for both the dynamic attenuator and a static beam-shaping (“bowtie”) filter. By redistributing flux, the dynamic attenuator reduces dose by 40% without increasing peak variance for the ideal detector. For non-ideal PCXDs, the impact of count rate loss is also reduced. The nonparalyzable detector shows little impact from count rate loss, but with the paralyzable model, count rate loss leads to noise streaks that can be controlled with the dynamic attenuator. With the hybrid model, the characteristic count rates required before noise streaks dominate the reconstruction are reduced by a factor of two to three. We conclude that the piecewise-linear attenuator can reduce the count rate requirements of the PCXD in addition to improving dose efficiency. The magnitude of this reduction depends on the detector, with paralyzable detectors showing much greater benefit than nonparalyzable detectors. PMID
Linear Approximation SAR Azimuth Processing Study
NASA Technical Reports Server (NTRS)
Lindquist, R. B.; Masnaghetti, R. K.; Belland, E.; Hance, H. V.; Weis, W. G.
1979-01-01
A segmented linear approximation of the quadratic phase function that is used to focus the synthetic antenna of a SAR was studied. Ideal focusing, using a quadratic varying phase focusing function during the time radar target histories are gathered, requires a large number of complex multiplications. These can be largely eliminated by using linear approximation techniques. The result is a reduced processor size and chip count relative to ideally focussed processing and a correspondingly increased feasibility for spaceworthy implementation. A preliminary design and sizing for a spaceworthy linear approximation SAR azimuth processor meeting requirements similar to those of the SEASAT-A SAR was developed. The study resulted in a design with approximately 1500 IC's, 1.2 cubic feet of volume, and 350 watts of power for a single look, 4000 range cell azimuth processor with 25 meters resolution.
Robust chaos and border-collision bifurcations in non-invertible piecewise-linear maps
NASA Astrophysics Data System (ADS)
Kowalczyk, P.
2005-03-01
This paper investigates border-collision bifurcations in piecewise-linear planar maps that are non-invertible in one region. Maps of this type arise as normal forms for grazing-sliding bifurcations in three-dimensional Filippov-type systems. A possible strategy is presented for classifying fixed and period-2 points, that are involved in such bifurcations. This allows one to determine a region of parameter space where a bifurcation leading to chaos might occur. The main part of the paper contains a careful proof of the onset of attractors which are robust to small parameter changes. An intricate structure is revealed of the limiting set on which the attractor lives, consisting of distinct continuous line segments. As parameters are varied, the attractor on the segments can change from being chaotic to periodic. Also, the mechanism by which the number of line segments can change is uncovered.
Goreac, Dan; Serea, Oana-Silvia
2012-10-15
We aim at characterizing domains of attraction for controlled piecewise deterministic processes using an occupational measure formulation and Zubov's approach. Firstly, we provide linear programming (primal and dual) formulations of discounted, infinite horizon control problems for PDMPs. These formulations involve an infinite-dimensional set of probability measures and are obtained using viscosity solutions theory. Secondly, these tools allow to construct stabilizing measures and to avoid the assumption of stability under concatenation for controls. The domain of controllability is then characterized as some level set of a convenient solution of the associated Hamilton-Jacobi integral-differential equation. The theoretical results are applied to PDMPs associated to stochastic gene networks. Explicit computations are given for Cook's model for gene expression.
Dynamical analysis of periodic bursting in piece-wise linear planar neuron model.
Ji, Ying; Zhang, Xiaofang; Liang, Minjie; Hua, Tingting; Wang, Yawei
2015-12-01
A piece-wise linear planar neuron model, namely, two-dimensional McKean model with periodic drive is investigated in this paper. Periodical bursting phenomenon can be observed in the numerical simulations. By assuming the formal solutions associated with different intervals of this non-autonomous system and introducing the generalized Jacobian matrix at the non-smooth boundaries, the bifurcation mechanism for the bursting solution induced by the slowly varying periodic drive is presented. It is shown that, the discontinuous Hopf bifurcation occurring at the non-smooth boundaries, i.e., the bifurcation taking place at the thresholds of the stimulation, leads the alternation between the rest state and spiking state. That is, different oscillation modes of this non-autonomous system convert periodically due to the non-smoothness of the vector field and the slow variation of the periodic drive as well.
Mixed-Mode Oscillations in a piecewise linear system with multiple time scale coupling
NASA Astrophysics Data System (ADS)
Fernández-García, S.; Krupa, M.; Clément, F.
2016-10-01
In this work, we analyze a four dimensional slow-fast piecewise linear system with three time scales presenting Mixed-Mode Oscillations. The system possesses an attractive limit cycle along which oscillations of three different amplitudes and frequencies can appear, namely, small oscillations, pulses (medium amplitude) and one surge (largest amplitude). In addition to proving the existence and attractiveness of the limit cycle, we focus our attention on the canard phenomena underlying the changes in the number of small oscillations and pulses. We analyze locally the existence of secondary canards leading to the addition or subtraction of one small oscillation and describe how this change is globally compensated for or not with the addition or subtraction of one pulse.
Nie, Xiaobing; Zheng, Wei Xing
2015-11-01
In this paper, we discuss the coexistence and dynamical behaviors of multiple equilibrium points for recurrent neural networks with a class of discontinuous nonmonotonic piecewise linear activation functions. It is proved that under some conditions, such n -neuron neural networks can have at least 5(n) equilibrium points, 3(n) of which are locally stable and the others are unstable, based on the contraction mapping theorem and the theory of strict diagonal dominance matrix. The investigation shows that the neural networks with the discontinuous activation functions introduced in this paper can have both more total equilibrium points and more locally stable equilibrium points than the ones with continuous Mexican-hat-type activation function or discontinuous two-level activation functions. An illustrative example with computer simulations is presented to verify the theoretical analysis.
Brooks III, E D; Szoke, A; Peterson, J L
2005-11-15
We describe a Monte Carlo solution for time dependent photon transport, in the difference formulation with the material in local thermodynamic equilibrium (LTE), that is piecewise linear in its treatment of the material state variable. Our method employs a Galerkin solution for the material energy equation while using Symbolic Implicit Monte Carlo (SIMC) to solve the transport equation. In constructing the scheme, one has the freedom to choose between expanding the material temperature, or the equivalent black body radiation energy density at the material temperature, in terms of finite element basis functions. The former provides a linear treatment of the material energy while the latter provides a linear treatment of the radiative coupling between zones. Subject to the conditional use of a lumped material energy in the vicinity of strong gradients, possible with a linear treatment of the material energy, our approach provides a robust solution for time dependent transport of thermally emitted radiation that can address a wide range of problems. It produces accurate results in the diffusion limit.
NASA Astrophysics Data System (ADS)
Ostrovski, Georg; van Strien, Sebastian
2011-02-01
In this paper we consider a class of piecewise affine Hamiltonian vector fields whose orbits are piecewise straight lines. We give a first classification result of such systems and show that the orbit-structure of the flow of such a differential equation is surprisingly rich.
NASA Astrophysics Data System (ADS)
Edwards, C. L.; Edwards, M. L.
2009-05-01
MEMS micro-mirror technology offers the opportunity to replace larger optical actuators with smaller, faster ones for lidar, network switching, and other beam steering applications. Recent developments in modeling and simulation of MEMS two-axis (tip-tilt) mirrors have resulted in closed-form solutions that are expressed in terms of physical, electrical and environmental parameters related to the MEMS device. The closed-form analytical expressions enable dynamic time-domain simulations without excessive computational overhead and are referred to as the Micro-mirror Pointing Model (MPM). Additionally, these first-principle models have been experimentally validated with in-situ static, dynamic, and stochastic measurements illustrating their reliability. These models have assumed that the mirror has a rectangular shape. Because the corners can limit the dynamic operation of a rectangular mirror, it is desirable to shape the mirror, e.g., mitering the corners. Presented in this paper is the formulation of a generalized electrostatic micromirror (GEM) model with an arbitrary convex piecewise linear shape that is readily implemented in MATLAB and SIMULINK for steady-state and dynamic simulations. Additionally, such a model permits an arbitrary shaped mirror to be approximated as a series of linearly tapered segments. Previously, "effective area" arguments were used to model a non-rectangular shaped mirror with an equivalent rectangular one. The GEM model shows the limitations of this approach and provides a pre-fabrication tool for designing mirror shapes.
NASA Astrophysics Data System (ADS)
Mohammadi, Masoumeh; Borzì, Alfio
2016-07-01
The Hermite spectral approximation of a hyperbolic Fokker-Planck (FP) optimality system arising in the control of an unbounded piecewise-deterministic process (PDP) is discussed. To control the probability density function (PDF) corresponding to the PDP process, an optimal control based on an FP strategy is considered. The resulting optimality system consists of a hyperbolic system with opposite-time orientation and an integral optimality condition equation. A Hermite spectral discretisation is investigated to approximate solutions to the optimality system in unbounded domains. It is proven that the proposed scheme satisfies the conservativity requirement of the PDFs. The spectral convergence rate of the discretisation scheme is proved and validated by numerical experiments.
Analysis of stable periodic orbits in the one dimensional linear piecewise-smooth discontinuous map.
Rajpathak, Bhooshan; Pillai, Harish K; Bandyopadhyay, Santanu
2012-09-01
In this paper, we consider one dimensional linear piecewise-smooth discontinuous maps. It is well known that stable periodic orbits exist for such maps, in some parameter region. It is also known that the corresponding bifurcation phenomena (termed as period adding bifurcation) exhibit a special structure. In the last couple of years, several authors have analyzed this structure using border collision bifurcation curves and given the characterization for various parameter regions. In this paper, we have analyzed a specific parameter range employing a different approach. We show that this approach enables one to pose some interesting questions like: what is the number of distinct periodic orbits of any given cardinality? We prove that there are precisely φ(n) distinct orbits of period n, where φ is the Euler's totient function. We propose an algorithm which calculates the location of fixed points of all these φ(n) distinct orbits and gives the precise range of existence of these orbits with respect to the parameters. Further, we show how the amount of computations required to find these ranges of existence can be optimized.
Modified SNOW 3G: Stream cipher algorithm using piecewise linear chaotic map
NASA Astrophysics Data System (ADS)
Wasi, Muhammad Arif Ali; Windarta, Susila
2016-02-01
SNOW 3G is a synchronous stream cipher developed by Thomas Johansson and Patrik Ekhdal at Lund University. In 2006, it was chosen as the main part of the second set of Universal Mobile Telecommunications System (UMTS) confidentiality and integrity algorithms [2]. In 2008, Patrik Böhm published a report entitled "Statistical Evaluation of Stream Cipher SNOW 3G". He tested the randomness properties of SNOW 3G key stream generator. Böhm using NIST statistical test suite as randomness test tool with three kinds of test, i.e. long key stream data set, short key stream data set, and initialization vector data set. The result of the report shows that from three kind of tests, only short key stream data set has not passed eight randomness tests. He state that the suggests SNOW 3G fail because there is a weakness in the initialization of the cipher. In this paper we modify SNOW 3G algorithm using piecewise linear chaotic map (PLCM) on the key initialization mode and keystream generation mode. We use the same statistical test that have been used by Böhm [5]. The experiment shows that modified SNOW 3G stream cipher algorithm has passed all the statistical test. The results prove that PLCM impact on algorithm's randomness.
Nie, Xiaobing; Zheng, Wei Xing
2016-03-01
This paper addresses the problem of coexistence and dynamical behaviors of multiple equilibria for competitive neural networks. First, a general class of discontinuous nonmonotonic piecewise linear activation functions is introduced for competitive neural networks. Then based on the fixed point theorem and theory of strict diagonal dominance matrix, it is shown that under some conditions, such n -neuron competitive neural networks can have 5(n) equilibria, among which 3(n) equilibria are locally stable and the others are unstable. More importantly, it is revealed that the neural networks with the discontinuous activation functions introduced in this paper can have both more total equilibria and locally stable equilibria than the ones with other activation functions, such as the continuous Mexican-hat-type activation function and discontinuous two-level activation function. Furthermore, the 3(n) locally stable equilibria given in this paper are located in not only saturated regions, but also unsaturated regions, which is different from the existing results on multistability of neural networks with multiple level activation functions. A simulation example is provided to illustrate and validate the theoretical findings.
NASA Astrophysics Data System (ADS)
Venkatesh, P. R.; Venkatesan, A.
2016-10-01
We report the occurrence of vibrational resonance in piecewise-linear non-autonomous system. Especially, we show that an optimal amplitude of the high frequency second harmonic driving enhances the response of a piece-wise linear non-autonomous Murali-Lakshmanan-Chua (MLC) system to a low frequency first harmonic signal. This phenomenon is illustrated with the analytical solutions of circuit equations characterising the system and finally compared with the numerical method. Further, it has been enunciated explicitly, the implementation of the fundamental NOR/NAND gate via vibrational resonance, both by numerical and analytical solutions. In addition, these logical behaviours (AND/NAND/OR/NOR) can be decided by the amplitude of the input square waves without altering the system parameters.
Rajpathak, Bhooshan; Pillai, Harish K; Bandyopadhyay, Santanu
2015-10-01
In this paper, we analytically examine the unstable periodic orbits and chaotic orbits of the 1-D linear piecewise-smooth discontinuous map. We explore the existence of unstable orbits and the effect of variation in parameters on the coexistence of unstable orbits. Further, we show that this structuring is different from the well known period adding cascade structure associated with the stable periodic orbits of the same map. Further, we analytically prove the existence of chaotic orbit for this map.
Rajpathak, Bhooshan Pillai, Harish K.; Bandyopadhyay, Santanu
2015-10-15
In this paper, we analytically examine the unstable periodic orbits and chaotic orbits of the 1-D linear piecewise-smooth discontinuous map. We explore the existence of unstable orbits and the effect of variation in parameters on the coexistence of unstable orbits. Further, we show that this structuring is different from the well known period adding cascade structure associated with the stable periodic orbits of the same map. Further, we analytically prove the existence of chaotic orbit for this map.
Harvey, Emily P; Ben-Tal, Alona
2016-02-01
Avian lungs are remarkably different from mammalian lungs in that air flows unidirectionally through rigid tubes in which gas exchange occurs. Experimental observations have been able to determine the pattern of gas flow in the respiratory system, but understanding how the flow pattern is generated and determining the factors contributing to the observed dynamics remains elusive. It has been hypothesized that the unidirectional flow is due to aerodynamic valving during inspiration and expiration, resulting from the anatomical structure and the fluid dynamics involved, however, theoretical studies to back up this hypothesis are lacking. We have constructed a novel mathematical model of the airflow in the avian respiratory system that can produce unidirectional flow which is robust to changes in model parameters, breathing frequency and breathing amplitude. The model consists of two piecewise linear ordinary differential equations with lumped parameters and discontinuous, flow-dependent resistances that mimic the experimental observations. Using dynamical systems techniques and numerical analysis, we show that unidirectional flow can be produced by either effective inspiratory or effective expiratory valving, but that both inspiratory and expiratory valving are required to produce the high efficiencies of flows observed in avian lungs. We further show that the efficacy of the inspiratory and expiratory valving depends on airsac compliances and airflow resistances that may not be located in the immediate area of the valving. Our model provides additional novel insights; for example, we show that physiologically realistic resistance values lead to efficiencies that are close to maximum, and that when the relative lumped compliances of the caudal and cranial airsacs vary, it affects the timing of the airflow across the gas exchange area. These and other insights obtained by our study significantly enhance our understanding of the operation of the avian respiratory
NASA Astrophysics Data System (ADS)
Bailey, Teresa S.
In this dissertation we discuss the development, implementation, analysis and testing of the Piecewise Linear Discontinuous Finite Element Method (PWLD) applied to the particle transport equation in two-dimensional cylindrical (RZ) and three-dimensional Cartesian (XYZ) geometries. We have designed this method to be applicable to radiative-transfer problems in radiation-hydrodynamics systems for arbitrary polygonal and polyhedral meshes. For RZ geometry, we have implemented this method in the Capsaicin radiative-transfer code being developed at Los Alamos National Laboratory. In XYZ geometry, we have implemented the method in the Parallel Deterministic Transport code being developed at Texas A&M University. We discuss the importance of the thick diffusion limit for radiative-transfer problems, and perform a thick diffusion-limit analysis on our discretized system for both geometries. This analysis predicts that the PWLD method will perform well in this limit for many problems of physical interest with arbitrary polygonal and polyhedral cells. Finally, we run a series of test problems to determine some useful properties of the method and verify the results of our thick diffusion limit analysis. Finally, we test our method on a variety of test problems and show that it compares favorably to existing methods. With these test problems, we also show that our method performs well in the thick diffusion limit as predicted by our analysis. Based on PWLD's solid finite-element foundation, the desirable properties it shows under analysis, and the excellent performance it demonstrates on test problems even with highly distorted spatial grids, we conclude that it is an excellent candidate for radiative-transfer problems that need a robust method that performs well in thick diffusive problems or on distorted grids.
Harvey, Emily P.; Ben-Tal, Alona
2016-01-01
Avian lungs are remarkably different from mammalian lungs in that air flows unidirectionally through rigid tubes in which gas exchange occurs. Experimental observations have been able to determine the pattern of gas flow in the respiratory system, but understanding how the flow pattern is generated and determining the factors contributing to the observed dynamics remains elusive. It has been hypothesized that the unidirectional flow is due to aerodynamic valving during inspiration and expiration, resulting from the anatomical structure and the fluid dynamics involved, however, theoretical studies to back up this hypothesis are lacking. We have constructed a novel mathematical model of the airflow in the avian respiratory system that can produce unidirectional flow which is robust to changes in model parameters, breathing frequency and breathing amplitude. The model consists of two piecewise linear ordinary differential equations with lumped parameters and discontinuous, flow-dependent resistances that mimic the experimental observations. Using dynamical systems techniques and numerical analysis, we show that unidirectional flow can be produced by either effective inspiratory or effective expiratory valving, but that both inspiratory and expiratory valving are required to produce the high efficiencies of flows observed in avian lungs. We further show that the efficacy of the inspiratory and expiratory valving depends on airsac compliances and airflow resistances that may not be located in the immediate area of the valving. Our model provides additional novel insights; for example, we show that physiologically realistic resistance values lead to efficiencies that are close to maximum, and that when the relative lumped compliances of the caudal and cranial airsacs vary, it affects the timing of the airflow across the gas exchange area. These and other insights obtained by our study significantly enhance our understanding of the operation of the avian respiratory
Bailey, T S; Adams, M L; Chang, J H
2008-10-01
We present a new spatial discretization of the discrete-ordinates transport equation in two-dimensional cylindrical (RZ) geometry for arbitrary polygonal meshes. This discretization is a discontinuous finite element method that utilizes the piecewise linear basis functions developed by Stone and Adams. We describe an asymptotic analysis that shows this method to be accurate for many problems in the thick diffusion limit on arbitrary polygons, allowing this method to be applied to radiative transfer problems with these types of meshes. We also present numerical results for multiple problems on quadrilateral grids and compare these results to the well-known bi-linear discontinuous finite element method.
Hydration thermodynamics beyond the linear response approximation.
Raineri, Fernando O
2016-10-19
The solvation energetics associated with the transformation of a solute molecule at infinite dilution in water from an initial state A to a final state B is reconsidered. The two solute states have different potentials energies of interaction, [Formula: see text] and [Formula: see text], with the solvent environment. Throughout the A [Formula: see text] B transformation of the solute, the solvation system is described by a Hamiltonian [Formula: see text] that changes linearly with the coupling parameter ξ. By focusing on the characterization of the probability density [Formula: see text] that the dimensionless perturbational solute-solvent interaction energy [Formula: see text] has numerical value y when the coupling parameter is ξ, we derive a hierarchy of differential equation relations between the ξ-dependent cumulant functions of various orders in the expansion of the appropriate cumulant generating function. On the basis of this theoretical framework we then introduce an inherently nonlinear solvation model for which we are able to find analytical results for both [Formula: see text] and for the solvation thermodynamic functions. The solvation model is based on the premise that there is an upper or a lower bound (depending on the nature of the interactions considered) to the amplitude of the fluctuations of Y in the solution system at equilibrium. The results reveal essential differences in behavior for the model when compared with the linear response approximation to solvation, particularly with regards to the probability density [Formula: see text]. The analytical expressions for the solvation properties show, however, that the linear response behavior is recovered from the new model when the room for the thermal fluctuations in Y is not restricted by the existence of a nearby bound. We compare the predictions of the model with the results from molecular dynamics computer simulations for aqueous solvation, in which either (1) the solute
Hydration thermodynamics beyond the linear response approximation.
Raineri, Fernando O
2016-10-19
The solvation energetics associated with the transformation of a solute molecule at infinite dilution in water from an initial state A to a final state B is reconsidered. The two solute states have different potentials energies of interaction, [Formula: see text] and [Formula: see text], with the solvent environment. Throughout the A [Formula: see text] B transformation of the solute, the solvation system is described by a Hamiltonian [Formula: see text] that changes linearly with the coupling parameter ξ. By focusing on the characterization of the probability density [Formula: see text] that the dimensionless perturbational solute-solvent interaction energy [Formula: see text] has numerical value y when the coupling parameter is ξ, we derive a hierarchy of differential equation relations between the ξ-dependent cumulant functions of various orders in the expansion of the appropriate cumulant generating function. On the basis of this theoretical framework we then introduce an inherently nonlinear solvation model for which we are able to find analytical results for both [Formula: see text] and for the solvation thermodynamic functions. The solvation model is based on the premise that there is an upper or a lower bound (depending on the nature of the interactions considered) to the amplitude of the fluctuations of Y in the solution system at equilibrium. The results reveal essential differences in behavior for the model when compared with the linear response approximation to solvation, particularly with regards to the probability density [Formula: see text]. The analytical expressions for the solvation properties show, however, that the linear response behavior is recovered from the new model when the room for the thermal fluctuations in Y is not restricted by the existence of a nearby bound. We compare the predictions of the model with the results from molecular dynamics computer simulations for aqueous solvation, in which either (1) the solute
Greenough, J A; Rider, W J
2003-11-05
A numerical study is undertaken comparing a fifth-order version of the weighted essentially non-oscillatory numerical (WENO5) method to a modern piecewise-linear, second-order, version of Godunov's (PLMDE) method for the compressible Euler Equations. A series of one-dimensional test problems are examined beginning with classical linear problems and ending with complex shock interactions. The problems considered are: (1) linear advection of a Gaussian pulse in density, (2) Sod's shock tube problem, (3) the ''peak'' shock tube problem, (4) a version of the Shu and Osher shock entropy wave interaction and (5) the Woodward and Colella interacting shock wave problem. For each problem and method, run times, density error norms and convergence rates are reported for each method as produced from a common code test-bed. The linear problem exhibits the advertised convergence rate for both methods as well as the expected large disparity in overall error levels; WENO5 has the smaller errors and an enormous advantage in overall efficiency (in accuracy per unit CPU time). For the nonlinear problems with discontinuities, however, we generally see both first-order self-convergence of error as compared to an exact solution, or when an analytic solution is not available, a converged solution generated on an extremely fine grid. The overall comparison of error levels shows some variation from problem to problem. For Sod's shock tube, PLMDE has nearly half the error, while on the peak problem the errors are nearly the same. For the interacting blast wave problem the two methods again produce a similar level of error with a slight edge for the PLMDE. On the other hand, for the Shu-Osher problem the errors are similar on the coarser grids, but favors WENO by a factor of nearly 1.5 on the finer grids used. In all cases holding mesh resolution constant though, PLMDE is less costly in terms of CPU time by approximately a factor six. If the CPU cost is taken as fixed, that is run times are
Hydration thermodynamics beyond the linear response approximation
NASA Astrophysics Data System (ADS)
Raineri, Fernando O.
2016-10-01
The solvation energetics associated with the transformation of a solute molecule at infinite dilution in water from an initial state A to a final state B is reconsidered. The two solute states have different potentials energies of interaction, {{\\Psi}\\text{A}} and {{\\Psi}\\text{B}} , with the solvent environment. Throughout the A \\to B transformation of the solute, the solvation system is described by a Hamiltonian H≤ft(ξ \\right) that changes linearly with the coupling parameter ξ. By focusing on the characterization of the probability density {{\\wp}ξ}≤ft( y\\right) that the dimensionless perturbational solute-solvent interaction energy Y=β ≤ft({{\\Psi}\\text{B}}-{{\\Psi}\\text{A}}\\right) has numerical value y when the coupling parameter is ξ, we derive a hierarchy of differential equation relations between the ξ-dependent cumulant functions of various orders in the expansion of the appropriate cumulant generating function. On the basis of this theoretical framework we then introduce an inherently nonlinear solvation model for which we are able to find analytical results for both {{\\wp}ξ} ≤ft( y\\right) and for the solvation thermodynamic functions. The solvation model is based on the premise that there is an upper or a lower bound (depending on the nature of the interactions considered) to the amplitude of the fluctuations of Y in the solution system at equilibrium. The results reveal essential differences in behavior for the model when compared with the linear response approximation to solvation, particularly with regards to the probability density {{\\wp}ξ} ≤ft( y\\right) . The analytical expressions for the solvation properties show, however, that the linear response behavior is recovered from the new model when the room for the thermal fluctuations in Y is not restricted by the existence of a nearby bound. We compare the predictions of the model with the results from molecular dynamics computer simulations for aqueous solvation, in
Schumacher, Robin; Wahl, S. Aljoscha
2015-01-01
The design of microbial production processes relies on rational choices for metabolic engineering of the production host and the process conditions. These require a systematic and quantitative understanding of cellular regulation. Therefore, a novel method for dynamic flux identification using quantitative metabolomics and 13C labeling to identify piecewise-affine (PWA) flux functions has been described recently. Obtaining flux estimates nevertheless still required frequent manual reinitalization to obtain a good reproduction of the experimental data and, moreover, did not optimize on all observables simultaneously (metabolites and isotopomer concentrations). In our contribution we focus on measures to achieve faster and robust dynamic flux estimation which leads to a high dimensional parameter estimation problem. Specifically, we address the following challenges within the PWA problem formulation: (1) Fast selection of sufficient domains for the PWA flux functions, (2) Control of over-fitting in the concentration space using shape-prescriptive modeling and (3) robust and efficient implementation of the parameter estimation using the hybrid implicit filtering algorithm. With the improvements we significantly speed up the convergence by efficiently exploiting that the optimization problem is partly linear. This allows application to larger-scale metabolic networks and demonstrates that the proposed approach is not purely theoretical, but also applicable in practice. PMID:26690237
Kohli, Nidhi; Sullivan, Amanda L; Sadeh, Shanna; Zopluoglu, Cengiz
2015-04-01
Effective instructional planning and intervening rely heavily on accurate understanding of students' growth, but relatively few researchers have examined mathematics achievement trajectories, particularly for students with special needs. We applied linear, quadratic, and piecewise linear mixed-effects models to identify the best-fitting model for mathematics development over elementary and middle school and to ascertain differences in growth trajectories of children with learning disabilities relative to their typically developing peers. The analytic sample of 2150 students was drawn from the Early Childhood Longitudinal Study - Kindergarten Cohort, a nationally representative sample of United States children who entered kindergarten in 1998. We first modeled students' mathematics growth via multiple mixed-effects models to determine the best fitting model of 9-year growth and then compared the trajectories of students with and without learning disabilities. Results indicate that the piecewise linear mixed-effects model captured best the functional form of students' mathematics trajectories. In addition, there were substantial achievement gaps between students with learning disabilities and students with no disabilities, and their trajectories differed such that students without disabilities progressed at a higher rate than their peers who had learning disabilities. The results underscore the need for further research to understand how to appropriately model students' mathematics trajectories and the need for attention to mathematics achievement gaps in policy.
Kohli, Nidhi; Sullivan, Amanda L; Sadeh, Shanna; Zopluoglu, Cengiz
2015-04-01
Effective instructional planning and intervening rely heavily on accurate understanding of students' growth, but relatively few researchers have examined mathematics achievement trajectories, particularly for students with special needs. We applied linear, quadratic, and piecewise linear mixed-effects models to identify the best-fitting model for mathematics development over elementary and middle school and to ascertain differences in growth trajectories of children with learning disabilities relative to their typically developing peers. The analytic sample of 2150 students was drawn from the Early Childhood Longitudinal Study - Kindergarten Cohort, a nationally representative sample of United States children who entered kindergarten in 1998. We first modeled students' mathematics growth via multiple mixed-effects models to determine the best fitting model of 9-year growth and then compared the trajectories of students with and without learning disabilities. Results indicate that the piecewise linear mixed-effects model captured best the functional form of students' mathematics trajectories. In addition, there were substantial achievement gaps between students with learning disabilities and students with no disabilities, and their trajectories differed such that students without disabilities progressed at a higher rate than their peers who had learning disabilities. The results underscore the need for further research to understand how to appropriately model students' mathematics trajectories and the need for attention to mathematics achievement gaps in policy. PMID:25746821
Wollaeger, Ryan T.; Wollaber, Allan B.; Urbatsch, Todd J.; Densmore, Jeffery D.
2016-05-04
Here, the non-linear thermal radiative-transfer equations can be solved in various ways. One popular way is the Fleck and Cummings Implicit Monte Carlo (IMC) method. The IMC method was originally formulated with piecewise-constant material properties. For domains with a coarse spatial grid and large temperature gradients, an error known as numerical teleportation may cause artificially non-causal energy propagation and consequently an inaccurate material temperature. Source tilting is a technique to reduce teleportation error by constructing sub-spatial-cell (or sub-cell) emission profiles from which IMC particles are sampled. Several source tilting schemes exist, but some allow teleportation error to persist. We examinemore » the effect of source tilting in problems with a temperature-dependent opacity. Within each cell, the opacity is evaluated continuously from a temperature profile implied by the source tilt. For IMC, this is a new approach to modeling the opacity. We find that applying both source tilting along with a source tilt-dependent opacity can introduce another dominant error that overly inhibits thermal wavefronts. We show that we can mitigate both teleportation and under-propagation errors if we discretize the temperature equation with a linear discontinuous (LD) trial space. Our method is for opacities ~ 1/T3, but we formulate and test a slight extension for opacities ~ 1/T3.5, where T is temperature. We find our method avoids errors that can be incurred by IMC with continuous source tilt constructions and piecewise-constant material temperature updates.« less
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.
Nie, Xiaobing; Zheng, Wei Xing
2015-05-01
This paper is concerned with the problem of coexistence and dynamical behaviors of multiple equilibrium points for neural networks with discontinuous non-monotonic piecewise linear activation functions and time-varying delays. The fixed point theorem and other analytical tools are used to develop certain sufficient conditions that ensure that the n-dimensional discontinuous neural networks with time-varying delays can have at least 5(n) equilibrium points, 3(n) of which are locally stable and the others are unstable. The importance of the derived results is that it reveals that the discontinuous neural networks can have greater storage capacity than the continuous ones. Moreover, different from the existing results on multistability of neural networks with discontinuous activation functions, the 3(n) locally stable equilibrium points obtained in this paper are located in not only saturated regions, but also unsaturated regions, due to the non-monotonic structure of discontinuous activation functions. A numerical simulation study is conducted to illustrate and support the derived theoretical results.
Generating exact solutions to Einstein's equation using linearized approximations
NASA Astrophysics Data System (ADS)
Harte, Abraham I.; Vines, Justin
2016-10-01
We show that certain solutions to the linearized Einstein equation can—by the application of a particular type of linearized gauge transformation—be straightforwardly transformed into solutions of the exact Einstein equation. In cases with nontrivial matter content, the exact stress-energy tensor of the transformed metric has the same eigenvalues and eigenvectors as the linearized stress-energy tensor of the initial approximation. When our gauge exists, the tensorial structure of transformed metric perturbations identically eliminates all nonlinearities in Einstein's equation. As examples, we derive the exact Kerr and gravitational plane wave metrics from standard harmonic-gauge approximations.
Lehtikangas, O.; Tarvainen, T.; Kim, A.D.; Arridge, S.R.
2015-02-01
The radiative transport equation can be used as a light transport model in a medium with scattering particles, such as biological tissues. In the radiative transport equation, the refractive index is assumed to be constant within the medium. However, in biomedical media, changes in the refractive index can occur between different tissue types. In this work, light propagation in a medium with piece-wise constant refractive index is considered. Light propagation in each sub-domain with a constant refractive index is modeled using the radiative transport equation and the equations are coupled using boundary conditions describing Fresnel reflection and refraction phenomena on the interfaces between the sub-domains. The resulting coupled system of radiative transport equations is numerically solved using a finite element method. The approach is tested with simulations. The results show that this coupled system describes light propagation accurately through comparison with the Monte Carlo method. It is also shown that neglecting the internal changes of the refractive index can lead to erroneous boundary measurements of scattered light.
Equilibria of an epidemic game with piecewise linear social distancing cost.
Reluga, Timothy C
2013-10-01
Around the world, infectious disease epidemics continue to threaten people's health. When epidemics strike, we often respond by changing our behaviors to reduce our risk of infection. This response is sometimes called "social distancing." Since behavior changes can be costly, we would like to know the optimal social distancing behavior. But the benefits of changes in behavior depend on the course of the epidemic, which itself depends on our behaviors. Differential population game theory provides a method for resolving this circular dependence. Here, I present the analysis of a special case of the differential SIR epidemic population game with social distancing when the relative infection rate is linear, but bounded below by zero. Equilibrium solutions are constructed in closed-form for an open-ended epidemic. Constructions are also provided for epidemics that are stopped by the deployment of a vaccination that becomes available a fixed-time after the start of the epidemic. This can be used to anticipate a window of opportunity during which mass vaccination can significantly reduce the cost of an epidemic.
How to Solve Schroedinger Problems by Approximating the Potential Function
Ledoux, Veerle; Van Daele, Marnix
2010-09-30
We give a survey over the efforts in the direction of solving the Schroedinger equation by using piecewise approximations of the potential function. Two types of approximating potentials have been considered in the literature, that is piecewise constant and piecewise linear functions. For polynomials of higher degree the approximating problem is not so easy to integrate analytically. This obstacle can be circumvented by using a perturbative approach to construct the solution of the approximating problem, leading to the so-called piecewise perturbation methods (PPM). We discuss the construction of a PPM in its most convenient form for applications and show that different PPM versions (CPM,LPM) are in fact equivalent.
NASA Astrophysics Data System (ADS)
Hoshino, Tasuku
This paper deals with an approximate linearization control of 2-DOF underactuated-by-1 nonlinear systems, proposing a novel linearization coordinate which reduces the approximation error over the state space around the operating point. The coordinate is analytically constructed in a systematic way by solving two first order linear partial differential equations and the solution is given in an infinite series of configuration variables. The resulting linearization feedback is highly nonlinear and the basin of attraction of the stabilized system using proposed coordinate is large, comparing with those of a conventional first order or other lower order linearization coordinates. The approximate linearization control based on the proposed coordinate is applied to the stabilization of a rotational inverted pendulum; the advantage is verified in simulations and experiments. Some perspectives on availability of the linearization coordinate are discussed and they are computed also for a mobile inverted pendulum, Acrobot, and for Pendubot as examples.
Identifying approximate linear models for simple nonlinear systems
NASA Technical Reports Server (NTRS)
Horta, L. G.; Juang, J.-N.
1985-01-01
This paper addresses the identification (realization) of approximate linear models from response data for certain nonlinear dynamic systems. Response characteristics for several typical nonlinear joints are analyzed mathematically and represented by series expansions. The parameters of the series expansion are then compared with the modal parameters of a linear model identified by the Eigensystem Realization Algorithm. The agreement of the identified model and the analytically derived representation is excellent for the cases studied. Also laboratory data from a model which exhibited stiffening behavior was analyzed using the Eigensystem Realization algorithm and Fast Fourier Transform. The laboratory experiment demonstrated the ability of the technique to recover the model characteristics using real data.
Linear approximation of Rayleigh-Brillouin scattering spectra.
Binietoglou, Ioannis; Giampouras, Paris; Belegante, Livio
2016-09-20
Rayleigh-Brillouin scattering is the basis of many remote sensing techniques, including high spectral resolution lidar measurements of aerosols and wind. Rayleigh-Brillouin spectra can be accurately estimated using physics-based models like the so-called Tenti's S6 and Pan's S7 models. Unfortunately, these are computationally expensive and can be the bottleneck for real-time lidar processing and iterative parameter estimation problems. This short article describes a very efficient linear approximation of the Rayleigh-Brillouin spectra based on Principal Component Analysis (PCA). Using PCA, the outputs of the above models can be approximated with very high accuracy using a single matrix multiplication. The described method can be applied to the output of any detailed scattering model, thus it can be used for a wide range of problems, e.g., for scattering from different gases (Air, N_{2}, O_{2},…) and for different ranges of temperature and pressure. The precision of the approximation can be adapted to the requirements of the studied problem, and can easily exceed the actual accuracy of the reference models. PMID:27661601
Inference for reaction networks using the linear noise approximation.
Fearnhead, Paul; Giagos, Vasilieos; Sherlock, Chris
2014-06-01
We consider inference for the reaction rates in discretely observed networks such as those found in models for systems biology, population ecology, and epidemics. Most such networks are neither slow enough nor small enough for inference via the true state-dependent Markov jump process to be feasible. Typically, inference is conducted by approximating the dynamics through an ordinary differential equation (ODE) or a stochastic differential equation (SDE). The former ignores the stochasticity in the true model and can lead to inaccurate inferences. The latter is more accurate but is harder to implement as the transition density of the SDE model is generally unknown. The linear noise approximation (LNA) arises from a first-order Taylor expansion of the approximating SDE about a deterministic solution and can be viewed as a compromise between the ODE and SDE models. It is a stochastic model, but discrete time transition probabilities for the LNA are available through the solution of a series of ordinary differential equations. We describe how a restarting LNA can be efficiently used to perform inference for a general class of reaction networks; evaluate the accuracy of such an approach; and show how and when this approach is either statistically or computationally more efficient than ODE or SDE methods. We apply the LNA to analyze Google Flu Trends data from the North and South Islands of New Zealand, and are able to obtain more accurate short-term forecasts of new flu cases than another recently proposed method, although at a greater computational cost.
NASA Astrophysics Data System (ADS)
Vjačeslavov, N. S.
1980-02-01
In this paper estimates are found for L_pR_n(f) - the least deviation in the L_p-metric, 0 < p\\leq\\infty, of a piecewise analytic function f from the rational functions of degree at most n. It is shown that these estimates are sharp in a well-defined sense.Bibliography: 12 titles.
Nie, Xiaobing; Zheng, Wei Xing; Cao, Jinde
2015-11-01
The problem of coexistence and dynamical behaviors of multiple equilibrium points is addressed for a class of memristive Cohen-Grossberg neural networks with non-monotonic piecewise linear activation functions and time-varying delays. By virtue of the fixed point theorem, nonsmooth analysis theory and other analytical tools, some sufficient conditions are established to guarantee that such n-dimensional memristive Cohen-Grossberg neural networks can have 5(n) equilibrium points, among which 3(n) equilibrium points are locally exponentially stable. It is shown that greater storage capacity can be achieved by neural networks with the non-monotonic activation functions introduced herein than the ones with Mexican-hat-type activation function. In addition, unlike most existing multistability results of neural networks with monotonic activation functions, those obtained 3(n) locally stable equilibrium points are located both in saturated regions and unsaturated regions. The theoretical findings are verified by an illustrative example with computer simulations.
Approximate inverse preconditioning of iterative methods for nonsymmetric linear systems
Benzi, M.; Tuma, M.
1996-12-31
A method for computing an incomplete factorization of the inverse of a nonsymmetric matrix A is presented. The resulting factorized sparse approximate inverse is used as a preconditioner in the iterative solution of Ax = b by Krylov subspace methods.
Approximate analytic solutions for singular non-linear oscillators
NASA Technical Reports Server (NTRS)
Bota, K. B.; Mickens, R. E.
1984-01-01
Mickens (1981, 1984) has considered analytic techniques for obtaining approximate solutions to one-dimensional nonlinear oscillatory systems x(double-dot) + x = lambda f(x, x/dot/, lambda) where lambda is a small positive parameter and f is a nonlinear polynomial function of its arguments. However, in certain cases there is interest in the analysis of physical systems for which the nonlinear function f(x, x/dot/, lambda) is singular for finite values of x or x(dot). The present investigation is concerned with the use of existing approximate analytic schemes to obtain solutions to singular nonlinear oscillatory differential equations.
Weighted least squares stationary approximations to linear systems.
NASA Technical Reports Server (NTRS)
Bierman, G. J.
1972-01-01
Investigation of the problem of replacing a certain time-varying linear system by a stationary one. Several quadratic criteria are proposed to aid in determining suitable candidate systems. One criterion for choosing the matrix B (in the stationary system) is initial-condition dependent, and another bounds the 'worst case' homogeneous system performance. Both of these criteria produce weighted least square fits.
Yamada, Shinsuke; Gotoh, Osamu; Yamana, Hayato
2006-01-01
Background Multiple sequence alignment (MSA) is a useful tool in bioinformatics. Although many MSA algorithms have been developed, there is still room for improvement in accuracy and speed. In the alignment of a family of protein sequences, global MSA algorithms perform better than local ones in many cases, while local ones perform better than global ones when some sequences have long insertions or deletions (indels) relative to others. Many recent leading MSA algorithms have incorporated pairwise alignment information obtained from a mixture of sources into their scoring system to improve accuracy of alignment containing long indels. Results We propose a novel group-to-group sequence alignment algorithm that uses a piecewise linear gap cost. We developed a program called PRIME, which employs our proposed algorithm to optimize the well-defined sum-of-pairs score. PRIME stands for Profile-based Randomized Iteration MEthod. We evaluated PRIME and some recent MSA programs using BAliBASE version 3.0 and PREFAB version 4.0 benchmarks. The results of benchmark tests showed that PRIME can construct accurate alignments comparable to the most accurate programs currently available, including L-INS-i of MAFFT, ProbCons, and T-Coffee. Conclusion PRIME enables users to construct accurate alignments without having to employ pairwise alignment information. PRIME is available at . PMID:17137519
Rational approximations to linear forms of exponentials and binomials
Chudnovsky, G. V.
1983-01-01
Mahler proved the following quantitative result supplementing the Lindemann-Weierstrass theorem: ǀΣi=0nCieriǀ > H-n-ε for any distinct rational numbers r0,r1,..., rn and rational integers C0,C1,...,Cn with H = max0≤i≤n ǀCiǀ. We improve Mahler's estimate by replacing exponentials eri by linearly independent linear forms Li = Σ Lijesij with rational Lij,siji = 0,1,...,n. Similar results are obtained for binomials (a/b)ri or Σ Lij(a/b)sij with integers a,b and logǀbǀ/logǀaǀ > 1 - ε. The simplest examples of new numbers with the irrationality exponent “2 + ε” are sinh 1 or sin 1. PMID:16593320
A Study of Linear Approximation Techniques for SAR Azimuth Processing
NASA Technical Reports Server (NTRS)
Martinson, L. W.; Perry, R. P.; Liu, B.
1979-01-01
The application of the step transform subarray processing techniques to synthetic aperture radar (SAR) was studied. The subarray technique permits the application of efficient digital transform computational techniques such as the fast Fourier transform to be applied while offering an effective tool for range migration compensation. Range migration compensation is applied at the subarray level, and with the subarray size based on worst case range migration conditions, a minimum control system is achieved. A baseline processor was designed for a four-look SAR system covering approximately 4096 by 4096 SAR sample field every 2.5 seconds. Implementation of the baseline system was projected using advanced low power technologies. A 20 swath is implemented with approximately 1000 circuits having a power dissipation of from 70 to 195 watts. The baseline batch step transform processor is compared to a continuous strip processor, and variations of the baseline are developed for a wide range of SAR parameters.
Manning, Jeffrey A; Garton, Edward O
2013-07-01
Habitat selection fundamentally drives the distribution of organisms across landscapes; density-dependent habitat selection (DDHS) is considered a central component of ecological theories explaining habitat use and population regulation. A preponderance of DDHS theories is based on ideal distributions, such that organisms select habitat according to either the ideal free, despotic, or pre-emptive distributions. Models that can be used to simultaneously test competing DDHS theories are desirable to help improve our understanding of habitat selection. We developed hierarchical, piecewise linear models that allow for simultaneous testing of DDHS theories and accommodate densities from multiple habitats and regional populations, environmental covariates, and random effects. We demonstrate the use of these models with data on mule deer (Odocoileus hemionus) abundance and net energy costs in different snow depths within winter ranges of five regional populations in western Idaho, USA. Regional population density explained 40% of the variation in population growth, and we found that deer were ideal free in winter ranges. Deer occupied habitats with lowest net energy costs at higher densities and at a higher rate than compared to habitats with intermediate and high energy costs. The proportion of a regional population in low energy cost habitat the previous year accounted for a significant amount of variation in population growth (17%), demonstrating the importance of winter habitat selection in regulating deer populations. These linear models are most appropriate for empirical data collected from centralized habitat patches within the local range of a species where individuals are either year-round residents or migratory (but have already arrived from migration).
NASA Technical Reports Server (NTRS)
Ito, Kazufumi; Teglas, Russell
1987-01-01
The numerical scheme based on the Legendre-tau approximation is proposed to approximate the feedback solution to the linear quadratic optimal control problem for hereditary differential systems. The convergence property is established using Trotter ideas. The method yields very good approximations at low orders and provides an approximation technique for computing closed-loop eigenvalues of the feedback system. A comparison with existing methods (based on averaging and spline approximations) is made.
Piecing Together Piecewise Functions.
ERIC Educational Resources Information Center
King, Sybrina L.
1997-01-01
Presents an activity to teach piecewise functions using wax paper and rectangular grids. Helps students understand the idea of different pieces by literally "piecing" together a new type of mathematical function. Also describes a followup activity and explains how piecewise functions can be graphed using graphing calculators. (NB)
RKH space approximations for the feedback operator in a linear hereditary control system
NASA Technical Reports Server (NTRS)
Reneke, J. A.; Fennell, R. E.
1987-01-01
Computational implementation of feedback control laws for linear hereditary systems requires the approximation of infinite dimensional feedback operators with finite dimensional operators. The dense subspaces of K-polygonal functions in reproducing kernel Hilbert spaces, RKH spaces, suggest finite dimensional approximations of the matrix representations of the control operators. A convergence theorem is developed for the approximations and the numerical implementation of the approximations is discussed.
Mechanical System Reliability and Cost Integration Using a Sequential Linear Approximation Method
NASA Technical Reports Server (NTRS)
Kowal, Michael T.
1997-01-01
The development of new products is dependent on product designs that incorporate high levels of reliability along with a design that meets predetermined levels of system cost. Additional constraints on the product include explicit and implicit performance requirements. Existing reliability and cost prediction methods result in no direct linkage between variables affecting these two dominant product attributes. A methodology to integrate reliability and cost estimates using a sequential linear approximation method is proposed. The sequential linear approximation method utilizes probability of failure sensitivities determined from probabilistic reliability methods as well a manufacturing cost sensitivities. The application of the sequential linear approximation method to a mechanical system is demonstrated.
Further results on the L1 analysis of sampled-data systems via kernel approximation approach
NASA Astrophysics Data System (ADS)
Kim, Jung Hoon; Hagiwara, Tomomichi
2016-08-01
This paper gives two methods for the L1 analysis of sampled-data systems, by which we mean computing the L∞-induced norm of sampled-data systems. This is achieved by developing what we call the kernel approximation approach in the setting of sampled-data systems. We first consider the lifting treatment of sampled-data systems and give an operator theoretic representation of their input/output relation. We further apply the fast-lifting technique by which the sampling interval [0, h) is divided into M subintervals with an equal width, and provide methods for computing the L∞-induced norm. In contrast to a similar approach developed earlier called the input approximation approach, we use an idea of kernel approximation, in which the kernel function of an input operator and the hold function of an output operator are approximated by piecewise constant or piecewise linear functions. Furthermore, it is shown that the approximation errors in the piecewise constant approximation or piecewise linear approximation scheme converge to 0 at the rate of 1/M or 1/M2, respectively. In comparison with the existing input approximation approach, in which the input function (rather than the kernel function) of the input operator is approximated by piecewise constant or piecewise linear functions, we show that the kernel approximation approach gives improved computation results. More precisely, even though the convergence rates in the kernel approximation approach remain qualitatively the same as those in the input approximation approach, the newly developed former approach could lead to quantitatively improved approximation errors than the latter approach particularly when the piecewise linear approximation scheme is taken. Finally, a numerical example is given to demonstrate the effectiveness of the kernel approximation approach with this scheme.
Nonlinear control via approximate input-output linearization - The ball and beam example
NASA Technical Reports Server (NTRS)
Hauser, John; Sastry, Shankar; Kokotovic, Petar
1992-01-01
A study is made of approximate input-output linearization of nonlinear systems which fail to have a well defined relative degree. For such systems, a method is provided for constructing approximate systems that are input-output linearizable. The analysis presented in this note is motivated through its application to a common undergraduate control laboratory experiment, the ball and beam system, where it is shown to be more effective for trajectory tracking than the standard Jacobian linearization.
NASA Technical Reports Server (NTRS)
Gibson, J. S.; Rosen, I. G.
1986-01-01
An abstract approximation framework is developed for the finite and infinite time horizon discrete-time linear-quadratic regulator problem for systems whose state dynamics are described by a linear semigroup of operators on an infinite dimensional Hilbert space. The schemes included the framework yield finite dimensional approximations to the linear state feedback gains which determine the optimal control law. Convergence arguments are given. Examples involving hereditary and parabolic systems and the vibration of a flexible beam are considered. Spline-based finite element schemes for these classes of problems, together with numerical results, are presented and discussed.
NASA Astrophysics Data System (ADS)
Volkov, V. V.; Erokhin, V. I.
2010-04-01
The properties of a mathematical programming problem that arises in finding a stable (in the sense of Tikhonov) solution to a system of linear algebraic equations with an approximately given augmented coefficient matrix are examined. Conditions are obtained that determine whether this problem can be reduced to the minimization of a smoothing functional or to the minimal matrix correction of the underlying system of linear algebraic equations. A method for constructing (exact or approximately given) model systems of linear algebraic equations with known Tikhonov solutions is described. Sharp lower bounds are derived for the maximal error in the solution of an approximately given system of linear algebraic equations under finite perturbations of its coefficient matrix. Numerical examples are given.
Nonlinear control via approximate input-output linearization - The ball and beam example
NASA Technical Reports Server (NTRS)
Hauser, John; Sastry, Shankar; Kokotovic, Petar
1989-01-01
This paper presents an approach for the approximate input-output linearization of nonlinear systems, particularly those for which relative degree is not well defined. It is shown that there is a great deal of freedom in the selection of an approximation and that, by designing a tracking controller based on the approximating system, tracking of reasonable trajectories can be achieved with small error. The approximating system is itself a nonlinear system, with the difference that it is input-output linearizable by state feedback. Some properties of the accuracy of the approximation are demonstrated and, in the context of the ball and beam example, it is shown to be far superior to the Jacobian approximation. The results are focused on finding regular SISO systems which are close to systems which are not regular and controlling these approximate regular systems.
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.
Piecewise Cubic Interpolation Package
1982-04-23
PCHIP (Piecewise Cubic Interpolation Package) is a set of subroutines for piecewise cubic Hermite interpolation of data. It features software to produce a monotone and "visually pleasing" interpolant to monotone data. Such an interpolant may be more reasonable than a cubic spline if the data contain both 'steep' and 'flat' sections. Interpolation of cumulative probability distribution functions is another application. In PCHIP, all piecewise cubic functions are represented in cubic Hermite form; that is, f(x)more » is determined by its values f(i) and derivatives d(i) at the breakpoints x(i), i=1(1)N. PCHIP contains three routines - PCHIM, PCHIC, and PCHSP to determine derivative values, six routines - CHFEV, PCHFE, CHFDV, PCHFD, PCHID, and PCHIA to evaluate, differentiate, or integrate the resulting cubic Hermite function, and one routine to check for monotonicity. A FORTRAN 77 version and SLATEC version of PCHIP are included.« less
Piecewise polynomial representations of genomic tracks.
Tarabichi, Maxime; Detours, Vincent; Konopka, Tomasz
2012-01-01
Genomic data from micro-array and sequencing projects consist of associations of measured values to chromosomal coordinates. These associations can be thought of as functions in one dimension and can thus be stored, analyzed, and interpreted as piecewise-polynomial curves. We present a general framework for building piecewise polynomial representations of genome-scale signals and illustrate some of its applications via examples. We show that piecewise constant segmentation, a typical step in copy-number analyses, can be carried out within this framework for both array and (DNA) sequencing data offering advantages over existing methods in each case. Higher-order polynomial curves can be used, for example, to detect trends and/or discontinuities in transcription levels from RNA-seq data. We give a concrete application of piecewise linear functions to diagnose and quantify alignment quality at exon borders (splice sites). Our software (source and object code) for building piecewise polynomial models is available at http://sourceforge.net/projects/locsmoc/.
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…
Heinemann, Tobias; Quataert, Eliot E-mail: eliot@berkeley.edu
2014-09-01
We derive the conductivity tensor for axisymmetric perturbations of a hot, collisionless, and charge-neutral plasma in the shearing sheet approximation. Our results generalize the well-known linear Vlasov theory for uniform plasmas to differentially rotating plasmas and can be used for wide range of kinetic stability calculations. We apply these results to the linear theory of the magneto-rotational instability (MRI) in collisionless plasmas. We show analytically and numerically how the general kinetic theory results derived here reduce in appropriate limits to previous results in the literature, including the low-frequency guiding center (or 'kinetic MHD') approximation, Hall magnetohydrodynamics (MHD), and the gyro-viscous approximation. We revisit the cold plasma model of the MRI and show that, contrary to previous results, an initially unmagnetized collisionless plasma is linearly stable to axisymmetric perturbations in the cold plasma approximation. In addition to their application to astrophysical plasmas, our results provide a useful framework for assessing the linear stability of differentially rotating plasmas in laboratory experiments.
Approximate solutions of non-linear circular orbit relative motion in curvilinear coordinates
NASA Astrophysics Data System (ADS)
Bombardelli, Claudio; Gonzalo, Juan Luis; Roa, Javier
2016-07-01
A compact, time-explicit, approximate solution of the highly non-linear relative motion in curvilinear coordinates is provided under the assumption of circular orbit for the chief spacecraft. The rather compact, three-dimensional solution is obtained by algebraic manipulation of the individual Keplerian motions in curvilinear, rather than Cartesian coordinates, and provides analytical expressions for the secular, constant and periodic terms of each coordinate as a function of the initial relative motion conditions or relative orbital elements. Numerical test cases are conducted to show that the approximate solution can be effectively employed to extend the classical linear Clohessy-Wiltshire solution to include non-linear relative motion without significant loss of accuracy up to a limit of 0.4-0.45 in eccentricity and 40-45° in relative inclination for the follower. A very simple, quadratic extension of the classical Clohessy-Wiltshire solution in curvilinear coordinates is also presented.
Linear Approximation to Optimal Control Allocation for Rocket Nozzles with Elliptical Constraints
NASA Technical Reports Server (NTRS)
Orr, Jeb S.; Wall, Johnm W.
2011-01-01
In this paper we present a straightforward technique for assessing and realizing the maximum control moment effectiveness for a launch vehicle with multiple constrained rocket nozzles, where elliptical deflection limits in gimbal axes are expressed as an ensemble of independent quadratic constraints. A direct method of determining an approximating ellipsoid that inscribes the set of attainable angular accelerations is derived. In the case of a parameterized linear generalized inverse, the geometry of the attainable set is computationally expensive to obtain but can be approximated to a high degree of accuracy with the proposed method. A linear inverse can then be optimized to maximize the volume of the true attainable set by maximizing the volume of the approximating ellipsoid. The use of a linear inverse does not preclude the use of linear methods for stability analysis and control design, preferred in practice for assessing the stability characteristics of the inertial and servoelastic coupling appearing in large boosters. The present techniques are demonstrated via application to the control allocation scheme for a concept heavy-lift launch vehicle.
Comparison of dynamical approximation schemes for non-linear gravitational clustering
NASA Technical Reports Server (NTRS)
Melott, Adrian L.
1994-01-01
We have recently conducted a controlled comparison of a number of approximations for gravitational clustering against the same n-body simulations. These include ordinary linear perturbation theory (Eulerian), the adhesion approximation, the frozen-flow approximation, the Zel'dovich approximation (describable as first-order Lagrangian perturbation theory), and its second-order generalization. In the last two cases we also created new versions of approximation by truncation, i.e., smoothing the initial conditions by various smoothing window shapes and varying their sizes. The primary tool for comparing simulations to approximation schemes was crosscorrelation of the evolved mass density fields, testing the extent to which mass was moved to the right place. The Zel'dovich approximation, 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 approximation 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 approximation produced the most accurate 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 approximation, depending upon the purpose. The theoretical implication is that pancaking is implicit in all cosmological gravitational clustering, at least from Gaussian initial conditions, even
Comparison of dynamical approximation schemes for non-linear gravitational clustering
NASA Astrophysics Data System (ADS)
Melott, Adrian L.
We have recently conducted a controlled comparison of a number of approximations for gravitational clustering against the same n-body simulations. These include ordinary linear perturbation theory (Eulerian), the adhesion approximation, the frozen-flow approximation, the Zel'dovich approximation (describable as first-order Lagrangian perturbation theory), and its second-order generalization. In the last two cases we also created new versions of approximation by truncation, i.e., smoothing the initial conditions by various smoothing window shapes and varying their sizes. The primary tool for comparing simulations to approximation schemes was crosscorrelation of the evolved mass density fields, testing the extent to which mass was moved to the right place. The Zel'dovich approximation, 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 approximation 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 approximation produced the most accurate 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 approximation, depending upon the purpose. The theoretical implication is that pancaking is implicit in all cosmological gravitational clustering, at least from Gaussian initial conditions, even
Sharir-Ivry, Avital; Varatharaj, Rajapandian; Shurki, Avital
2015-01-13
Various aspects of the linear response approximation (LRA) approach were examined when calculating reaction barriers within an enzyme and its different mutants. Scaling the electrostatic interactions is shown to slightly affect the absolute values of the barriers but not the overall trend when comparing wild-type and mutants. Convergence of the overall energetics was shown to depend on the sampling. Finally, the contribution of particular residues was shown to be significant, despite its small value. PMID:26574227
NASA Technical Reports Server (NTRS)
Milman, Mark H.
1987-01-01
The fundamental control synthesis issue of establishing a priori convergence rates of approximation schemes for feedback controllers for a class of distributed parameter systems is addressed within the context of hereditary systems. Specifically, a factorization approach is presented for deriving approximations to the optimal feedback gains for the linear regulator-quadratic cost problem associated with time-varying functional differential equations with control delays. The approach is based on a discretization of the state penalty which leads to a simple structure for the feedback control law. General properties of the Volterra factors of Hilbert-Schmidt operators are then used to obtain convergence results for the controls, trajectories and feedback kernels. Two algorithms are derived from the basic approximation scheme, including a fast algorithm, in the time-invariant case. A numerical example is also considered.
NASA Astrophysics Data System (ADS)
Kaoui, Fawzi; Rocca, Dario
2016-01-01
A new approach was recently presented to compute correlation energies within the random phase approximation using Lanczos chains and an optimal basis set (Rocca 2014 J. Chem. Phys. 140 18A501). This novel method avoids the explicit calculation of conduction states and represents linear response functions on a compact auxiliary basis set obtained from the diagonalization of an approximate dielectric matrix that contains only the kinetic energy contribution. Here, we extend this formalism, originally implemented for molecular systems, to treat periodic solids. In particular, the approximate dielectric matrix used to build the auxiliary basis set is generalized to avoid unphysical negative gaps, that make the model inefficient. The numerical convergence of the method is discussed and the accuracy is demonstrated considering a set including three covalently bonded (C, Si, and SiC) and three weakly bonded (Ne, Ar, and Kr) solids.
Approximate linear confidence and curvature of a kinetic model of dodecanedioic acid in humans.
Panunzi, Simona; De Gaetano, Andrea; Mingrone, Geltrude
2005-11-01
Dicarboxylic acids with an even number of carbon atoms have been proposed as an alternate energy substrate for enteral or parenteral nutrition in the acutely ill patient, due to their water solubility and their yielding TCA cycle intermediates upon beta-oxidation. In the present work, a nonlinear compartmental model of the kinetics of dodecanedioic acid is developed, and its parameters are estimated from time concentration experimental observations obtained from six healthy volunteers undergoing a per os administration of 3 g of the substance. Although the model is linear in the transfer of the free substance from plasma to the tissues, the exchange between gut and plasma compartments is represented as a saturable function. Albumin binding is then incorporated to obtain the final model in terms of the measured total concentrations. Estimates of the model's structural parameters were computed for each experimental subject, and the usual single-subject approximate confidence regions for the parameters were derived by inversion of the Hessian at the optimum. To verify the applicability of this approximation, the nonlinearity of the expectation surface at the optimum was measured by computing the normal (intrinsic) component of curvature. Because the model curvature was excessive in all subjects, the usual approximation could not be trusted to provide acceptable approximations to the parameter confidence regions. A suitable Monte Carlo simulation yielded empirical joint parameter distributions from which the approximate parameter variances could finally be obtained.
NASA Astrophysics Data System (ADS)
Atalla, Viktor; Zhang, Igor Ying; Hofmann, Oliver T.; Ren, Xinguo; Rinke, Patrick; Scheffler, Matthias
2016-07-01
We obtain the exchange parameter of hybrid functionals by imposing the fundamental condition of a piecewise linear total energy with respect to electron number. For the Perdew-Burke-Ernzerhof (PBE) hybrid family of exchange-correlation functionals (i.e., for an approximate generalized Kohn-Sham theory) this implies that (i) the highest occupied molecular orbital corresponds to the ionization potential (I ), (ii) the energy of the lowest unoccupied molecular orbital corresponds to the electron affinity (A ), and (iii) the energies of the frontier orbitals are constant as a function of their occupation. In agreement with a previous study [N. Sai et al., Phys. Rev. Lett. 106, 226403 (2011), 10.1103/PhysRevLett.106.226403], we find that these conditions are met for high values of the exact exchange admixture α and illustrate their importance for the tetrathiafulvalene-tetracyanoquinodimethane complex for which standard density functional theory functionals predict artificial electron transfer. We further assess the performance for atomization energies and weak interaction energies. We find that atomization energies are significantly underestimated compared to PBE or PBE0, whereas the description of weak interaction energies improves significantly if a 1 /R6 van der Waals correction scheme is employed.
Parallel FE Approximation of the Even/Odd Parity Form of the Linear Boltzmann Equation
Drumm, Clifton R.; Lorenz, Jens
1999-07-21
A novel solution method has been developed to solve the linear Boltzmann equation on an unstructured triangular mesh. Instead of tackling the first-order form of the equation, this approach is based on the even/odd-parity form in conjunction with the conventional mdtigroup discrete-ordinates approximation. The finite element method is used to treat the spatial dependence. The solution method is unique in that the space-direction dependence is solved simultaneously, eliminating the need for the conventional inner iterations, and the method is well suited for massively parallel computers.
Brett, Tobias; Galla, Tobias
2013-06-21
We develop a systematic approach to the linear-noise approximation for stochastic reaction systems with distributed delays. Unlike most existing work our formalism does not rely on a master equation; instead it is based upon a dynamical generating functional describing the probability measure over all possible paths of the dynamics. We derive general expressions for the chemical Langevin equation for a broad class of non-Markovian systems with distributed delay. Exemplars of a model of gene regulation with delayed autoinhibition and a model of epidemic spread with delayed recovery provide evidence of the applicability of our results.
Tejero, E. M.; Gatling, G.
2009-03-15
A method for approximating arbitrary axial magnetic field profiles for a given solenoidal electromagnet coil array is described. The method casts the individual contributions from each coil as a truncated orthonormal basis for the space within the array. This truncated basis allows for the linear decomposition of an arbitrary profile function, which returns the appropriate currents for each coil to best reproduce the desired profile. We present the mathematical details of the method along with a detailed example of its use. The results from the method are used in a simulation and compared with magnetic field measuremen0008.
Delayed-exponential approximation of a linear homogeneous diffusion model of neuron.
Pacut, A; Dabrowski, L
1988-01-01
The diffusion models of neuronal activity are general yet conceptually simple and flexible enough to be useful in a variety of modeling problems. Unfortunately, even simple diffusion models lead to tedious numerical calculations. Consequently, the existing neural net models use characteristics of a single neuron taken from the "pre-diffusion" era of neural modeling. Simplistic elements of neural nets forbid to incorporate a single learning neuron structure into the net model. The above drawback cannot be overcome without the use of the adequate structure of the single neuron as an element of a net. A linear (not necessarily homogeneous) diffusion model of a single neuron is a good candidate for such a structure, it must, however, be simplified. In the paper the structure of the diffusion model of neuron is discussed and a linear homogeneous model with reflection is analyzed. For this model an approximation is presented, which is based on the approximation of the first passage time distribution of the Ornstein-Uhlenbeck process by the delayed (shifted) exponential distribution. The resulting model has a simple structure and has a prospective application in neural modeling and in analysis of neural nets.
NASA Astrophysics Data System (ADS)
Meruane, V.; Ortiz-Bernardin, A.
2015-03-01
Supervised learning algorithms have been proposed as a suitable alternative to model updating methods in structural damage assessment, being Artificial Neural Networks the most frequently used. Notwithstanding, the slow learning speed and the large number of parameters that need to be tuned within the training stage have been a major bottleneck in their application. This article presents a new algorithm for real-time damage assessment that uses a linear approximation method in conjunction with antiresonant frequencies that are identified from transmissibility functions. The linear approximation is handled by a statistical inference model based on the maximum-entropy principle. The merits of this new approach are twofold: training is avoided and data is processed in a period of time that is comparable to the one of Neural Networks. The performance of the proposed methodology is validated by considering three experimental structures: an eight-degree-of-freedom (DOF) mass-spring system, a beam, and an exhaust system of a car. To demonstrate the potential of the proposed algorithm over existing ones, the obtained results are compared with those of a model updating method based on parallel genetic algorithms and a multilayer feedforward neural network approach.
A novel impact identification algorithm based on a linear approximation with maximum entropy
NASA Astrophysics Data System (ADS)
Sanchez, N.; Meruane, V.; Ortiz-Bernardin, A.
2016-09-01
This article presents a novel impact identification algorithm that uses a linear approximation handled by a statistical inference model based on the maximum-entropy principle, termed linear approximation with maximum entropy (LME). Unlike other regression algorithms as artificial neural networks (ANNs) and support vector machines, the proposed algorithm requires only parameter to be selected and the impact is identified after solving a convex optimization problem that has a unique solution. In addition, with LME data is processed in a period of time that is comparable to the one of other algorithms. The performance of the proposed methodology is validated by considering an experimental aluminum plate. Time varying strain data is measured using four piezoceramic sensors bonded to the plate. To demonstrate the potential of the proposed approach over existing ones, results obtained via LME are compared with those of ANN and least square support vector machines. The results demonstrate that with a low number of sensors it is possible to accurately locate and quantify impacts on a structure and that LME outperforms other impact identification algorithms.
NASA Astrophysics Data System (ADS)
Schuster, Thomas; Schöpfer, Frank
2010-08-01
The method of approximate inverse is a mollification method for stably solving inverse problems. In its original form it has been developed to solve operator equations in L2-spaces and general Hilbert spaces. We show that the method of approximate inverse can be extended to solve linear, ill-posed problems in Banach spaces. This paper is restricted to function spaces. The method itself consists of evaluations of dual pairings of the given data with reconstruction kernels that are associated with mollifiers and the dual of the operator. We first define what we mean by a mollifier in general Banach spaces and then investigate two settings more exactly: the case of Lp-spaces and the case of the Banach space of continuous functions on a compact set. For both settings we present the criteria turning the method of approximate inverse into a regularization method and prove convergence with rates. As an application we refer to x-ray diffractometry which is a technique of non-destructive testing that is concerned with computing the stress tensor of a specimen. Since one knows that the stress tensor is smooth, x-ray diffractometry can appropriately be modelled by a Banach space setting using continuous functions.
Coherent-potential approximation in the tight-binding linear muffin-tin orbital method
NASA Astrophysics Data System (ADS)
Singh, Prabhakar P.; Gonis, A.
1993-07-01
We describe a consistent approach for applying the coherent-potential approximation (CPA) to the various representations of the linear muffin-tin orbital method. Unlike the previous works of Kudrnovský et al. [Phys. Rev. B 35, 2487 (1987); 41, 7515 (1990)], our results for the ensemble-averaged Green functions in the tight-binding representation yield E- and r-dependent quantities that are consistent with the traditional applications of the single-site CPA. To illustrate the reliability and the usefulness of our approach we compare the nonspherically averaged charge densities, calculated in real space, of ordered NiPt in L10 structure and the substitutionally disordered Ni0.5Pt0.5 on a face-centered-cubic lattice.
Approximating high-dimensional dynamics by barycentric coordinates with linear programming
Hirata, Yoshito Aihara, Kazuyuki; Suzuki, Hideyuki; Shiro, Masanori; Takahashi, Nozomu; Mas, Paloma
2015-01-15
The increasing development of novel methods and techniques facilitates the measurement of high-dimensional time series but challenges our ability for accurate 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 accurately model a high-dimensional time series. Our method extends the barycentric coordinates to high-dimensional phase space by employing linear programming, and allowing the approximation 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 accurately 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.
Are Bilinear Quadrilaterals Better Than Linear Triangles?
D'Azevedo, E.F.
1993-01-01
This paper compares the theoretical effectiveness of bilinear approximation over quadrilaterals with linear approximation over triangles. Anisotropic mesh transformation is used to generate asymptotically optimally efficient meshes for piecewise linear interpolation over triangles and bilinear interpolation over quadrilaterals. For approximating a convex function, although bilinear quadrilaterals are more efficient, linear triangles are more accurate and may be preferred in finite element computations; whereas for saddle-shaped functions, quadrilaterals may offer a higher order approximation on a well-designed mesh. A surprising finding is different grid orientations may yield an order of magnitude improvement in approximation accuracy.
NASA Astrophysics Data System (ADS)
Sun, Xiong; Wang, Haobin; Miller, William H.
1998-11-01
A linearized approximation to the semiclassical initial value representation (SC-IVR), referred to herein as the LSC-IVR, was used by us in a recent paper [J. Chem. Phys. 108, 9726 (1998)] to calculate reactive flux correlation functions for a model of a chemical reaction on a single potential energy surface. This paper shows how the LSC-IVR—which is much easier to apply than the full SC-IVR because it linearizes the phase difference between interfering classical trajectories—can be applied to electronically nonadiabatic processes, i.e., those involving transitions between different potential-energy surfaces. Applications to several model problems are presented to show its usefulness: These are the nonadiabatic scattering problems used by Tully to test surface-hopping models, and also the spin-boson model of coupled electronic states in a condensed phase environment. Though not as accurate as the full SC-IVR, the LSC-IVR does a reasonably good job for all these applications, even describing correctly Stuckelberg oscillations (interference between nonadiabatic transitions) and the transition between coherent and incoherent behavior in the spin-boson example.
Sun, X.; Wang, H.; Miller, W.H. |
1998-11-01
A linearized approximation to the semiclassical initial value representation (SC-IVR), referred to herein as the LSC-IVR, was used by us in a recent paper [J. Chem. Phys. {bold 108}, 9726 (1998)] to calculate reactive flux correlation functions for a model of a chemical reaction on a single potential energy surface. This paper shows how the LSC-IVR{emdash}which is much easier to apply than the full SC-IVR because it linearizes the phase difference between interfering classical trajectories{emdash}can be applied to {ital electronically nonadiabatic} processes, i.e., those involving transitions between different potential-energy surfaces. Applications to several model problems are presented to show its usefulness: These are the nonadiabatic scattering problems used by Tully to test surface-hopping models, and also the spin{endash}boson model of coupled electronic states in a condensed phase environment. Though not as accurate as the full SC-IVR, the LSC-IVR does a reasonably good job for all these applications, even describing correctly Stuckelberg oscillations (interference between nonadiabatic transitions) and the transition between coherent and incoherent behavior in the spin{endash}boson example. {copyright} {ital 1998 American Institute of Physics.} thinsp
Linear-scaling implementation of the direct random-phase approximation
Kállay, Mihály
2015-05-28
We report the linear-scaling implementation of the direct random-phase approximation (dRPA) for closed-shell molecular systems. As a bonus, linear-scaling algorithms are also presented for the second-order screened exchange extension of dRPA as well as for the second-order Møller–Plesset (MP2) method and its spin-scaled variants. Our approach is based on an incremental scheme which is an extension of our previous local correlation method [Rolik et al., J. Chem. Phys. 139, 094105 (2013)]. The approach extensively uses local natural orbitals to reduce the size of the molecular orbital basis of local correlation domains. In addition, we also demonstrate that using natural auxiliary functions [M. Kállay, J. Chem. Phys. 141, 244113 (2014)], the size of the auxiliary basis of the domains and thus that of the three-center Coulomb integral lists can be reduced by an order of magnitude, which results in significant savings in computation time. The new approach is validated by extensive test calculations for energies and energy differences. Our benchmark calculations also demonstrate that the new method enables dRPA calculations for molecules with more than 1000 atoms and 10 000 basis functions on a single processor.
Helbig, N.; Fuks, J. I.; Verstraete, M. J.; Marques, M. A. L.; Tokatly, I. V.; Rubio, A.
2011-03-15
We present a local density approximation (LDA) for one-dimensional (1D) systems interacting via the soft-Coulomb interaction based on quantum Monte Carlo calculations. Results for the ground-state energies and ionization potentials of finite 1D systems show excellent agreement with exact calculations obtained by exploiting the mapping of an N-electron system in d dimensions onto a single electron in Nxd dimensions, properly symmetrized by the Young diagrams. We conclude that 1D LDA is of the same quality as its three-dimensional (3D) counterpart, and we infer conclusions about 3D LDA. The linear and nonlinear time-dependent responses of 1D model systems using LDA, exact exchange, and the exact solution are investigated and show very good agreement in both cases, except for the well-known problem of missing double excitations. Consequently, the 3D LDA is expected to be of good quality beyond the linear response. In addition, the 1D LDA should prove useful in modeling the interaction of atoms with strong laser fields, where this specific 1D model is often used.
Chen, Zikuan; Calhoun, Vince
2015-05-01
The underlying source of brain imaging by T2*-weighted magnetic resonance imaging (T2*MRI) is the intracranial inhomogeneous tissue magnetic susceptibility (denoted by χ) that causes an inhomogeneous field map (via magnetization) in a main field. By decomposing T2*MRI into two steps, we understand that the 1st step from a χ source to a field map is a linear but non-isomorphic spatial mapping, and the 2nd step from the field map to a T2* image is a nonlinear mapping due to the trigonometric behavior of spin precession signals. The magnitude and phase calculations from a complex T2* image introduce additional nonlinearities. In this report, we look into the magnitude and phase behaviors of a T2* image (signal) by theoretical approximation and Monte Carlo simulation. We perform the 1st-order Taylor expansion on intravoxel dephasing formula of a T2* signal and show that the T2* magnitude is a quadratic mapping of the field map and T2* phase is a linear isomorphic mapping. By Monte Carlo simulation of T2*MRI for a span of echo times (with B0=3T and TE=[0,120] ms), we first confirm the quadratic magnitude and linear phase behaviors in small phase angle regime (via TE <30ms), and then provide more general magnitude and phase nonlinear behaviors in large phase angle scenarios (via TE >30ms). By solving the inverse problem of T2*MRI, we demonstrate χ tomography and conclude that the χ source can be reliably reconstructed from a T2* phase image in a small phase angle regime. PMID:25623809
Structural interfaces in linear elasticity. Part I: Nonlocality and gradient approximations
NASA Astrophysics Data System (ADS)
Bertoldi, K.; Bigoni, D.; Drugan, W. J.
2007-01-01
Many biological and optimal materials, at multiple scales, consist of what can be idealized as continuous bodies joined by structural interfaces. Mechanical characterization of the microstructure defining the interface can nowadays be accurately done; however, such interfaces are usually analyzed employing models where those properties are overly simplified. To introduce into the analysis the microstructure properties, a new model of structural interfaces is proposed and developed: a true structure is introduced in the transition zone, joining continuous bodies, with geometrical and material properties directly obtained from those of the interfacial microstructure. First, the case of an elliptical inclusion connected by a structural interface to an infinite matrix is solved analytically, showing that nonlocal effects follow directly from the introduction of the structure, related to the inclination of the connecting elements. Second, starting from a discrete structure, a continuous model of a structural interface is derived. The usual zero-thickness linear interface model is shown to be a special case of this more general continuous structural interface model. Then, a gradient approximation of the interface constitutive law is rigorously derived: it is the first example of the analytical derivation of a nonlocal interface model from the microstructure properties. The effects introduced in the mechanical behavior by both the continuous model and its gradient approximation are illustrated by solving, for the first time, the problem of a circular inclusion connected to an infinite matrix by a structural interface and subject to remote uniform stress.
Gaussian functional approximation to 't Hooft's extension of the linear Σ model
NASA Astrophysics Data System (ADS)
Nakamura, Issei; Dmitrašinović, V.
2012-03-01
We apply a self-consistent relativistic mean-field variational “Gaussian functional” (or optimized one-loop perturbation theory, or Hartree+RPA) approximation to the extended Nf=2 linear σ model with spontaneously and explicitly broken chiral SUR(2)×SUL(2)×UA(1)≡O(4)×O(2) symmetry. We set up the self-consistency, or gap equations that dress up the bare fields with “cactus tree” loop diagrams, and the Bethe-Salpeter equations that provide further dressing with one-loop irreducible diagrams. In a previous publication [V. Dmitrašinović and I. Nakamura, J. Math. Phys. (N.Y.)JMAPAQ0022-2488 44, 2839 (2003).10.1063/1.1576907] we have already shown the ability of this approximation to create composite (i.e., bound and/or resonance) states. With explicit SUR(2)×SUL(2)×UA(1) chiral symmetry breaking first we consider how the UA(1) symmetry induced scalar-pseudoscalar meson mass relation that is known to hold in fermionic chiral models is modified by the bosonic gap equations. Then we solve the gap and Bethe-Salpeter equations numerically and discuss the solutions’ properties and the particle content of the theory. We show that in the strong-coupling regime two, sometimes even three solutions to the η meson channel Bethe-Salpeter equation may coexist.
Wallace, E W J; Gillespie, D T; Sanft, K R; Petzold, L R
2012-08-01
The linear noise approximation (LNA) is a way of approximating the stochastic time evolution of a well-stirred chemically reacting system. It can be obtained either as the lowest order correction to the deterministic chemical reaction rate equation (RRE) in van Kampen's system-size expansion of the chemical master equation (CME), or by linearising the two-term-truncated chemical Kramers-Moyal equation. However, neither of those derivations sheds much light on the validity of the LNA. The problematic character of the system-size expansion of the CME for some chemical systems, the arbitrariness of truncating the chemical Kramers-Moyal equation at two terms, and the sometimes poor agreement of the LNA with the solution of the CME, have all raised concerns about the validity and usefulness of the LNA. Here, the authors argue that these concerns can be resolved by viewing the LNA as an approximation of the chemical Langevin equation (CLE). This view is already implicit in Gardiner's derivation of the LNA from the truncated Kramers-Moyal equation, as that equation is mathematically equivalent to the CLE. However, the CLE can be more convincingly derived in a way that does not involve either the truncated Kramers-Moyal equation or the system-size expansion. This derivation shows that the CLE will be valid, at least for a limited span of time, for any system that is sufficiently close to the thermodynamic (large-system) limit. The relatively easy derivation of the LNA from the CLE shows that the LNA shares the CLE's conditions of validity, and it also suggests that what the LNA really gives us is a description of the initial departure of the CLE from the RRE as we back away from the thermodynamic limit to a large but finite system. The authors show that this approach to the LNA simplifies its derivation, clarifies its limitations, and affords an easier path to its solution.
NASA Astrophysics Data System (ADS)
Liu, Y.; Li, T.; Zhu, C.; Zhang, R.; Wu, Y.
2015-12-01
Three-dimensional (3-D) electromagnetic (EM) forward modelling and inversion continues to be an important issue for the correct interpretation of EM data.To this end,approximate solutions have been developed that allow the construction of relatively fast forward modelling and inversion schemes.We have developed an improved quasi-linear approximation which is more appropriate in solving the linear equation for greatly shortening calculation time.We achieved this by using green's function properties.Then we introduced the improved quasi-linear approximation to spectral induced polarization (SIP) to tackle the problem of the resolution and the efficiency.The localized quasi-linear (LQL) approximation theory is appropriate for multisource array-type surveys assuming that the normal field is slowly varying within the inhomogeneity domain.However,the normal field of attenuates severely which dose not satisfy the assumption of the LQL approximation.As a consenquence,the imaginary part is not accurate when LQL approximation is adopted for the simulation.The improved quasi-linear approximation provide a new approach with the same resolution of QL approximation and much less calculation time.We have also constructed three-dimensional SIP forward modeling based on improved quasi-linear approximation method.It only takes 0.8s for forward modeling when inhomogeneity domain is divided into 2000 blocks.Beyond that, we have introduced the Cole-Cole model to the algorithm and complete the three-dimensional complex resistivity conjugate gradient inversion with parameter restraint.The model trial results show that this method can obtain good inversion results in physical parameters such as zero frequency resistivity, polarization.The results demonstrate the stability and the efficiency of the improved quasi-linear approximation and the method may be a practical solution for3-D EM forward modelling and inversion of SIP.
MAP estimators for piecewise continuous inversion
NASA Astrophysics Data System (ADS)
Dunlop, M. M.; Stuart, A. M.
2016-10-01
We study the inverse problem of estimating a field u a from data comprising a finite set of nonlinear functionals of u a , subject to additive noise; we denote this observed data by y. Our interest is in the reconstruction of piecewise continuous fields u a in which the discontinuity set is described by a finite number of geometric parameters a. Natural applications include groundwater flow and electrical impedance tomography. We take a Bayesian approach, placing a prior distribution on u a and determining the conditional distribution on u a given the data y. It is then natural to study maximum a posterior (MAP) estimators. Recently (Dashti et al 2013 Inverse Problems 29 095017) it has been shown that MAP estimators can be characterised as minimisers of a generalised Onsager-Machlup functional, in the case where the prior measure is a Gaussian random field. We extend this theory to a more general class of prior distributions which allows for piecewise continuous fields. Specifically, the prior field is assumed to be piecewise Gaussian with random interfaces between the different Gaussians defined by a finite number of parameters. We also make connections with recent work on MAP estimators for linear problems and possibly non-Gaussian priors (Helin and Burger 2015 Inverse Problems 31 085009) which employs the notion of Fomin derivative. In showing applicability of our theory we focus on the groundwater flow and EIT models, though the theory holds more generally. Numerical experiments are implemented for the groundwater flow model, demonstrating the feasibility of determining MAP estimators for these piecewise continuous models, but also that the geometric formulation can lead to multiple nearby (local) MAP estimators. We relate these MAP estimators to the behaviour of output from MCMC samples of the posterior, obtained using a state-of-the-art function space Metropolis-Hastings method.
Groups of piecewise projective homeomorphisms
Monod, Nicolas
2013-01-01
The group of piecewise projective homeomorphisms of the line provides straightforward torsion-free counterexamples to the so-called von Neumann conjecture. The examples are so simple that many additional properties can be established.
Boundary parametric approximation to the linearized scalar potential magnetostatic field problem
Bramble, J.H.; Pasciak, J.E.
1984-01-01
We consider the linearized scalar potential formulation of the magnetostatic field problem in this paper. Our approach involves a reformulation of the continuous problem as a parametric boundary problem. By the introduction of a spherical interface and the use of spherical harmonics, the infinite boundary conditions can also be satisfied in the parametric framework. That is, the field in the exterior of a sphere is expanded in a harmonic series of eigenfunctions for the exterior harmonic problem. The approach is essentially a finite element method coupled with a spectral method via a boundary parametric procedure. The reformulated problem is discretized by finite element techniques which lead to a discrete parametric problem which can be solved by well conditioned iteration involving only the solution of decoupled Neumann type elliptic finite element systems and L/sup 2/ projection onto subspaces of spherical harmonics. Error and stability estimates given show exponential convergence in the degree of the spherical harmonics and optimal order convergence with respect to the finite element approximation for the resulting fields in L/sup 2/. 24 references.
How reliable is the linear noise approximation of gene regulatory networks?
2013-01-01
Background The linear noise approximation (LNA) is commonly used to predict how noise is regulated and exploited at the cellular level. These predictions are exact for reaction networks composed exclusively of first order reactions or for networks involving bimolecular reactions and large numbers of molecules. It is however well known that gene regulation involves bimolecular interactions with molecule numbers as small as a single copy of a particular gene. It is therefore questionable how reliable are the LNA predictions for these systems. Results We implement in the software package intrinsic Noise Analyzer (iNA), a system size expansion based method which calculates the mean concentrations and the variances of the fluctuations to an order of accuracy higher than the LNA. We then use iNA to explore the parametric dependence of the Fano factors and of the coefficients of variation of the mRNA and protein fluctuations in models of genetic networks involving nonlinear protein degradation, post-transcriptional, post-translational and negative feedback regulation. We find that the LNA can significantly underestimate the amplitude and period of noise-induced oscillations in genetic oscillators. We also identify cases where the LNA predicts that noise levels can be optimized by tuning a bimolecular rate constant whereas our method shows that no such regulation is possible. All our results are confirmed by stochastic simulations. Conclusion The software iNA allows the investigation of parameter regimes where the LNA fares well and where it does not. We have shown that the parametric dependence of the coefficients of variation and Fano factors for common gene regulatory networks is better described by including terms of higher order than LNA in the system size expansion. This analysis is considerably faster than stochastic simulations due to the extensive ensemble averaging needed to obtain statistically meaningful results. Hence iNA is well suited for performing
Piecewise nonlinear image registration using DCT basis functions
NASA Astrophysics Data System (ADS)
Gan, Lin; Agam, Gady
2015-03-01
The deformation field in nonlinear image registration is usually modeled by a global model. Such models are often faced with the problem that a locally complex deformation cannot be accurately modeled by simply increasing degrees of freedom (DOF). In addition, highly complex models require additional regularization which is usually ineffective when applied globally. Registering locally corresponding regions addresses this problem in a divide and conquer strategy. In this paper we propose a piecewise image registration approach using Discrete Cosine Transform (DCT) basis functions for a nonlinear model. The contributions of this paper are three-folds. First, we develop a multi-level piecewise registration framework that extends the concept of piecewise linear registration and works with any nonlinear deformation model. This framework is then applied to nonlinear DCT registration. Second, we show how adaptive model complexity and regularization could be applied for local piece registration, thus accounting for higher variability. Third, we show how the proposed piecewise DCT can overcome the fundamental problem of a large curvature matrix inversion in global DCT when using high degrees of freedoms. The proposed approach can be viewed as an extension of global DCT registration where the overall model complexity is increased while achieving effective local regularization. Experimental evaluation results provide comparison of the proposed approach to piecewise linear registration using an affine transformation model and a global nonlinear registration using DCT model. Preliminary results show that the proposed approach achieves improved performance.
Piecewise-Planar Parabolic Reflectarray Antenna
NASA Technical Reports Server (NTRS)
Hodges, Richard; Zawadzki, Mark
2009-01-01
The figure shows a dual-beam, dualpolarization Ku-band antenna, the reflector of which comprises an assembly of small reflectarrays arranged in a piecewise- planar approximation of a parabolic reflector surface. The specific antenna design is intended to satisfy requirements for a wide-swath spaceborne radar altimeter, but the general principle of piecewise-planar reflectarray approximation of a parabolic reflector also offers advantages for other applications in which there are requirements for wideswath antennas that can be stowed compactly and that perform equally in both horizontal and vertical polarizations. The main advantages of using flat (e.g., reflectarray) antenna surfaces instead of paraboloidal or parabolic surfaces is that the flat ones can be fabricated at lower cost and can be stowed and deployed more easily. Heretofore, reflectarray antennas have typically been designed to reside on single planar surfaces and to emulate the focusing properties of, variously, paraboloidal (dish) or parabolic antennas. In the present case, one approximates the nominal parabolic shape by concatenating several flat pieces, while still exploiting the principles of the planar reflectarray for each piece. Prior to the conception of the present design, the use of a single large reflectarray was considered, but then abandoned when it was found that the directional and gain properties of the antenna would be noticeably different for the horizontal and vertical polarizations.
Meshkov, E.E.; Mokhov, V.N.
1983-01-01
Stability problems and the development of small perturbations in gasdynamics are ordinarily investigated by using the solution of linearized equations. The applicability of the linear approximation is usually determined by the smallness of the perturbation. However, a linear approximation turns out to be false in a number of cases. The authors consider a plane problem in which a characteristic surface curved along a sinusoid moves over a substance at a constant velocity. In this case, the change in surface shape with time is determined by the Huygens principle. Also considered is the one-dimensional flow of an ideal gas with adiabatic index ..nu.. in which there is a small sinusoidal perturbation at the initial time. These examples are encountered locally in the majority of problems on flow stability in gasdynamics. It is shown that the shape of the reflected wave front deviates from the sinusoidal with time and the formation of singularities and the deviation from the sinusoidal shape slow down as the amplitude of the initial perturbation diminishes. The authors conclude that utilization of a linearized approximation to solve the gasdynamics equations is possible only up to a certain time. Consequently, application of asymptotic formulas obtained on the basis of a linear approximation for a finite magnitude of the perbation requires an additional foundation.
NASA Astrophysics Data System (ADS)
Simpson, D. J. W.
2016-09-01
An attractor of a piecewise-smooth continuous system of differential equations can bifurcate from a stable equilibrium to a more complicated invariant set when it collides with a switching manifold under parameter variation. Here numerical evidence is provided to show that this invariant set can be chaotic. The transition occurs locally (in a neighbourhood of a point) and instantaneously (for a single critical parameter value). This phenomenon is illustrated for the normal form of a boundary equilibrium bifurcation in three dimensions using parameter values adapted from of a piecewise-linear model of a chaotic electrical circuit. The variation of a secondary parameter reveals a period-doubling cascade to chaos with windows of periodicity. The dynamics is well approximated by a one-dimensional unimodal map which explains the bifurcation structure. The robustness of the attractor is also investigated by studying the influence of nonlinear terms.
Are bilinear quadrilaterals better than linear triangles?
D`Azevedo, E.F.
1993-08-01
This paper compares the theoretical effectiveness of bilinear approximation over quadrilaterals with linear approximation over triangles. Anisotropic mesh transformation is used to generate asymptotically optimally efficient meshes for piecewise linear interpolation over triangles and bilinear interpolation over quadrilaterals. The theory and numerical results suggest triangles may have a slight advantage over quadrilaterals for interpolating convex data function but bilinear approximation may offer a higher order approximation for saddle-shaped functions on a well-designed mesh. This work is a basic study on optimal meshes with the intention of gaining insight into the more complex meshing problems in finite element analysis.
Ergodic theory and Diophantine approximation for translation surfaces and linear forms
NASA Astrophysics Data System (ADS)
Athreya, Jayadev; Parrish, Andrew; Tseng, Jimmy
2016-08-01
We derive results on the distribution of directions of saddle connections on translation surfaces using only the Birkhoff ergodic theorem applied to the geodesic flow on the moduli space of translation surfaces. Our techniques, together with an approximation argument, also give an alternative proof of a weak version of a classical theorem in multi-dimensional Diophantine approximation due to Schmidt (1960 Can. J. Math. 12 619–31, 1964 Trans. Am. Math. Soc. 110 493–518). The approximation argument allows us to deduce the Birkhoff genericity of almost all lattices in a certain submanifold of the space of unimodular lattices from the Birkhoff genericity of almost all lattices in the whole space and similarly for the space of affine unimodular lattices.
The lens effect of a big spherical inhomogeneity in the linear approximation
Moreno, J.; Portilla, M. )
1990-04-01
The paper addresses a large gravitational lens, of dimensions comparable with observer-lens-source distances, for arbitrary lens-observer-source angles. The lens is approximated by a small, pressureless, spsherically symmetric perturbation in a Einstein-de Sitter universe. The deflection angle contains essential terms which do not appear when the lens is approximated by an isolated body in a Minkowskian space. These terms should be considered to study the optical appearance of the inhomogeneity. The lens equation explicitly conserves brightness over the whole celestial sphere of the observer. 8 refs.
Construction of approximate analytical solutions to a new class of non-linear oscillator equation
NASA Technical Reports Server (NTRS)
Mickens, R. E.; Oyedeji, K.
1985-01-01
The principle of harmonic balance is invoked in the development of an approximate analytic model for a class of nonlinear oscillators typified by a mass attached to a stretched wire. By assuming that harmonic balance will hold, solutions are devised for a steady state limit cycle and/or limit point motion. A method of slowly varying amplitudes then allows derivation of approximate solutions by determining the form of the exact solutions and substituting into them the lowest order terms of their respective Fourier expansions. The latter technique is actually a generalization of the method proposed by Kryloff and Bogoliuboff (1943).
Approximation functions for airblast environments from buried charges
Reichenbach, H.; Behrens, K.; Kuhl, A.L.
1993-11-01
In EMI report E 1/93, ``Airblast Environments from Buried HE-Charges,`` fit functions were used for the compact description of blastwave parameters. The coefficients of these functions were approximated by means of second order polynomials versus DOB. In most cases, the agreement with the measured data was satisfactory; to reduce remaining noticeable deviations, an approximation by polygons (i.e., piecewise-linear approximation) was used instead of polynomials. The present report describes the results of the polygon approximation and compares them to previous data. We conclude that the polygon representation leads to a better agreement with the measured data.
Approximating electronically excited states with equation-of-motion linear coupled-cluster theory
Byrd, Jason N. Rishi, Varun; Perera, Ajith; Bartlett, Rodney J.
2015-10-28
A new perturbative approach to canonical equation-of-motion coupled-cluster theory is presented using coupled-cluster perturbation theory. A second-order Møller-Plesset partitioning of the Hamiltonian is used to obtain the well known equation-of-motion many-body perturbation theory equations and two new equation-of-motion methods based on the linear coupled-cluster doubles and linear coupled-cluster singles and doubles wavefunctions. These new methods are benchmarked against very accurate theoretical and experimental spectra from 25 small organic molecules. It is found that the proposed methods have excellent agreement with canonical equation-of-motion coupled-cluster singles and doubles state for state orderings and relative excited state energies as well as acceptable quantitative agreement for absolute excitation energies compared with the best estimate theory and experimental spectra.
Balagurov, B. Ya.
2011-11-15
A general approach is proposed for calculating the conductivity of anisotropic composites with a low concentration of inclusions of an arbitrary shape. The contribution to effective conductivity {sigma}{sub e} , which is linear in concentration, is expressed in terms of the polarizability of the inclusion defined in a certain transformed system in which the inclusion is surrounded by an isotropic matrix. A transition to this system is performed using a symmetry transformation that does not change the equations for direct current.
Key-frame retrieval from MPEG video based on linear approximation of content curve
NASA Astrophysics Data System (ADS)
Kim, Tae-hee; Lee, Woong-hee; Jeong, Dong-seok
2003-01-01
In general, video is too much lengthy for browsing the contents. So, there are many efforts being made for browsing the content of the video fast and effectively. Video summary is the one of techniques related to those efforts. Video summary comprises a number of key-frames. Therefore, we propose a method to extract key-frames from the video in MPEG compressed domain. Proposed method extracts the simple 2D content curve reflecting the variation of the video content from the MPEG video in the compressed domain, approximates the curve to polygonal lines and then extracts key-frames from the approximated lines effectively and rapidly. Also, proposed method let the user set the number of key-frames.
Ringham, Brandy M; Kreidler, Sarah M; Muller, Keith E; Glueck, Deborah H
2016-07-30
Multilevel and longitudinal studies are frequently subject to missing data. For example, biomarker studies for oral cancer may involve multiple assays for each participant. Assays may fail, resulting in missing data values that can be assumed to be missing completely at random. Catellier and Muller proposed a data analytic technique to account for data missing at random in multilevel and longitudinal studies. They suggested modifying the degrees of freedom for both the Hotelling-Lawley trace F statistic and its null case reference distribution. We propose parallel adjustments to approximate power for this multivariate test in studies with missing data. The power approximations use a modified non-central F statistic, which is a function of (i) the expected number of complete cases, (ii) the expected number of non-missing pairs of responses, or (iii) the trimmed sample size, which is the planned sample size reduced by the anticipated proportion of missing data. The accuracy of the method is assessed by comparing the theoretical results to the Monte Carlo simulated power for the Catellier and Muller multivariate test. Over all experimental conditions, the closest approximation to the empirical power of the Catellier and Muller multivariate test is obtained by adjusting power calculations with the expected number of complete cases. The utility of the method is demonstrated with a multivariate power analysis for a hypothetical oral cancer biomarkers study. We describe how to implement the method using standard, commercially available software products and give example code. Copyright © 2015 John Wiley & Sons, Ltd. PMID:26603500
Ringham, Brandy M; Kreidler, Sarah M; Muller, Keith E; Glueck, Deborah H
2016-07-30
Multilevel and longitudinal studies are frequently subject to missing data. For example, biomarker studies for oral cancer may involve multiple assays for each participant. Assays may fail, resulting in missing data values that can be assumed to be missing completely at random. Catellier and Muller proposed a data analytic technique to account for data missing at random in multilevel and longitudinal studies. They suggested modifying the degrees of freedom for both the Hotelling-Lawley trace F statistic and its null case reference distribution. We propose parallel adjustments to approximate power for this multivariate test in studies with missing data. The power approximations use a modified non-central F statistic, which is a function of (i) the expected number of complete cases, (ii) the expected number of non-missing pairs of responses, or (iii) the trimmed sample size, which is the planned sample size reduced by the anticipated proportion of missing data. The accuracy of the method is assessed by comparing the theoretical results to the Monte Carlo simulated power for the Catellier and Muller multivariate test. Over all experimental conditions, the closest approximation to the empirical power of the Catellier and Muller multivariate test is obtained by adjusting power calculations with the expected number of complete cases. The utility of the method is demonstrated with a multivariate power analysis for a hypothetical oral cancer biomarkers study. We describe how to implement the method using standard, commercially available software products and give example code. Copyright © 2015 John Wiley & Sons, Ltd.
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.
Losa, C.; Doessing, T.; Pastore, A.; Vigezzi, E.; Broglia, R. A.
2010-06-15
We present a calculation of the properties of vibrational states in deformed, axially-symmetric even-even nuclei, within the framework of a fully self-consistent quasiparticle random phase approximation (QRPA). The same Skyrme energy density and density-dependent pairing functionals are used to calculate the mean field and the residual interaction in the particle-hole and particle-particle channels. We have tested our software in the case of spherical nuclei against fully self-consistent calculations published in the literature, finding excellent agreement. We investigate the consequences of neglecting the spin-orbit and Coulomb residual interactions in QRPA. Furthermore we discuss the improvement obtained in the QRPA result associated with the removal of spurious modes. Isoscalar and isovector responses in the deformed {sup 24-26}Mg, {sup 34}Mg isotopes are presented and compared to experimental findings.
Zimmer, Christoph; Sahle, Sven
2015-10-01
Estimating model parameters from experimental data is a crucial technique for working with computational models in systems biology. Since stochastic models are increasingly important, parameter estimation methods for stochastic modelling are also of increasing interest. This study presents an extension to the 'multiple shooting for stochastic systems (MSS)' method for parameter estimation. The transition probabilities of the likelihood function are approximated with normal distributions. Means and variances are calculated with a linear noise approximation on the interval between succeeding measurements. The fact that the system is only approximated on intervals which are short in comparison with the total observation horizon allows to deal with effects of the intrinsic stochasticity. The study presents scenarios in which the extension is essential for successfully estimating the parameters and scenarios in which the extension is of modest benefit. Furthermore, it compares the estimation results with reversible jump techniques showing that the approximation does not lead to a loss of accuracy. Since the method is not based on stochastic simulations or approximative sampling of distributions, its computational speed is comparable with conventional least-squares parameter estimation methods.
Validity of the kink approximation to the tunneling action
NASA Astrophysics Data System (ADS)
Dutta, Koushik; Hector, Cecelie; Konstandin, Thomas; Vaudrevange, Pascal M.; Westphal, Alexander
2012-12-01
Coleman tunneling in a general scalar potential with two nondegenerate minima is known to have an approximation in terms of a piecewise linear triangular-shaped potential with sharp “kinks” at the place of the local minima. This approximate potential has a regime where the existence of the bounce solution needs the scalar field to “wait” for some amount of Euclidean time at one of the kinks. We discuss under which conditions a kink approximation of locally smooth “cap” regions provides a good estimate for the bounce action.
NASA Astrophysics Data System (ADS)
Hainzl, Christian; Seyrich, Jonathan
2016-05-01
In this paper we report on the results of a numerical study of the nonlinear time-dependent Bardeen-Cooper-Schrieffer (BCS) equations, often also denoted as Bogoliubov-de-Gennes (BdG) equations, for a one-dimensional system of fermions with contact interaction. We show that, even above the critical temperature, the full equations and their linear approximation give rise to completely different evolutions. In contrast to its linearization, the full nonlinear equation does not show any diffusive behavior in the order parameter. This means that the order parameter does not follow a Ginzburg-Landau-type of equation, in accordance with a recent theoretical result in [R.L. Frank, C. Hainzl, B. Schlein, R. Seiringer, to appear in Lett. Math. Phys., arXiv:1504.05885 (2016)]. We include a full description on the numerical implementation of the partial differential BCS/BdG equations.
Catt, B; Snyder, M
2014-06-15
Purpose: To investigate the use of the linear Boltzmann transport equation as a dose calculation tool which can account for interface effects, while still having faster computation times than Monte Carlo methods. In particular, we introduce a forward scattering approximation, in hopes of improving calculation time without a significant hindrance to accuracy. Methods: Two coupled Boltzmann transport equations were constructed, one representing the fluence of photons within the medium, and the other, the fluence of electrons. We neglect the scattering term within the electron transport equation, resulting in an extreme forward scattering approximation to reduce computational complexity. These equations were then solved using a numerical technique for solving partial differential equations, known as a finite difference scheme, where the fluence at each discrete point in space is calculated based on the fluence at the previous point in the particle's path. Using this scheme, it is possible to develop a solution to the Boltzmann transport equations by beginning with boundary conditions and iterating across the entire medium. The fluence of electrons can then be used to find the dose at any point within the medium. Results: Comparisons with Monte Carlo simulations indicate that even simplistic techniques for solving the linear Boltzmann transport equation yield expected interface effects, which many popular dose calculation algorithms are not capable of predicting. Implementation of a forward scattering approximation does not appear to drastically reduce the accuracy of this algorithm. Conclusion: Optimized implementations of this algorithm have been shown to be very accurate when compared with Monte Carlo simulations, even in build up regions where many models fail. Use of a forward scattering approximation could potentially give a reasonably accurate dose distribution in a shorter amount of time for situations where a completely accurate dose distribution is not
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
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.
NASA Astrophysics Data System (ADS)
Dagrau, Franck; Coulouvrat, François; Marchiano, Régis; Héron, Nicolas
2008-06-01
Dassault Aviation as a civil aircraft manufacturer is studying the feasibility of a supersonic business jet with the target of an "acceptable" sonic boom at the ground level, and in particular in case of focusing. A sonic boom computational process has been performed, that takes into account meteorological effects and aircraft manoeuvres. Turn manoeuvres and aircraft acceleration create zones of convergence of rays (caustics) which are the place of sound amplification. Therefore two elements have to be evaluated: firstly the geometrical position of the caustics, and secondly the noise level in the neighbourhood of the caustics. The modelling of the sonic boom propagation is based essentially on the assumptions of geometrical acoustics. Ray tracing is obtained according to Fermat's principle as paths that minimise the propagation time between the source (the aircraft) and the receiver. Wave amplitude and time waveform result from the solution of the inviscid Burgers' equation written along each individual ray. The "age variable" measuring the cumulative nonlinear effects is linked to the ray tube area. Caustics are located as the place where the ray tube area vanishes. Since geometrical acoustics does not take into account diffraction effects, it breaks down in the neighbourhood of caustics where it would predict unphysical infinite pressure amplitude. The aim of this study is to describe an original method for computing the focused noise level. The approach involves three main steps that can be summarised as follows. The propagation equation is solved by a forward marching procedure split into three successive steps: linear propagation in a homogeneous medium, linear perturbation due to the weak heterogeneity of the medium, and non-linear effects. The first step is solved using an "exact" angular spectrum algorithm. Parabolic approximation is applied only for the weak perturbation due to the heterogeneities. Finally, non linear effects are performed by solving the
Designing Chaotic Systems by Piecewise Affine Systems
NASA Astrophysics Data System (ADS)
Wu, Tiantian; Li, Qingdu; Yang, Xiao-Song
Based on mathematical analysis, this paper provides a methodology to ensure the existence of homoclinic orbits in a class of three-dimensional piecewise affine systems. In addition, two chaotic generators are provided to illustrate the effectiveness of the method.
NASA Astrophysics Data System (ADS)
Mourenas, D.; Artemyev, A. V.; Agapitov, O. V.; Krasnoselskikh, V.; Li, W.
2014-12-01
The distribution of trapped energetic electrons inside the Earth's radiation belts is the focus of intense studies aiming at better describing the evolution of the space environment in the presence of various disturbances induced by the solar wind or by an enhanced lightning activity. Such studies are usually performed by means of comparisons with full numerical simulations solving the Fokker-Planck quasi-linear diffusion equation for the particle distribution function. Here we present for the first time approximate but realistic analytical solutions for the electron distribution, which are shown to be in good agreement with exact numerical solutions in situations where resonant scattering of energetic electrons by whistler mode hiss, lightning-generated or chorus waves, is the dominant process. Quiet time distributions are well recovered, as well as the evolution of energized relativistic electron distributions during disturbed geomagnetic conditions. It is further shown that careful comparisons between the analytical solutions and measured distributions may allow to infer important bounce- and drift-averaged wave characteristics (such as wave amplitude). It could also help to improve the global understanding of underlying physical phenomena.
NASA Astrophysics Data System (ADS)
Yao, Weigang; Liou, Meng-Sing
2016-08-01
To preserve nonlinearity of a full-order system over a range of parameters of interest, we propose an accurate and robust nonlinear modeling approach by assembling a set of piecewise linear local solutions expanded about some sampling states. The work by Rewienski and White [1] on micromachined devices inspired our use of piecewise linear local solutions to study nonlinear unsteady aerodynamics. These local approximations are assembled via nonlinear weights of radial basis functions. The efficacy of the proposed procedure is validated for a two-dimensional airfoil moving with different pitching motions, specifically AGARD's CT2 and CT5 problems [27], in which the flows exhibit different nonlinear behaviors. Furthermore, application of the developed aerodynamic model to a two-dimensional aero-elastic system proves the approach is capable of predicting limit cycle oscillations (LCOs) by using AGARD's CT6 [28] as a benchmark test. All results, based on inviscid solutions, confirm that our nonlinear model is stable and accurate, against the full model solutions and measurements, and for predicting not only aerodynamic forces but also detailed flowfields. Moreover, the model is robust for inputs that considerably depart from the base trajectory in form and magnitude. This modeling provides a very efficient way for predicting unsteady flowfields with varying parameters because it needs only a tiny fraction of the cost of a full-order modeling for each new condition-the more cases studied, the more savings rendered. Hence, the present approach is especially useful for parametric studies, such as in the case of design optimization and exploration of flow phenomena.
Mixed finite element methods for linear elasticity with weakly imposed symmetry
NASA Astrophysics Data System (ADS)
Arnold, Douglas N.; Falk, Richard S.; Winther, Ragnar
2007-12-01
In this paper, we construct new finite element methods for the approximation of the equations of linear elasticity in three space dimensions that produce direct approximations to both stresses and displacements. The methods are based on a modified form of the Hellinger-Reissner variational principle that only weakly imposes the symmetry condition on the stresses. Although this approach has been previously used by a number of authors, a key new ingredient here is a constructive derivation of the elasticity complex starting from the de Rham complex. By mimicking this construction in the discrete case, we derive new mixed finite elements for elasticity in a systematic manner from known discretizations of the de Rham complex. These elements appear to be simpler than the ones previously derived. For example, we construct stable discretizations which use only piecewise linear elements to approximate the stress field and piecewise constant functions to approximate the displacement field.
Rogers, H R; van den Berg, C M
1988-04-01
Borate anions, B(OH)(-)(4), are known to associate with alkali and alkaline-earth metal cations in sea-water. The borate cation ion-pairs are of the general form MB(OH)((n-1)+)(4), where M(n+) is the cation. In this work, the cation borate stability constants (K*(MB)) have been evaluated for Na(+), Li(+), Mg(2+), Ca(2+) and Sr(2+) where K*(MB) = [MB(OH(4))((n-1)+)]/[M(n+)][B(OH)(-)(4)]. The K*(MB) values were obtained from values found for the stability constant of boric acid (K*(B)) in various electrolyte media at 25 degrees and an ionic strength of 0.7. Acid-base potentiometric titrations were performed in the electrolyte media with a standard Pt/H(2) electrode and a junctionless Ag/AgCl reference electrode to monitor the emf. A non-approximative equation was used to linearize the titration data. The values obtained were: K*(Lib) = 0.89 +/- 0.02, K*(NaB) = 0.44 +/- 0.01, K*(MgB) = 13.6 +/- 0.7, K*(CaB) = 11.4 +/- 0.15, K*(SrB) = 3.47 +/- 0.06. The values for K*(MB) correlate with the charge-density parameter z(2)/(r + 0.85), where r is the radius of the cation. The speciation of boron in sea-water was predicted from the K*(MB), data for the major cations present.
H∞ filtering for piecewise homogeneous Markovian jump nonlinear systems
NASA Astrophysics Data System (ADS)
Zhao, Feng; Zhang, Qingling; Wang, Guoliang
2016-10-01
This paper concerns the problem of H∞ filtering for piecewise homogeneous Markovian jump nonlinear systems. Different from the existing studies in the literatures, the existence of variations in transition rates for Markovian jump nonlinear systems is considered. The purpose of the paper is to design mode-dependent and mode-independent filters, such that the dynamics of the filtering errors are stochastic integral input-to-state stable with H∞ performance index. Using the linear matrix inequality method and the Lyapunov functional method, sufficient conditions for the solution to the H∞ filtering problem are derived. Finally, three examples are proposed to illustrate the effectiveness of the given theoretical results.
Mixing with piecewise isometries on a hemispherical shell
NASA Astrophysics Data System (ADS)
Park, Paul P.; Umbanhowar, Paul B.; Ottino, Julio M.; Lueptow, Richard M.
2016-07-01
We introduce mixing with piecewise isometries (PWIs) on a hemispherical shell, which mimics features of mixing by cutting and shuffling in spherical shells half-filled with granular media. For each PWI, there is an inherent structure on the hemispherical shell known as the exceptional set E, and a particular subset of E, E+, provides insight into how the structure affects mixing. Computer simulations of PWIs are used to visualize mixing and approximations of E+ to demonstrate their connection. While initial conditions of unmixed materials add a layer of complexity, the inherent structure of E+ defines fundamental aspects of mixing by cutting and shuffling.
Moment inversion problem for piecewise D-finite functions
NASA Astrophysics Data System (ADS)
Batenkov, Dmitry
2009-10-01
We consider the problem of exact reconstruction of univariate functions with jump discontinuities at unknown positions from their moments. These functions are assumed to satisfy an a priori unknown linear homogeneous differential equation with polynomial coefficients on each continuity interval. Therefore, they may be specified by a finite amount of information. This reconstruction problem has practical importance in signal processing and other applications. It is somewhat of a 'folklore' that the sequence of the moments of such 'piecewise D-finite' functions satisfies a linear recurrence relation of bounded order and degree. We derive this recurrence relation explicitly. It turns out that the coefficients of the differential operator which annihilates every piece of the function, as well as the locations of the discontinuities, appear in this recurrence in a precisely controlled manner. This leads to the formulation of a generic algorithm for reconstructing a piecewise D-finite function from its moments. We investigate the conditions for solvability of resulting linear systems in the general case, as well as analyse a few particular examples. We provide results of numerical simulations for several types of signals, which test the sensitivity of the proposed algorithm to noise.
Calculating the derivative of piecewise functions
NASA Astrophysics Data System (ADS)
Tomas Johansson, B.
2016-01-01
Exercises involving the calculation of the derivative of piecewise defined functions are common in calculus, with the aim of consolidating beginners' knowledge of applying the definition of the derivative. In such exercises, the piecewise function is commonly made up of two smooth pieces joined together at one point. A strategy which avoids using the definition of the derivative is to find the derivative function of each smooth piece and check whether these functions agree at the chosen point. Showing that this strategy works together with investigating discontinuities of the derivative is usually beyond a calculus course. However, we shall show that elementary arguments can be used to clarify the calculation and behaviour of the derivative for piecewise functions.
Regular and chaotic dynamics of a piecewise smooth bouncer
Langer, Cameron K. Miller, Bruce N.
2015-07-15
The dynamical properties of a particle in a gravitational field colliding with a rigid wall moving with piecewise constant velocity are studied. The linear nature of the wall's motion permits further analytical investigation than is possible for the system's sinusoidal counterpart. We consider three distinct approaches to modeling collisions: (i) elastic, (ii) inelastic with constant restitution coefficient, and (iii) inelastic with a velocity-dependent restitution function. We confirm the existence of distinct unbounded orbits (Fermi acceleration) in the elastic model, and investigate regular and chaotic behavior in the inelastic cases. We also examine in the constant restitution model trajectories wherein the particle experiences an infinite number of collisions in a finite time, i.e., the phenomenon of inelastic collapse. We address these so-called “sticking solutions” and their relation to both the overall dynamics and the phenomenon of self-reanimating chaos. Additionally, we investigate the long-term behavior of the system as a function of both initial conditions and parameter values. We find the non-smooth nature of the system produces novel bifurcation phenomena not seen in the sinusoidal model, including border-collision bifurcations. The analytical and numerical investigations reveal that although our piecewise linear bouncer is a simplified version of the sinusoidal model, the former not only captures essential features of the latter but also exhibits behavior unique to the discontinuous dynamics.
Meneses, Domingos De Sousa; Rousseau, Benoit; Echegut, Patrick; Matzen, Guy
2007-06-01
A new expression of dielectric function model based on piecewise polynomials is introduced. Its association with spline and more recent shape preserving interpolation algorithms allows easy reproduction of every kind of experimental spectra and thus retrieval of all the linear optical functions of a material. Based on a pure mathematical framework, the expression of the model is always applicable and does not necessitate any knowledge of the microscopic mechanisms of absorption responsible for the optical response. The potential of piecewise polynomial dielectric functions is shown through synthetic examples and the analysis of experimental spectra.
NASA Astrophysics Data System (ADS)
Merrikh-Bayat, Farshad
2012-04-01
Oustaloup recursive approximation (ORA) is widely used to find a rational integer-order approximation for fractional-order integrators and differentiators of the form sv, v ∈ (-1, 1). In this method the lower bound, the upper bound and the order of approximation should be determined beforehand, which is currently performed by trial and error and may be inefficient in some cases. The aim of this paper is to provide efficient rules for determining the suitable value of these parameters when a fractional-order PID controller is used in a stable linear feedback system. Two numerical examples are also presented to confirm the effectiveness of the proposed formulas.
Optical spectroscopies of materials from orbital-dependent approximations
NASA Astrophysics Data System (ADS)
Dabo, Ismaila; Ferretti, Andrea; Cococcioni, Matteo; Marzari, Nicola
2013-03-01
Electronic-structure calculations based upon density-functional theory (DFT) have been fruitful in diverse areas of materials science. Despite their exceptional success and widespread use, a range of spectroscopic properties fall beyond the scope of existing DFT approximations. Failures of DFT calculations in describing electronic and optical phenomena take root in the lack of piecewise linearity of approximate functionals. This known deficiency reverberates negatively on the spectroscopic description of systems involving fractionally occupied or spatially delocalized electronic states, such as donor-acceptor organic heterojunctions and heavy-metal organometallic complexes. In this talk, I will present a class of orbital-dependent density-functional theory (OD-DFT) methods that are derived from a multidensity formulation of the electronic-structure problem and that restore the piecewise linearity of the total energy via Koopmans' theorem. Such OD-DFT electronic-structure approximations are apt at describing full orbital spectra within a few tenths of an electron-volt relative to experimental photoemission spectroscopies and with the additional benefit of providing appreciably improved total energies for molecular systems with fractional occupations.
Canard-like phenomena in piecewise-smooth Van der Pol systems.
Roberts, Andrew; Gendinning, Paul
2014-06-01
We show that a nonlinear, piecewise-smooth, planar dynamical system can exhibit canard phenomena. Canard phenomena in nonlinear, piecewise-smooth systems can be qualitatively more similar to the phenomena in smooth systems than piecewise-linear systems, since the nonlinearity allows for canards to transition from small cycles to canards "with heads." The canards are born of a bifurcation that occurs as the slow-nullcline coincides with the splitting manifold. However, there are conditions under which this bifurcation leads to a phenomenon called super-explosion, the instantaneous transition from a globally attracting periodic orbit to relaxations oscillations. Also, we demonstrate that the bifurcation-whether leading to canards or super-explosion-can be subcritical. PMID:24985452
Canard-like phenomena in piecewise-smooth Van der Pol systems
NASA Astrophysics Data System (ADS)
Roberts, Andrew; Gendinning, Paul
2014-06-01
We show that a nonlinear, piecewise-smooth, planar dynamical system can exhibit canard phenomena. Canard phenomena in nonlinear, piecewise-smooth systems can be qualitatively more similar to the phenomena in smooth systems than piecewise-linear systems, since the nonlinearity allows for canards to transition from small cycles to canards "with heads." The canards are born of a bifurcation that occurs as the slow-nullcline coincides with the splitting manifold. However, there are conditions under which this bifurcation leads to a phenomenon called super-explosion, the instantaneous transition from a globally attracting periodic orbit to relaxations oscillations. Also, we demonstrate that the bifurcation—whether leading to canards or super-explosion—can be subcritical.
DiFrancesco, D; Noble, D
1980-01-01
1. Regular perturbation theory was used to obtain analytical solutions for the time course of membrane current decay following voltage-clamp depolarizing pulses when both time-dependent K conductance mechanisms and the process of K accumulation in extracellular spaces are present. These solutions apply when the current and K concentration changes are small enough for linear relations to be assumed between current and K concentration. 2. In the case of a single Hodgkin-Huxley type conductance variable with time constant tau chi the presence of an accumulation process which, by itself, would produce a current decay with time constant tau alpha, induces the appearance of two infinite sets of components with decreasing time constants (1/(n+1/tau chi) and 1/(1/tau alpha + n/tau chi), where n is integer), and decreasing magnitudes. 3. The analytical solutions are used to investigate the range of conditions over which semi-exponential (curve-stripping) analysis of current decay tails may give useful information on the kinetics of current change. It is shown that, except at very large decay tail amplitudes, the method may give a good estimate of the true time constants of conductance decay even when the currents are assumed to be strongly dependent on external K concentration. 4. The method introduces error in current amplitude, but over the range in which curve-stripping gives useful results, the direct distortion of activation curves by variations in external K concentration is fairly small. However, as the current decay becomes grossly distorted in its time course by accumulation, so does the activation curve. The effects are very similar both to those obtained using numerical computation without linearization, and to those obtained experimentally. 5. Even with a large dependence of current on external K concentration the linear model does not reproduce i chi, fast as a perturbation of i chi, slow by K accumulation. PMID:7463358
NASA Astrophysics Data System (ADS)
Grima, Ramon
2015-10-01
It is well known that the linear-noise approximation (LNA) agrees with the chemical master equation, up to second-order moments, for chemical systems composed of zero and first-order reactions. Here we show that this is also a property of the LNA for a subset of chemical systems with second-order reactions. This agreement is independent of the number of interacting molecules.
Grima, Ramon
2015-10-01
It is well known that the linear-noise approximation (LNA) agrees with the chemical master equation, up to second-order moments, for chemical systems composed of zero and first-order reactions. Here we show that this is also a property of the LNA for a subset of chemical systems with second-order reactions. This agreement is independent of the number of interacting molecules.
Martínez, Ana
2009-03-12
Carotenoids are one particular type of conjugated chromophores with a great capacity for accepting electrons. The question posed here is how the capacity to accept electrons is related to extension of the conjugation. If there is a connection, any chromophore should represent a good antiradical, a point of interest for those investigating the biological effects of antioxidants. In order to analyze the relationship between the extension of the conjugation and the absorbance and electron-donor properties described in this paper, full geometry optimizations at the BPW91/D5DVZ level of theory are reported for a number of linear conjugated polyene systems. Maximum wavelengths were obtained using the TDDFT methodology. From these results, it is possible to conclude that large conjugated chromophores have a great capacity for accepting electrons but diminished power for donating electrons. Apparently, any chromophore should be a good antiradical, but various mechanisms exist for scavenging free radicals. In the case of linear polyene-conjugated molecules, indigo, blue, and green chromophores represent good antiradicals because they are also good antioxidants (effective electron donors). Yellow and red chromophores represent good antiradicals because they are good antireductants (effective electron acceptors). In the case of the molecules reported in this paper, the ionization energy and the electron affinity come close to the work function of graphite. This may be important for future applications, where the movement of the electrons is crucial.
NASA Astrophysics Data System (ADS)
Caizergues, C.; Blenski, T.; Piron, R.
2016-03-01
We report results on the self-consistent linear response theory of quantum average-atoms in plasmas. The approach is based on the two first orders of the cluster expansion of the plasma susceptibility. A change of variable is applied, which allows us to handle the diverging free-free transitions contribution in the self-consistent induced electron density and potential. The method is first tested on the case of rare gas isolated neutral atoms. A test of the Ehrenfest-type sum rule is then performed in a case of an actual average-atom in a plasma. At frequencies much higher than the plasma frequency, the sum rule seems to be fulfilled within the accuracy of the numerical methods. Close to the plasma frequency, the method seems not to account for the cold-plasma dielectric function renormalization in the sum rule, which was correctly reproduced in the case of the Thomas-Fermi-Bloch self-consistent linear response. This suggests the need for a better accounting for the outgoing waves in the asymptotic boundary conditions.
NASA Astrophysics Data System (ADS)
Zhang, Jun; Guo, Yu-Feng; Xu, Yue; Lin, Hong; Yang, Hui; Hong, Yang; Yao, Jia-Fei
2015-02-01
A novel one-dimensional (1D) analytical model is proposed for quantifying the breakdown voltage of a reduced surface field (RESURF) lateral power device fabricated on silicon on an insulator (SOI) substrate. We assume that the charges in the depletion region contribute to the lateral PN junctions along the diagonal of the area shared by the lateral and vertical depletion regions. Based on the assumption, the lateral PN junction behaves as a linearly graded junction, thus resulting in a reduced surface electric field and high breakdown voltage. Using the proposed model, the breakdown voltage as a function of device parameters is investigated and compared with the numerical simulation by the TCAD tools. The analytical results are shown to be in fair agreement with the numerical results. Finally, a new RESURF criterion is derived which offers a useful scheme to optimize the structure parameters. This simple 1D model provides a clear physical insight into the RESURF effect and a new explanation on the improvement in breakdown voltage in an SOI RESURF device. Project supported by the National Natural Science Foundation of China (Grant No. 61076073) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20133223110003).
NASA Astrophysics Data System (ADS)
Gulati, Mamta; Saini, Tarun Deep
2016-07-01
The short-wave asymptotics (WKB) of spiral density waves in self-gravitating stellar discs is well suited for the study of the dynamics of tightly-wound wavepackets. But the textbook WKB theory is not well adapted to the study of the linear eigenmodes in a collisionless self-gravitating disc because of the transcendental nature of the dispersion relation. We present a modified WKB theory of spiral density waves, for collisionless discs in the epicyclic limit, in which the perturbed gravitational potential is related to the perturbed surface density by the Poisson integral in Kalnaj's logarithmic spiral form. An integral equation is obtained for the surface density perturbation, which is seen to also reduce to the standard WKB dispersion relation. Although our formulation is general and applies to all discs, we present our analysis only for nearly Keplerian, low-mass, self-gravitating discs revolving around massive central objects, and derive an integral equation governing the slow precessional modes of such discs. For a prograde disc, the integral kernel turns out be real and symmetric, implying that all slow modes are stable. We apply the slow mode integral equation to two unperturbed disc profiles, the Jalali-Tremaine annular discs, and the Kuzmin disc. We determine eigenvalues and eigenfunctions for both m = 1 and m = 2 slow modes for these profiles and discuss their properties. Our results compare well with those of Jalali-Tremaine.
A piecewise linear approach to volume tracking a triple point
NASA Astrophysics Data System (ADS)
Choi, Benjamin Y.; Bussmann, Markus
2007-02-01
An approach to volume tracking three materials is presented that, in contrast with the so-called onion-skin methodology, assumes the existence of a triple point at which two interfaces between three materials intersect. The reconstruction of any cell that contains three materials is iterative: the approach is to locate a point of intersection between two interfaces that minimizes a given error expression. The advantages and limitations of the algorithm are presented via a series of advection tests that demonstrate that triple points can be reconstructed and advected just as well as simpler interfaces in typical applications.
Piecewise Linear Membership Function Generator-Divider Approach
NASA Technical Reports Server (NTRS)
Hart, Ron; Martinez, Gene; Yuan, Bo; Zrilic, Djuro; Ramirez, Jaime
1997-01-01
In this paper a simple, inexpensive, membership function circuit for fuzzy controllers is presented. The proposed circuit may be used to generate a general trapezoidal membership function. The slope and horizontal shift are fully programmable parameters.
NASA Astrophysics Data System (ADS)
Isaeva, Julia; Martens, Magni; Sæbø, Solve; Wyller, John A.; Martens, Harald
2012-05-01
A generic mathematical phenomenon (line curvature) is described quantitatively and linguistically: a range of very different in form and representation models z=Fm(x), m=1,2,…,38, each yielding smooth, but curved relationships z=f(x) with 0 or 1 inflection points, were collected from different fields of science, ranging from systems biology and statistics to trigonometry and psychophysics (Isaeva et al. (2011)). The behavioural repertoire of each of the models was realised by exhaustive statistically designed computer experiments, yielding a total of about 50,000 curves z=f(x), each recorded at 100 x-values. A modelome of curvature was formed by this set of arched or sigmoid curves and was preprocessed and combined in a joint metamodel based on a bi-linear subspace analysis. To describe a total of 99.9% of the variability in the curves, 12 eigenvectors were needed. These 12 common curve descriptors were successfully related back to the original model input parameters in each of the individual models. Furthermore, to give verbal meaning to the per se meaningless axes in this 12-dimensional eigenvector space, a total of 64 curve images were selected by a statistical design, printed and submitted to descriptive sensory analysis, using a panel of ten trained judges. A quantitative map between the eigenvector space and the sensory space was successfully established and then used for predicting what the human descriptive profiling would be for each of the 50,000 curves. Thus, a first version of a complete “modelome” of the mathematical phenomenon “line curvature” has been established by multivariate metamodelling and described in terms of quantitative maps both to the original model parameters in the 38 individual models and to human verbal description of curve shapes.
NASA Astrophysics Data System (ADS)
Peng, Degao; van Aggelen, Helen; Steinmann, Stephan; Yang, Yang; Yang, Weitao; Duke University Team
2014-03-01
The particle-particle random-phase approximation (pp-RPA) recently attracts extensive interests in quantum chemistry recently. Pp-RPA is a versatile model to calculate ground-state correlation energies, and double ionization potential/double electron affinity. We inspect particle-particle random-phase approximation in different perspectives to further understand its theoretical fundamentals. Viewed as summation of all ladder diagrams, the pp-RPA correlation energy is proved to be analytically equivalent to the ladder coupled-cluster doubles (ladder-CCD) theory. With this equivalence, we can make use of various well-established coupled-cluster techniques to study pp-RPA. Furthermore, we establish linear-response time-dependent density-functional theory with pairing fields (TDDFT-PF), where pp-RPA can be interpreted as the mean-field approximation to a general theory. TDDFT-PF is closely related to the density-functional theory of superconductors, but is applied to normal systems to capture exact N plus/minus 2 excitations. In the linear-response regime, both the adiabatic and non-adiabatic TDDFT-PF equations are established. This sets the fundamentals for further density-functional developments aiming for pp-RPA. These theoretical perspectives will be very helpful for future study.
Stauber, Douglas A.
1985-01-01
A Born approximation is used to linearize the relationship, in the horizontal-wavenumber and frequency domains, between lateral perturbations of modulus and density in a layered half-space and the acoustic wave field observed at the surface when a plane wave is incident from below. The resulting equations can be used to perform a linear inversion of observed acoustic wave fields to obtain lateral perturbations in modulus and density. Since modulus and density effects are separated, gravity observations can be included in the inversion procedure without any assumptions about the relationship between density and acoustic velocity. Tests with synthetic data sets reveal that the inversion method gives useful results when the spatial scales of the inhomogeneities are smaller than several acoustic wavelengths. Refs.
Schurkus, Henry F; Ochsenfeld, Christian
2016-01-21
An atomic-orbital (AO) reformulation of the random-phase approximation (RPA) correlation energy is presented allowing to reduce the steep computational scaling to linear, so that large systems can be studied on simple desktop computers with fully numerically controlled accuracy. Our AO-RPA formulation introduces a contracted double-Laplace transform and employs the overlap-metric resolution-of-the-identity. First timings of our pilot code illustrate the reduced scaling with systems comprising up to 1262 atoms and 10 090 basis functions. .
NASA Astrophysics Data System (ADS)
Pedersen, Thomas Bondo; Sánchez de Merás, Alfredo M. J.; Koch, Henrik
2004-05-01
A new implementation of the approximate coupled cluster singles and doubles CC2 linear response model using Cholesky decomposition of the two-electron integrals is presented. Significantly reducing storage demands and computational effort without sacrificing accuracy compared to the conventional model, the algorithm is well suited for large-scale applications. Extensive basis set convergence studies are presented for the static and frequency-dependent electric dipole polarizability of benzene and C60, and for the optical rotation of CNOFH2 and (-)-trans-cyclooctene (TCO). The origin-dependence of the optical rotation is calculated and shown to persist for CC2 even at basis set convergence.
NASA Astrophysics Data System (ADS)
Schurkus, Henry F.; Ochsenfeld, Christian
2016-01-01
An atomic-orbital (AO) reformulation of the random-phase approximation (RPA) correlation energy is presented allowing to reduce the steep computational scaling to linear, so that large systems can be studied on simple desktop computers with fully numerically controlled accuracy. Our AO-RPA formulation introduces a contracted double-Laplace transform and employs the overlap-metric resolution-of-the-identity. First timings of our pilot code illustrate the reduced scaling with systems comprising up to 1262 atoms and 10 090 basis functions.
Akcelik, Volkan; Flath, Pearl; Ghattas, Omar; Hill, Judith C; Van Bloemen Waanders, Bart; Wilcox, Lucas
2011-01-01
We consider the problem of estimating the uncertainty in large-scale linear statistical inverse problems with high-dimensional parameter spaces within the framework of Bayesian inference. When the noise and prior probability densities are Gaussian, the solution to the inverse problem is also Gaussian, and is thus characterized by the mean and covariance matrix of the posterior probability density. Unfortunately, explicitly computing the posterior covariance matrix requires as many forward solutions as there are parameters, and is thus prohibitive when the forward problem is expensive and the parameter dimension is large. However, for many ill-posed inverse problems, the Hessian matrix of the data misfit term has a spectrum that collapses rapidly to zero. We present a fast method for computation of an approximation to the posterior covariance that exploits the lowrank structure of the preconditioned (by the prior covariance) Hessian of the data misfit. Analysis of an infinite-dimensional model convection-diffusion problem, and numerical experiments on large-scale 3D convection-diffusion inverse problems with up to 1.5 million parameters, demonstrate that the number of forward PDE solves required for an accurate low-rank approximation is independent of the problem dimension. This permits scalable estimation of the uncertainty in large-scale ill-posed linear inverse problems at a small multiple (independent of the problem dimension) of the cost of solving the forward problem.
Zuehlsdorff, T J; Hine, N D M; Payne, M C; Haynes, P D
2015-11-28
We present a solution of the full time-dependent density-functional theory (TDDFT) eigenvalue equation in the linear response formalism exhibiting a linear-scaling computational complexity with system size, without relying on the simplifying Tamm-Dancoff approximation (TDA). The implementation relies on representing the occupied and unoccupied subspaces with two different sets of in situ optimised localised functions, yielding a very compact and efficient representation of the transition density matrix of the excitation with the accuracy associated with a systematic basis set. The TDDFT eigenvalue equation is solved using a preconditioned conjugate gradient algorithm that is very memory-efficient. The algorithm is validated on a small test molecule and a good agreement with results obtained from standard quantum chemistry packages is found, with the preconditioner yielding a significant improvement in convergence rates. The method developed in this work is then used to reproduce experimental results of the absorption spectrum of bacteriochlorophyll in an organic solvent, where it is demonstrated that the TDA fails to reproduce the main features of the low energy spectrum, while the full TDDFT equation yields results in good qualitative agreement with experimental data. Furthermore, the need for explicitly including parts of the solvent into the TDDFT calculations is highlighted, making the treatment of large system sizes necessary that are well within reach of the capabilities of the algorithm introduced here. Finally, the linear-scaling properties of the algorithm are demonstrated by computing the lowest excitation energy of bacteriochlorophyll in solution. The largest systems considered in this work are of the same order of magnitude as a variety of widely studied pigment-protein complexes, opening up the possibility of studying their properties without having to resort to any semiclassical approximations to parts of the protein environment. PMID:26627950
NASA Astrophysics Data System (ADS)
Zuehlsdorff, T. J.; Hine, N. D. M.; Payne, M. C.; Haynes, P. D.
2015-11-01
We present a solution of the full time-dependent density-functional theory (TDDFT) eigenvalue equation in the linear response formalism exhibiting a linear-scaling computational complexity with system size, without relying on the simplifying Tamm-Dancoff approximation (TDA). The implementation relies on representing the occupied and unoccupied subspaces with two different sets of in situ optimised localised functions, yielding a very compact and efficient representation of the transition density matrix of the excitation with the accuracy associated with a systematic basis set. The TDDFT eigenvalue equation is solved using a preconditioned conjugate gradient algorithm that is very memory-efficient. The algorithm is validated on a small test molecule and a good agreement with results obtained from standard quantum chemistry packages is found, with the preconditioner yielding a significant improvement in convergence rates. The method developed in this work is then used to reproduce experimental results of the absorption spectrum of bacteriochlorophyll in an organic solvent, where it is demonstrated that the TDA fails to reproduce the main features of the low energy spectrum, while the full TDDFT equation yields results in good qualitative agreement with experimental data. Furthermore, the need for explicitly including parts of the solvent into the TDDFT calculations is highlighted, making the treatment of large system sizes necessary that are well within reach of the capabilities of the algorithm introduced here. Finally, the linear-scaling properties of the algorithm are demonstrated by computing the lowest excitation energy of bacteriochlorophyll in solution. The largest systems considered in this work are of the same order of magnitude as a variety of widely studied pigment-protein complexes, opening up the possibility of studying their properties without having to resort to any semiclassical approximations to parts of the protein environment.
Zuehlsdorff, T. J. Payne, M. C.; Hine, N. D. M.; Haynes, P. D.
2015-11-28
We present a solution of the full time-dependent density-functional theory (TDDFT) eigenvalue equation in the linear response formalism exhibiting a linear-scaling computational complexity with system size, without relying on the simplifying Tamm-Dancoff approximation (TDA). The implementation relies on representing the occupied and unoccupied subspaces with two different sets of in situ optimised localised functions, yielding a very compact and efficient representation of the transition density matrix of the excitation with the accuracy associated with a systematic basis set. The TDDFT eigenvalue equation is solved using a preconditioned conjugate gradient algorithm that is very memory-efficient. The algorithm is validated on a small test molecule and a good agreement with results obtained from standard quantum chemistry packages is found, with the preconditioner yielding a significant improvement in convergence rates. The method developed in this work is then used to reproduce experimental results of the absorption spectrum of bacteriochlorophyll in an organic solvent, where it is demonstrated that the TDA fails to reproduce the main features of the low energy spectrum, while the full TDDFT equation yields results in good qualitative agreement with experimental data. Furthermore, the need for explicitly including parts of the solvent into the TDDFT calculations is highlighted, making the treatment of large system sizes necessary that are well within reach of the capabilities of the algorithm introduced here. Finally, the linear-scaling properties of the algorithm are demonstrated by computing the lowest excitation energy of bacteriochlorophyll in solution. The largest systems considered in this work are of the same order of magnitude as a variety of widely studied pigment-protein complexes, opening up the possibility of studying their properties without having to resort to any semiclassical approximations to parts of the protein environment.
NASA Technical Reports Server (NTRS)
Smith, Ralph C.
1994-01-01
A Galerkin method for systems of PDE's in circular geometries is presented with motivating problems being drawn from structural, acoustic, and structural acoustic applications. Depending upon the application under consideration, piecewise splines or Legendre polynomials are used when approximating the system dynamics with modifications included to incorporate the analytic solution decay near the coordinate singularity. This provides an efficient method which retains its accuracy throughout the circular domain without degradation at singularity. Because the problems under consideration are linear or weakly nonlinear with constant or piecewise constant coefficients, transform methods for the problems are not investigated. While the specific method is developed for the two dimensional wave equations on a circular domain and the equation of transverse motion for a thin circular plate, examples demonstrating the extension of the techniques to a fully coupled structural acoustic system are used to illustrate the flexibility of the method when approximating the dynamics of more complex systems.
Piecewise power laws in individual learning curves.
Donner, Yoni; Hardy, Joseph L
2015-10-01
The notion that human learning follows a smooth power law (PL) of diminishing gains is well-established in psychology. This characteristic is observed when multiple curves are averaged, potentially masking more complex dynamics underpinning the curves of individual learners. Here, we analyzed 25,280 individual learning curves, each comprising 500 measurements of cognitive performance taken from four cognitive tasks. A piecewise PL (PPL) model explained the individual learning curves significantly better than a single PL, controlling for model complexity. The PPL model allows for multiple PLs connected at different points in the learning process. We also explored the transition dynamics between PL curve component pieces. Performance in later pieces typically surpassed that in earlier pieces, after a brief drop in performance at the transition point. The transition rate was negatively associated with age, even after controlling for overall performance. Our results suggest at least two processes at work in individual learning curves: locally, a gradual, smooth improvement, with diminishing gains within a specific strategy, which is modeled well as a PL; and globally, a discrete sequence of strategy shifts, in which each strategy is better in the long term than the ones preceding it. The piecewise extension of the classic PL of practice has implications for both individual skill acquisition and theories of learning.
Piecewise power laws in individual learning curves.
Donner, Yoni; Hardy, Joseph L
2015-10-01
The notion that human learning follows a smooth power law (PL) of diminishing gains is well-established in psychology. This characteristic is observed when multiple curves are averaged, potentially masking more complex dynamics underpinning the curves of individual learners. Here, we analyzed 25,280 individual learning curves, each comprising 500 measurements of cognitive performance taken from four cognitive tasks. A piecewise PL (PPL) model explained the individual learning curves significantly better than a single PL, controlling for model complexity. The PPL model allows for multiple PLs connected at different points in the learning process. We also explored the transition dynamics between PL curve component pieces. Performance in later pieces typically surpassed that in earlier pieces, after a brief drop in performance at the transition point. The transition rate was negatively associated with age, even after controlling for overall performance. Our results suggest at least two processes at work in individual learning curves: locally, a gradual, smooth improvement, with diminishing gains within a specific strategy, which is modeled well as a PL; and globally, a discrete sequence of strategy shifts, in which each strategy is better in the long term than the ones preceding it. The piecewise extension of the classic PL of practice has implications for both individual skill acquisition and theories of learning. PMID:25711183
Renormalizable two-parameter piecewise isometries
NASA Astrophysics Data System (ADS)
Lowenstein, J. H.; Vivaldi, F.
2016-06-01
We exhibit two distinct renormalization scenarios for two-parameter piecewise isometries, based on 2 π / 5 rotations of a rhombus and parameter-dependent translations. Both scenarios rely on the recently established renormalizability of a one-parameter triangle map, which takes place if and only if the parameter belongs to the algebraic number field K = Q ( √{ 5 }) associated with the rotation matrix. With two parameters, features emerge which have no counterpart in the single-parameter model. In the first scenario, we show that renormalizability is no longer rigid: whereas one of the two parameters is restricted to K , the second parameter can vary continuously over a real interval without destroying self-similarity. The mechanism involves neighbouring atoms which recombine after traversing distinct return paths. We show that this phenomenon also occurs in the simpler context of Rauzy-Veech renormalization of interval exchange transformations, here regarded as parametric piecewise isometries on a real interval. We explore this analogy in some detail. In the second scenario, which involves two-parameter deformations of a three-parameter rhombus map, we exhibit a weak form of rigidity. The phase space splits into several (non-convex) invariant components, on each of which the renormalization still has a free parameter. However, the foliations of the different components are transversal in parameter space; as a result, simultaneous self-similarity of the component maps requires that both of the original parameters belong to the field K .
NASA Astrophysics Data System (ADS)
Lorin, E.; Yang, X.; Antoine, X.
2016-06-01
The paper is devoted to develop efficient domain decomposition methods for the linear Schrödinger equation beyond the semiclassical regime, which does not carry a small enough rescaled Planck constant for asymptotic methods (e.g. geometric optics) to produce a good accuracy, but which is too computationally expensive if direct methods (e.g. finite difference) are applied. This belongs to the category of computing middle-frequency wave propagation, where neither asymptotic nor direct methods can be directly used with both efficiency and accuracy. Motivated by recent works of the authors on absorbing boundary conditions (Antoine et al. (2014) [13] and Yang and Zhang (2014) [43]), we introduce Semiclassical Schwarz Waveform Relaxation methods (SSWR), which are seamless integrations of semiclassical approximation to Schwarz Waveform Relaxation methods. Two versions are proposed respectively based on Herman-Kluk propagation and geometric optics, and we prove the convergence and provide numerical evidence of efficiency and accuracy of these methods.
Yoshikawa, Takeshi; Nakai, Hiromi
2015-01-30
Graphical processing units (GPUs) are emerging in computational chemistry to include Hartree-Fock (HF) methods and electron-correlation theories. However, ab initio calculations of large molecules face technical difficulties such as slow memory access between central processing unit and GPU and other shortfalls of GPU memory. The divide-and-conquer (DC) method, which is a linear-scaling scheme that divides a total system into several fragments, could avoid these bottlenecks by separately solving local equations in individual fragments. In addition, the resolution-of-the-identity (RI) approximation enables an effective reduction in computational cost with respect to the GPU memory. The present study implemented the DC-RI-HF code on GPUs using math libraries, which guarantee compatibility with future development of the GPU architecture. Numerical applications confirmed that the present code using GPUs significantly accelerated the HF calculations while maintaining accuracy.
A Lyapunov method for stability analysis of piecewise-affine systems over non-invariant domains
NASA Astrophysics Data System (ADS)
Rubagotti, Matteo; Zaccarian, Luca; Bemporad, Alberto
2016-05-01
This paper analyses stability of discrete-time piecewise-affine systems, defined on possibly non-invariant domains, taking into account the possible presence of multiple dynamics in each of the polytopic regions of the system. An algorithm based on linear programming is proposed, in order to prove exponential stability of the origin and to find a positively invariant estimate of its region of attraction. The results are based on the definition of a piecewise-affine Lyapunov function, which is in general discontinuous on the boundaries of the regions. The proposed method is proven to lead to feasible solutions in a broader range of cases as compared to a previously proposed approach. Two numerical examples are shown, among which a case where the proposed method is applied to a closed-loop system, to which model predictive control was applied without a-priori guarantee of stability.
Approximate Analysis of Semiconductor Laser Arrays
NASA Technical Reports Server (NTRS)
Marshall, William K.; Katz, Joseph
1987-01-01
Simplified equation yields useful information on gains and output patterns. Theoretical method based on approximate waveguide equation enables prediction of lateral modes of gain-guided planar array of parallel semiconductor lasers. Equation for entire array solved directly using piecewise approximation of index of refraction by simple functions without customary approximation based on coupled waveguid modes of individual lasers. Improved results yield better understanding of laser-array modes and help in development of well-behaved high-power semiconductor laser arrays.
NASA Astrophysics Data System (ADS)
Skulovich, Olya; Bent, Russell; Judi, David; Perelman, Lina Sela; Ostfeld, Avi
2015-06-01
Despite their potential catastrophic impact, transients are often ignored or presented ad hoc when designing water distribution systems. To address this problem, we introduce a new piece-wise function fitting model that is integrated with mixed integer programming to optimally place and size surge tanks for transient control. The key features of the algorithm are a model-driven discretization of the search space, a linear approximation nonsmooth system response surface to transients, and a mixed integer linear programming optimization. Results indicate that high quality solutions can be obtained within a reasonable number of function evaluations and demonstrate the computational effectiveness of the approach through two case studies. The work investigates one type of surge control devices (closed surge tank) for a specified set of transient events. The performance of the algorithm relies on the assumption that there exists a smooth relationship between the objective function and tank size. Results indicate the potential of the approach for the optimal surge control design in water systems.
Lee, Ming-Wei; Hung, Cheng-Hung; Liao, Jung-Li; Cheng, Nan-Yu; Hou, Ming-Feng; Tseng, Sheng-Hao
2014-10-01
In this paper, we demonstrate that a scanning MEMS mirror can be employed to create a linear gradient line source that is equivalent to a planar source. This light source setup facilitates the use of diffusion models of increased orders of approximation having closed form solution, and thus enhance the efficiency and accuracy in sample optical properties recovery. In addition, compared with a regular planar light source, the linear gradient line source occupies much less source area and has an elevated measurement efficiency. We employed a δ-P1 diffusion equation with a closed form solution and carried out a phantom study to understand the performance of this new method in determining the absorption and scattering properties of turbid samples. Moreover, our Monte Carlo simulation results indicated that this geometry had probing depths comparable to those of the conventional diffuse reflectance measurement geometry with a source-detector separation of 3 mm. We expect that this new source setup would facilitate the investigating of superficial volumes of turbid samples in the wavelength regions where tissue absorption coefficients are comparable to scattering coefficients.
NASA Astrophysics Data System (ADS)
Tetsu, Hiroyuki; Nakamoto, Taishi
2016-03-01
Radiation is an important process of energy transport, a force, and a basis for synthetic observations, so radiation hydrodynamics (RHD) calculations have occupied an important place in astrophysics. However, although the progress in computational technology is remarkable, their high numerical cost is still a persistent problem. In this work, we compare the following schemes used to solve the nonlinear simultaneous equations of an RHD algorithm with the flux-limited diffusion approximation: the Newton-Raphson (NR) method, operator splitting, and linearization (LIN), from the perspective of the computational cost involved. For operator splitting, in addition to the traditional simple operator splitting (SOS) scheme, we examined the scheme developed by Douglas & Rachford (DROS). We solve three test problems (the thermal relaxation mode, the relaxation and the propagation of linear waves, and radiating shock) using these schemes and then compare their dependence on the time step size. As a result, we find the conditions of the time step size necessary for adopting each scheme. The LIN scheme is superior to other schemes if the ratio of radiation pressure to gas pressure is sufficiently low. On the other hand, DROS can be the most efficient scheme if the ratio is high. Although the NR scheme can be adopted independently of the regime, especially in a problem that involves optically thin regions, the convergence tends to be worse. In all cases, SOS is not practical.
Casimir Effect for the Piecewise Uniform String
NASA Astrophysics Data System (ADS)
Brevik, Iver
The Casimir energy for the transverse oscillations of a piecewise uniform closed string is calculated. In its simplest version the string consists of two parts I and II having in general different tension and mass density, but is always obeying the condition that the velocity of sound is equal to the velocity of light. The model, first introduced by Brevik and Nielsen in 1990, possesses attractive formal properties implying that it becomes easily regularizable by several methods, the most powerful one being the contour integration method.We also consider the case where the string is divided into 2N pieces, of alternating type-I and type-II material. The free energy at finite temperature, as well as the Hagedorn temperature, are found. Finally, we make some remarks on the relationship between this kind of theory and the theory of quantum star graphs, recently considered by Fulling et al..
NASA Astrophysics Data System (ADS)
Hayes, Scott T.
2005-01-01
A method is developed for producing deterministic chaotic motion from the linear superposition of a bi-infinite sequence of randomly polarized basis functions. The resultant waveform is also formally a random process in the usual sense. In the example given, a threedimensional embedding produces an idealized version of Lorenz motion. The one-dimensional approximate return map is piecewise linear; a tent or shift, depending on the Poincaré section. The results are presented in an informal style so that they are accessible to a wide audience interested in both theory and applications of symbolic dynamics communication.
Geometric constrained variational calculus I: Piecewise smooth extremals
NASA Astrophysics Data System (ADS)
Massa, Enrico; Bruno, Danilo; Luria, Gianvittorio; Pagani, Enrico
2015-05-01
A geometric setup for constrained variational calculus is presented. The analysis deals with the study of the extremals of an action functional defined on piecewise differentiable curves, subject to differentiable, non-holonomic constraints. Special attention is paid to the tensorial aspects of the theory. As far as the kinematical foundations are concerned, a fully covariant scheme is developed through the introduction of the concept of infinitesimal control. The standard classification of the extremals into normal and abnormal ones is discussed, pointing out the existence of an algebraic algorithm assigning to each admissible curve a corresponding abnormality index, related to the co-rank of a suitable linear map. Attention is then shifted to the study of the first variation of the action functional. The analysis includes a revisitation of Pontryagin's equations and of the Lagrange multipliers method, as well as a reformulation of Pontryagin's algorithm in Hamiltonian terms. The analysis is completed by a general result, concerning the existence of finite deformations with fixed endpoints.
An algorithm for linearizing convex extremal problems
Gorskaya, Elena S
2010-06-09
This paper suggests a method of approximating the solution of minimization problems for convex functions of several variables under convex constraints is suggested. The main idea of this approach is the approximation of a convex function by a piecewise linear function, which results in replacing the problem of convex programming by a linear programming problem. To carry out such an approximation, the epigraph of a convex function is approximated by the projection of a polytope of greater dimension. In the first part of the paper, the problem is considered for functions of one variable. In this case, an algorithm for approximating the epigraph of a convex function by a polygon is presented, it is shown that this algorithm is optimal with respect to the number of vertices of the polygon, and exact bounds for this number are obtained. After this, using an induction procedure, the algorithm is generalized to certain classes of functions of several variables. Applying the suggested method, polynomial algorithms for an approximate calculation of the L{sub p}-norm of a matrix and of the minimum of the entropy function on a polytope are obtained. Bibliography: 19 titles.
An algorithm for linearizing convex extremal problems
NASA Astrophysics Data System (ADS)
Gorskaya, Elena S.
2010-06-01
This paper suggests a method of approximating the solution of minimization problems for convex functions of several variables under convex constraints is suggested. The main idea of this approach is the approximation of a convex function by a piecewise linear function, which results in replacing the problem of convex programming by a linear programming problem. To carry out such an approximation, the epigraph of a convex function is approximated by the projection of a polytope of greater dimension. In the first part of the paper, the problem is considered for functions of one variable. In this case, an algorithm for approximating the epigraph of a convex function by a polygon is presented, it is shown that this algorithm is optimal with respect to the number of vertices of the polygon, and exact bounds for this number are obtained. After this, using an induction procedure, the algorithm is generalized to certain classes of functions of several variables. Applying the suggested method, polynomial algorithms for an approximate calculation of the L_p-norm of a matrix and of the minimum of the entropy function on a polytope are obtained. Bibliography: 19 titles.
A piecewise lookup table for calculating nonbonded pairwise atomic interactions.
Luo, Jinping; Liu, Lijun; Su, Peng; Duan, Pengbo; Lu, Daihui
2015-11-01
A critical challenge for molecular dynamics simulations of chemical or biological systems is to improve the calculation efficiency while retaining sufficient accuracy. The main bottleneck in improving the efficiency is the evaluation of nonbonded pairwise interactions. We propose a new piecewise lookup table method for rapid and accurate calculation of interatomic nonbonded pairwise interactions. The piecewise lookup table allows nonuniform assignment of table nodes according to the slope of the potential function and the pair interaction distribution. The proposed method assigns the nodes more reasonably than in general lookup tables, and thus improves the accuracy while requiring fewer nodes. To obtain the same level of accuracy, our piecewise lookup table accelerates the calculation via the efficient usage of cache memory. This new method is straightforward to implement and should be broadly applicable. Graphical Abstract Illustration of piecewise lookup table method. PMID:26481475
Piccardo, Matteo; Bloino, Julien; Barone, Vincenzo
2015-01-01
Models going beyond the rigid-rotor and the harmonic oscillator levels are mandatory for providing accurate theoretical predictions for several spectroscopic properties. Different strategies have been devised for this purpose. Among them, the treatment by perturbation theory of the molecular Hamiltonian after its expansion in power series of products of vibrational and rotational operators, also referred to as vibrational perturbation theory (VPT), is particularly appealing for its computational efficiency to treat medium-to-large systems. Moreover, generalized (GVPT) strategies combining the use of perturbative and variational formalisms can be adopted to further improve the accuracy of the results, with the first approach used for weakly coupled terms, and the second one to handle tightly coupled ones. In this context, the GVPT formulation for asymmetric, symmetric, and linear tops is revisited and fully generalized to both minima and first-order saddle points of the molecular potential energy surface. The computational strategies and approximations that can be adopted in dealing with GVPT computations are pointed out, with a particular attention devoted to the treatment of symmetry and degeneracies. A number of tests and applications are discussed, to show the possibilities of the developments, as regards both the variety of treatable systems and eligible methods. © 2015 Wiley Periodicals, Inc. PMID:26345131
Numerical time-dependent solutions of the Schrödinger equation with piecewise continuous potentials
NASA Astrophysics Data System (ADS)
van Dijk, Wytse
2016-06-01
We consider accurate numerical solutions of the one-dimensional time-dependent Schrödinger equation when the potential is piecewise continuous. Spatial step sizes are defined for each of the regions between the discontinuities and a matching condition at the boundaries of the regions is employed. The Numerov method for spatial integration is particularly appropriate to this approach. By employing Padé approximants for the time-evolution operator, we obtain solutions with significantly improved precision without increased CPU time. This approach is also appropriate for adaptive changes in spatial step size even when there is no discontinuity of the potential.
Finite volume element approximation of an inhomogeneous Brusselator model with cross-diffusion
NASA Astrophysics Data System (ADS)
Lin, Zhigui; Ruiz-Baier, Ricardo; Tian, Canrong
2014-01-01
This paper is concerned with the study of pattern formation for an inhomogeneous Brusselator model with cross-diffusion, modeling an autocatalytic chemical reaction taking place in a three-dimensional domain. For the spatial discretization of the problem we develop a novel finite volume element (FVE) method associated to a piecewise linear finite element approximation of the cross-diffusion system. We study the main properties of the unique equilibrium of the related dynamical system. A rigorous linear stability analysis around the spatially homogeneous steady state is provided and we address in detail the formation of Turing patterns driven by the cross-diffusion effect. In addition we focus on the spatial accuracy of the FVE method, and a series of numerical simulations confirm the expected behavior of the solutions. In particular we show that, depending on the spatial dimension, the magnitude of the cross-diffusion influences the selection of spatial patterns.
Design of low-power hybrid digital pulse width modulator with piecewise calibration scheme
NASA Astrophysics Data System (ADS)
Zhen, Shaowei; Hou, Sijian; Gan, Wubing; Chen, Jingbo; Luo, Ping; Zhang, Bo
2015-12-01
A low-power hybrid digital pulse width modulator (DPWM) is proposed in the paper. Owing to the piecewise calibration scheme, the delay time of delay line is locked to target frequency. The delay line consists of two piecewise lines with different control codes. The delay time of each cell in one sub-delay-line is longer than the last significant bit (LSB) of DPWM, while the delay time of each cell in the other sub-delay-line is shorter than LSB. Optimum linearity is realised with minimum standard cells. Simulation results show that the differential nonlinearity and integral nonlinearity are improved from 5.1 to 0.4 and from 5 to 1.3, respectively. The DPWM is fully synthesised and fabricated in a 90-nm CMOS process. The proposed DPWM occupies a silicon area of 0.01 mm2, with 31.5 μw core power consumption. Experimental results are shown to demonstrate the 2-MHz, 10-bit resolution implementation. Pulse width histogram is firstly introduced to characterise the linearity of the DPWM.
A piecewise-focused high DQE detector for MV imaging
Star-Lack, Josh; Shedlock, Daniel; Swahn, Dennis; Humber, Dave; Wang, Adam; Hirsh, Hayley; Zentai, George; Sawkey, Daren; Kruger, Isaac; Sun, Mingshan; Abel, Eric; Virshup, Gary; Shin, Mihye; Fahrig, Rebecca
2015-01-01
Purpose: Electronic portal imagers (EPIDs) with high detective quantum efficiencies (DQEs) are sought to facilitate the use of the megavoltage (MV) radiotherapy treatment beam for image guidance. Potential advantages include high quality (treatment) beam’s eye view imaging, and improved cone-beam computed tomography (CBCT) generating images with more accurate electron density maps with immunity to metal artifacts. One approach to increasing detector sensitivity is to couple a thick pixelated scintillator array to an active matrix flat panel imager (AMFPI) incorporating amorphous silicon thin film electronics. Cadmium tungstate (CWO) has many desirable scintillation properties including good light output, a high index of refraction, high optical transparency, and reasonable cost. However, due to the 0 1 0 cleave plane inherent in its crystalline structure, the difficulty of cutting and polishing CWO has, in part, limited its study relative to other scintillators such as cesium iodide and bismuth germanate (BGO). The goal of this work was to build and test a focused large-area pixelated “strip” CWO detector. Methods: A 361 × 52 mm scintillator assembly that contained a total of 28 072 pixels was constructed. The assembly comprised seven subarrays, each 15 mm thick. Six of the subarrays were fabricated from CWO with a pixel pitch of 0.784 mm, while one array was constructed from BGO for comparison. Focusing was achieved by coupling the arrays to the Varian AS1000 AMFPI through a piecewise linear arc-shaped fiber optic plate. Simulation and experimental studies of modulation transfer function (MTF) and DQE were undertaken using a 6 MV beam, and comparisons were made between the performance of the pixelated strip assembly and the most common EPID configuration comprising a 1 mm-thick copper build-up plate attached to a 133 mg/cm2 gadolinium oxysulfide scintillator screen (Cu-GOS). Projection radiographs and CBCT images of phantoms were acquired. The work
Structural properties of fluids interacting via piece-wise constant potentials with a hard core
NASA Astrophysics Data System (ADS)
Santos, Andrés; Yuste, Santos B.; de Haro, Mariano López; Bárcenas, Mariana; Orea, Pedro
2013-08-01
The structural properties of fluids whose molecules interact via potentials with a hard core plus two piece-wise constant sections of different widths and heights are presented. These follow from the more general development previously introduced for potentials with a hard core plus n piece-wise constant sections [A. Santos, S. B. Yuste, and M. Lopez de Haro, Condens. Matter Phys. 15, 23602 (2012)], 10.5488/CMP.15.23602 in which use was made of a semi-analytic rational-function approximation method. The results of illustrative cases comprising eight different combinations of wells and shoulders are compared both with simulation data and with those that follow from the numerical solution of the Percus-Yevick and hypernetted-chain integral equations. It is found that the rational-function approximation generally predicts a more accurate radial distribution function than the Percus-Yevick theory and is comparable or even superior to the hypernetted-chain theory. This superiority over both integral equation theories is lost, however, at high densities, especially as the widths of the wells and/or the barriers increase.
One Line or Two? Perspectives on Piecewise Regression
Technology Transfer Automated Retrieval System (TEKTRAN)
Sometimes we are faced with data that could reasonably be represented either as a single line, or as two or more line segments. How do we identify the best breakpoint(s), and decide how many segments are “really” present? Most of us were taught to distrust piecewise regression because it can be ea...
Implementation of nonlinear registration of brain atlas based on piecewise grid system
NASA Astrophysics Data System (ADS)
Liu, Rong; Gu, Lixu; Xu, Jianrong
2007-12-01
In this paper, a multi-step registration method of brain atlas and clinical Magnetic Resonance Imaging (MRI) data based on Thin-Plate Splines (TPS) and Piecewise Grid System (PGS) is presented. The method can help doctors to determine the corresponding anatomical structure between patient image and the brain atlas by piecewise nonlinear registration. Since doctors mostly pay attention to particular Region of Interest (ROI), and a global nonlinear registration is quite time-consuming which is not suitable for real-time clinical application, we propose a novel method to conduct linear registration in global area before nonlinear registration is performed in selected ROI. The homogenous feature points are defined to calculate the transform matrix between patient data and the brain atlas to conclude the mapping function. Finally, we integrate the proposed approach into an application of neurosurgical planning and guidance system which lends great efficiency in both neuro-anatomical education and guiding of neurosurgical operations. The experimental results reveal that the proposed approach can keep an average registration error of 0.25mm in near real-time manner.
NASA Astrophysics Data System (ADS)
Tatsii, R. M.; Pazen, O. Yu.
2016-03-01
A constructive scheme for the construction of a solution of a mixed problem for the heat conduction equation with piecewise-continuous coefficients coordinate-dependent in the final interval is suggested and validated in the present work. The boundary conditions are assumed to be most general. The scheme is based on: the reduction method, the concept of quasi-derivatives, the currently accepted theory of the systems of linear differential equations, the Fourier method, and the modified method of eigenfunctions. The method based on this scheme should be related to direct exact methods of solving mixed problems that do not employ the procedures of constructing Green's functions or integral transformations. Here the theorem of eigenfunction expansion is adapted for the case of coefficients that have discontinuity points of the 1st kind. The results obtained can be used, for example, in investigating the process of heat transfer in a multilayer slab under conditions of ideal thermal contact between the layers. A particular case of piecewise-continuous coefficients is considered. A numerical example of calculation of a temperature field in a real four-layer building slab under boundary conditions of the 3rd kind (conditions of convective heat transfer) that model the phenomenon of fire near one of the external surfaces is given.
Gardini, Laura; Fournier-Prunaret, Danièle; Chargé, Pascal
2011-06-01
In recent years, the study of chaotic and complex phenomena in electronic circuits has been widely developed due to the increasing number of applications. In these studies, associated with the use of chaotic sequences, chaos is required to be robust (not occurring only in a set of zero measure and persistent to perturbations of the system). These properties are not easy to be proved, and numerical simulations are often used. In this work, we consider a simple electronic switching circuit, proposed as chaos generator. The object of our study is to determine the ranges of the parameters at which the dynamics are chaotic, rigorously proving that chaos is robust. This is obtained showing that the model can be studied via a two-dimensional piecewise smooth map in triangular form and associated with a one-dimensional piecewise linear map. The bifurcations in the parameter space are determined analytically. These are the border collision bifurcation curves, the degenerate flip bifurcations, which only are allowed to occur to destabilize the stable cycles, and the homoclinic bifurcations occurring in cyclical chaotic regions leading to chaos in 1-piece.
Existence of almost periodic solutions for forced perturbed systems with piecewise constant argument
NASA Astrophysics Data System (ADS)
Xia, Yonghui; Huang, Zhenkun; Han, Maoan
2007-09-01
Certain almost periodic forced perturbed systems with piecewise argument are considered in this paper. By using the contraction mapping principle and some new analysis technique, some sufficient conditions are obtained for the existence and uniqueness of almost periodic solution of these systems. Furthermore, we study the harmonic and subharmonic solutions of these systems. The obtained results generalize the previous known results such as [A.M. Fink, Almost Periodic Differential Equation, Lecture Notes in Math., volE 377, Springer-Verlag, Berlin, 1974; C.Y. He, Almost Periodic Differential Equations, Higher Education Press, Beijing, 1992 (in Chinese); Z.S. Lin, The existence of almost periodic solution of linear system, Acta Math. Sinica 22 (5) (1979) 515-528 (in Chinese); C.Y. He, Existence of almost periodic solutions of perturbation systems, Ann. Differential Equations 9 (2) (1992) 173-181; Y.H. Xia, M. Lin, J. Cao, The existence of almost periodic solutions of certain perturbation system, J. Math. Anal. Appl. 310 (1) (2005) 81-96]. Finally, a tangible example and its numeric simulations show the feasibility of our results, the comparison between non-perturbed system and perturbed system, the relation between systems with and without piecewise argument.
NASA Astrophysics Data System (ADS)
Ahlkrona, Josefin; Lötstedt, Per; Kirchner, Nina; Zwinger, Thomas
2016-03-01
We propose and implement a new method, called the Ice Sheet Coupled Approximation Levels (ISCAL) method, for simulation of ice sheet flow in large domains during long time-intervals. The method couples the full Stokes (FS) equations with the Shallow Ice Approximation (SIA). The part of the domain where SIA is applied is determined automatically and dynamically based on estimates of the modeling error. For a three dimensional model problem, ISCAL computes the solution substantially faster with a low reduction in accuracy compared to a monolithic FS. Furthermore, ISCAL is shown to be able to detect rapid dynamic changes in the flow. Three different error estimations are applied and compared. Finally, ISCAL is applied to the Greenland Ice Sheet on a quasi-uniform grid, proving ISCAL to be a potential valuable tool for the ice sheet modeling community.
Estimating piecewise exponential frailty model with changing prior for baseline hazard function
NASA Astrophysics Data System (ADS)
Thamrin, Sri Astuti; Lawi, Armin
2016-02-01
Piecewise exponential models provide a very flexible framework for modelling univariate survival data. It can be used to estimate the effects of different covariates which are influenced by the survival data. Although in a strict sense it is a parametric model, a piecewise exponential hazard can approximate any shape of a parametric baseline hazard. In the parametric baseline hazard, the hazard function for each individual may depend on a set of risk factors or explanatory variables. However, it usually does not explain all such variables which are known or measurable, and these variables become interesting to be considered. This unknown and unobservable risk factor of the hazard function is often termed as the individual's heterogeneity or frailty. This paper analyses the effects of unobserved population heterogeneity in patients' survival times. The issue of model choice through variable selection is also considered. A sensitivity analysis is conducted to assess the influence of the prior for each parameter. We used the Markov Chain Monte Carlo method in computing the Bayesian estimator on kidney infection data. The results obtained show that the sex and frailty are substantially associated with survival in this study and the models are relatively quite sensitive to the choice of two different priors.
Piecewise aggregate representations and lower-bound distance functions for multivariate time series
NASA Astrophysics Data System (ADS)
Li, Hailin
2015-06-01
Dimensionality reduction is one of the most important methods to improve the efficiency of the techniques that are applied to the field of multivariate time series data mining. Due to multivariate time series with the variable-based and time-based dimensions, the reduction techniques must take both of them into consideration. To achieve this goal, we use a center sequence to represent a multivariate time series so that the new sequence can be seen as a univariate time series. Thus two sophisticated piecewise aggregate representations, including piecewise aggregate approximation and symbolization applied to univariate time series, are used to further represent the extended sequence that is derived from the center one. Furthermore, some distance functions are designed to measure the similarity between two representations. Through being proven by some related mathematical analysis, the proposed functions are lower bound on Euclidean distance and dynamic time warping. In this way, false dismissals can be avoided when they are used to index the time series. In addition, multivariate time series with different lengths can be transformed into the extended sequences with equal length, and their corresponding distance functions can measure the similarity between two unequal-length multivariate time series. The experimental results demonstrate that the proposed methods can reduce the dimensionality, and their corresponding distance functions satisfy the lower-bound condition, which can speed up the calculation of similarity search and indexing in the multivariate time series datasets.
Hergt, Steven; Shah, Abhay; Schäfer, Gerhard
2013-07-12
The orbital motion is derived for a nonspinning test mass in the relativistic, gravitational field of a rotationally deformed body not restricted to the equatorial plane or spherical orbit. The gravitational field of the central body is represented by the Kerr metric, expanded to second post-Newtonian order including the linear and quadratic spin terms. The orbital period, the intrinsic periastron advance, and the precession of the orbital plane are derived with the aid of novel canonical variables and action-based methods.
Stability of bumps in piecewise smooth neural fields with nonlinear adaptation
NASA Astrophysics Data System (ADS)
Kilpatrick, Zachary P.; Bressloff, Paul C.
2010-06-01
We study the linear stability of stationary bumps in piecewise smooth neural fields with local negative feedback in the form of synaptic depression or spike frequency adaptation. The continuum dynamics is described in terms of a nonlocal integrodifferential equation, in which the integral kernel represents the spatial distribution of synaptic weights between populations of neurons whose mean firing rate is taken to be a Heaviside function of local activity. Discontinuities in the adaptation variable associated with a bump solution means that bump stability cannot be analyzed by constructing the Evans function for a network with a sigmoidal gain function and then taking the high-gain limit. In the case of synaptic depression, we show that linear stability can be formulated in terms of solutions to a system of pseudo-linear equations. We thus establish that sufficiently strong synaptic depression can destabilize a bump that is stable in the absence of depression. These instabilities are dominated by shift perturbations that evolve into traveling pulses. In the case of spike frequency adaptation, we show that for a wide class of perturbations the activity and adaptation variables decouple in the linear regime, thus allowing us to explicitly determine stability in terms of the spectrum of a smooth linear operator. We find that bumps are always unstable with respect to this class of perturbations, and destabilization of a bump can result in either a traveling pulse or a spatially localized breather.
Stochastic Schrödinger evolution over piecewise enlarged filtrations
NASA Astrophysics Data System (ADS)
Mengütürk, Levent Ali
2016-03-01
This paper constructs a nonlinear filtering framework that admits appearances of new information processes at random times by introducing piecewise enlargements of filtrations and proposes a new energy-based Schrodinger evolution expressed as a stochastic differential equation on a complex Hilbert space. Each information process is modeled as the sum of a random variable taking the eigenvalues of a Hamiltonian and an independent Brownian bridge noise. It is shown that under a piecewise enlarged filtration, the wave function is a jump-diffusion process until it collapses at some terminal time. In between discontinuities, the dynamics of the state vector are governed by different Wiener processes and diffusion coefficients. This motivates the introduction of an inclusive chain of Kolmogorov probability spaces or a *-isomorphic chain of commutative von Neumann probability spaces, on which the quantum system evolves differently based on the number of active information processes. The expectation of the Hamiltonian at a given state is the solution of a second-order nonlinear differential equation determined by one of the possible regimes that the quantum system belongs to. It is shown that the collapse rate is a submartingale with positive jumps and the Shannon entropy process is a supermartingale with expected negative jumps when passing to higher-order probability spaces. The framework is extended to the case when the Hamiltonian is modeled as a function of a set of commutative operators, where each operator is associated with a different piecewise enlarged filtration.
Periodic Solutions for Nonlinear Integro-Differential Systems with Piecewise Constant Argument
2014-01-01
We investigate the existence of the periodic solutions of a nonlinear integro-differential system with piecewise alternately advanced and retarded argument of generalized type, in short DEPCAG; that is, the argument is a general step function. We consider the critical case, when associated linear homogeneous system admits nontrivial periodic solutions. Criteria of existence of periodic solutions of such equations are obtained. In the process we use Green's function for periodic solutions and convert the given DEPCAG into an equivalent integral equation. Then we construct appropriate mappings and employ Krasnoselskii's fixed point theorem to show the existence of a periodic solution of this type of nonlinear differential equations. We also use the contraction mapping principle to show the existence of a unique periodic solution. Appropriate examples are given to show the feasibility of our results. PMID:24574895
Periodic solutions for nonlinear integro-differential systems with piecewise constant argument.
Chiu, Kuo-Shou
2014-01-01
We investigate the existence of the periodic solutions of a nonlinear integro-differential system with piecewise alternately advanced and retarded argument of generalized type, in short DEPCAG; that is, the argument is a general step function. We consider the critical case, when associated linear homogeneous system admits nontrivial periodic solutions. Criteria of existence of periodic solutions of such equations are obtained. In the process we use Green's function for periodic solutions and convert the given DEPCAG into an equivalent integral equation. Then we construct appropriate mappings and employ Krasnoselskii's fixed point theorem to show the existence of a periodic solution of this type of nonlinear differential equations. We also use the contraction mapping principle to show the existence of a unique periodic solution. Appropriate examples are given to show the feasibility of our results. PMID:24574895
Kumar, R Suresh; Sugumaran, G; Raja, R; Zhu, Quanxin; Raja, U Karthik
2016-02-01
This paper analyzes the global asymptotic stability of a class of neural networks with time delay in the leakage term and time-varying delays under impulsive perturbations. Here the time-varying delays are assumed to be piecewise. In this method, the interval of the variation is divided into two subintervals by its central point. By developing a new Lyapunov-Krasovskii functional and checking its variation in between the two subintervals, respectively, and then we present some sufficient conditions to guarantee the global asymptotic stability of the equilibrium point for the considered neural network. The proposed results which do not require the boundedness, differentiability and monotonicity of the activation functions, can be easily verified via the linear matrix inequality (LMI) control toolbox in MATLAB. Finally, a numerical example and its simulation are given to show the conditions obtained are new and less conservative than some existing ones in the literature.
Wavelet Sparse Approximate Inverse Preconditioners
NASA Technical Reports Server (NTRS)
Chan, Tony F.; Tang, W.-P.; Wan, W. L.
1996-01-01
There is an increasing interest in using sparse approximate inverses as preconditioners for Krylov subspace iterative methods. Recent studies of Grote and Huckle and Chow and Saad also show that sparse approximate inverse preconditioner can be effective for a variety of matrices, e.g. Harwell-Boeing collections. Nonetheless a drawback is that it requires rapid decay of the inverse entries so that sparse approximate inverse is possible. However, for the class of matrices that, come from elliptic PDE problems, this assumption may not necessarily hold. Our main idea is to look for a basis, other than the standard one, such that a sparse representation of the inverse is feasible. A crucial observation is that the kind of matrices we are interested in typically have a piecewise smooth inverse. We exploit this fact, by applying wavelet techniques to construct a better sparse approximate inverse in the wavelet basis. We shall justify theoretically and numerically that our approach is effective for matrices with smooth inverse. We emphasize that in this paper we have only presented the idea of wavelet approximate inverses and demonstrated its potential but have not yet developed a highly refined and efficient algorithm.
An algorithm for the numerical solution of linear differential games
Polovinkin, E S; Ivanov, G E; Balashov, M V; Konstantinov, R V; Khorev, A V
2001-10-31
A numerical algorithm for the construction of stable Krasovskii bridges, Pontryagin alternating sets, and also of piecewise program strategies solving two-person linear differential (pursuit or evasion) games on a fixed time interval is developed on the basis of a general theory. The aim of the first player (the pursuer) is to hit a prescribed target (terminal) set by the phase vector of the control system at the prescribed time. The aim of the second player (the evader) is the opposite. A description of numerical algorithms used in the solution of differential games of the type under consideration is presented and estimates of the errors resulting from the approximation of the game sets by polyhedra are presented.
NASA Astrophysics Data System (ADS)
Isegawa, Miho; Truhlar, Donald G.
2013-04-01
Time-dependent density functional theory (TDDFT) holds great promise for studying photochemistry because of its affordable cost for large systems and for repeated calculations as required for direct dynamics. The chief obstacle is uncertain accuracy. There have been many validation studies, but there are also many formulations, and there have been few studies where several formulations were applied systematically to the same problems. Another issue, when TDDFT is applied with only a single exchange-correlation functional, is that errors in the functional may mask successes or failures of the formulation. Here, to try to sort out some of the issues, we apply eight formulations of adiabatic TDDFT to the first valence excitations of ten molecules with 18 density functionals of diverse types. The formulations examined are linear response from the ground state (LR-TDDFT), linear response from the ground state with the Tamm-Dancoff approximation (TDDFT-TDA), the original collinear spin-flip approximation with the Tamm-Dancoff (TD) approximation (SF1-TDDFT-TDA), the original noncollinear spin-flip approximation with the TDA approximation (SF1-NC-TDDFT-TDA), combined self-consistent-field (SCF) and collinear spin-flip calculations in the original spin-projected form (SF2-TDDFT-TDA) or non-spin-projected (NSF2-TDDFT-TDA), and combined SCF and noncollinear spin-flip calculations (SF2-NC-TDDFT-TDA and NSF2-NC-TDDFT-TDA). Comparing LR-TDDFT to TDDFT-TDA, we observed that the excitation energy is raised by the TDA; this brings the excitation energies underestimated by full linear response closer to experiment, but sometimes it makes the results worse. For ethylene and butadiene, the excitation energies are underestimated by LR-TDDFT, and the error becomes smaller making the TDA. Neither SF1-TDDFT-TDA nor SF2-TDDFT-TDA provides a lower mean unsigned error than LR-TDDFT or TDDFT-TDA. The comparison between collinear and noncollinear kernels shows that the noncollinear kernel
Optimal approximation method to characterize the resource trade-off functions for media servers
NASA Astrophysics Data System (ADS)
Chang, Ray-I.
1999-08-01
We have proposed an algorithm to smooth the transmission of the pre-recorded VBR media stream. It takes O(n) time complexity, where n is large, this algorithm is not suitable for online resource management and admission control in media servers. To resolve this drawback, we have explored the optimal tradeoff among resources by an O(nlogn) algorithm. Based on the pre-computed resource tradeoff function, the resource management and admission control procedure is as simple as table hashing. However, this approach requires O(n) space to store and maintain the resource tradeoff function. In this paper, while giving some extra resources, a linear-time algorithm is proposed to approximate the resource tradeoff function by piecewise line segments. We can prove that the number of line segments in the obtained approximation function is minimized for the given extra resources. The proposed algorithm has been applied to approximate the bandwidth-buffer-tradeoff function of the real-world Star War movie. While an extra 0.1 Mbps bandwidth is given, the storage space required for the approximation function is over 2000 times smaller than that required for the original function. While an extra 10 KB buffer is given, the storage space for the approximation function is over 2200 over times smaller than that required for the original function. The proposed algorithm is really useful for resource management and admission control in real-world media servers.
Ultrasonic linear measurement system
NASA Technical Reports Server (NTRS)
Marshall, Scot H. (Inventor)
1991-01-01
An ultrasonic linear measurement system uses the travel time of surface waves along the perimeter of a three-dimensional curvilinear body to determine the perimeter of the curvilinear body. The system can also be used piece-wise to measure distances along plane surfaces. The system can be used to measure perimeters where use of laser light, optical means or steel tape would be extremely difficult, time consuming or impossible. It can also be used to determine discontinuities in surfaces of known perimeter or dimension.
Piecewise-parabolic methods for astrophysical fluid dynamics
Woodward, P.R.
1983-11-01
A general description of some modern numerical techniques for the simulation of astrophysical fluid flow is presented. The methods are introduced with a thorough discussion of the especially simple case of advection. Attention is focused on the piecewise-parabolic method (PPM). A description of the SLIC method for treating multifluid problems is also given. The discussion is illustrated by a number of advection and hydrodynamics test problems. Finally, a study of Kelvin-Helmholtz instability of supersonic jets using PPM with SLIC fluid interfaces is presented.
Approximating Functions with Exponential Functions
ERIC Educational Resources Information Center
Gordon, Sheldon P.
2005-01-01
The possibility of approximating a function with a linear combination of exponential functions of the form e[superscript x], e[superscript 2x], ... is considered as a parallel development to the notion of Taylor polynomials which approximate a function with a linear combination of power function terms. The sinusoidal functions sin "x" and cos "x"…
Towards a Theory of Sampled-Data Piecewise-Deterministic Markov Processes
NASA Technical Reports Server (NTRS)
Herencia-Zapana, Heber; Gonzalez, Oscar R.; Gray, W. Steven
2006-01-01
The analysis and design of practical control systems requires that stochastic models be employed. Analysis and design tools have been developed, for example, for Markovian jump linear continuous and discrete-time systems, piecewise-deterministic processes (PDP's), and general stochastic hybrid systems (GSHS's). These model classes have been used in many applications, including fault tolerant control and networked control systems. This paper presents initial results on the analysis of a sampled-data PDP representation of a nonlinear sampled-data system with a jump linear controller. In particular, it is shown that the state of the sampled-data PDP satisfies the strong Markov property. In addition, a relation between the invariant measures of a sampled-data system driven by a stochastic process and its associated discrete-time representation are presented. As an application, when the plant is linear with no external input, a sufficient testable condition for the convergence in distribution to the invariant delta Dirac measure is given.
Automated piecewise power-law modeling of biological systems.
Machina, Anna; Ponosov, Arkady; Voit, Eberhard O
2010-09-01
Recent trends suggest that future biotechnology will increasingly rely on mathematical models of the biological systems under investigation. In particular, metabolic engineering will make wider use of metabolic pathway models in stoichiometric or fully kinetic format. A significant obstacle to the use of pathway models is the identification of suitable process descriptions and their parameters. We recently showed that, at least under favorable conditions, Dynamic Flux Estimation (DFE) permits the numerical characterization of fluxes from sets of metabolic time series data. However, DFE does not prescribe how to convert these numerical results into functional representations. In some cases, Michaelis-Menten rate laws or canonical formats are well suited, in which case the estimation of parameter values is easy. However, in other cases, appropriate functional forms are not evident, and exhaustive searches among all possible candidate models are not feasible. We show here how piecewise power-law functions of one or more variables offer an effective default solution for the almost unbiased representation of uni- and multivariate time series data. The results of an automated algorithm for their determination are piecewise power-law fits, whose accuracy is only limited by the available data. The individual power-law pieces may lead to discontinuities at break points or boundaries between sub-domains. In many practical applications, these boundary gaps do not cause problems. Potential smoothing techniques, based on differential inclusions and Filippov's theory, are discussed in Appendix A. PMID:20060428
High resolution A/D conversion based on piecewise conversion at lower resolution
Terwilliger, Steve
2012-06-05
Piecewise conversion of an analog input signal is performed utilizing a plurality of relatively lower bit resolution A/D conversions. The results of this piecewise conversion are interpreted to achieve a relatively higher bit resolution A/D conversion without sampling frequency penalty.
Weighted piecewise LDA for solving the small sample size problem in face verification.
Kyperountas, Marios; Tefas, Anastasios; Pitas, Ioannis
2007-03-01
A novel algorithm that can be used to boost the performance of face-verification methods that utilize Fisher's criterion is presented and evaluated. The algorithm is applied to similarity, or matching error, data and provides a general solution for overcoming the "small sample size" (SSS) problem, where the lack of sufficient training samples causes improper estimation of a linear separation hyperplane between the classes. Two independent phases constitute the proposed method. Initially, a set of weighted piecewise discriminant hyperplanes are used in order to provide a more accurate discriminant decision than the one produced by the traditional linear discriminant analysis (LDA) methodology. The expected classification ability of this method is investigated throughout a series of simulations. The second phase defines proper combinations for person-specific similarity scores and describes an outlier removal process that further enhances the classification ability. The proposed technique has been tested on the M2VTS and XM2VTS frontal face databases. Experimental results indicate that the proposed framework greatly improves the face-verification performance.
Fast wavelet based sparse approximate inverse preconditioner
Wan, W.L.
1996-12-31
Incomplete LU factorization is a robust preconditioner for both general and PDE problems but unfortunately not easy to parallelize. Recent study of Huckle and Grote and Chow and Saad showed that sparse approximate inverse could be a potential alternative while readily parallelizable. However, for special class of matrix A that comes from elliptic PDE problems, their preconditioners are not optimal in the sense that independent of mesh size. A reason may be that no good sparse approximate inverse exists for the dense inverse matrix. Our observation is that for this kind of matrices, its inverse entries typically have piecewise smooth changes. We can take advantage of this fact and use wavelet compression techniques to construct a better sparse approximate inverse preconditioner. We shall show numerically that our approach is effective for this kind of matrices.
A cosmological hydrodynamic code based on the piecewise parabolic method
NASA Astrophysics Data System (ADS)
Gheller, Claudio; Pantano, Ornella; Moscardini, Lauro
1998-04-01
We present a hydrodynamical code for cosmological simulations which uses the piecewise parabolic method (PPM) to follow the dynamics of the gas component and an N-body particle-mesh algorithm for the evolution of the collisionless component. The gravitational interaction between the two components is regulated by the Poisson equation, which is solved by a standard fast Fourier transform (FFT) procedure. In order to simulate cosmological flows, we have introduced several modifications to the original PPM scheme, which we describe in detail. Various tests of the code are presented, including adiabatic expansion, single and multiple pancake formation, and three-dimensional cosmological simulations with initial conditions based on the cold dark matter scenario.
Duality in parameter space and approximation of measures for mixing repellers
NASA Astrophysics Data System (ADS)
Abenda, S.; Turchetti, G.
1990-10-01
For one-dimensional expanding maps T with an invariant measure μ we consider, in a parameter space, the envelope ℰ n of the real lines associated to any couple of points of the orbit, connected by n iterations of T. If the map has s inverses and is piecewise linear, then the sets ℰ n are just the union of s n points and converge to the invariant Cantor set of T. A correspondence between all the sets and their measures is established and allows one to associate the atomic measure on ℰ1 to the completly continuous measure on the Cantor set. If the map is nonlinear, hyperbolic, and has s inverses, the sets ℰ n are homeomorphic to the Cantor set; they converge to the Cantor set of T and their measures converge to the measure of the Cantor set when n→∞. The correspondence between the sets ℰ n allows one to define converging approximation schemes for the map an its measure: one replaces each of the s n disjoint sets with a point in a convenient neighborhood and a probability equal to its measure and transforms it back to the original set ℰ1. All the approximations with linear Cantor systems previously proposed are recovered, the converging proprties being straightforward in the present scheme. Moreover, extensions to higher dimensionality and to nondisconnected repellers arte possible and are briefly examined.
Exponential Approximations Using Fourier Series Partial Sums
NASA Technical Reports Server (NTRS)
Banerjee, Nana S.; Geer, James F.
1997-01-01
The problem of accurately 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 approximate 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 approximated to within O(N(sup -M-2), and the associated jump of the k(sup th) derivative of f is approximated 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 approximations to f. Each of these new approximations 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 approximations, 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.
Kernel approximation for solving few-body integral equations
NASA Astrophysics Data System (ADS)
Christie, I.; Eyre, D.
1986-06-01
This paper investigates an approximate method for solving integral equations that arise in few-body problems. The method is to replace the kernel by a degenerate kernel defined on a finite dimensional subspace of piecewise Lagrange polynomials. Numerical accuracy of the method is tested by solving the two-body Lippmann-Schwinger equation with non-separable potentials, and the three-body Amado-Lovelace equation with separable two-body potentials.
DC-ELF characterization of random mixtures of piecewise nonlinear media.
Bianco, B; Chiabrera, A; Giordano, S
2000-02-01
Biological tissues are ensembles of linear and nonlinear, symmetric and asymmetric constituents. As far as their electromagnetic characterization is concerned, they can be modeled as microscopic mixtures of the corresponding material media. Any medium volume can be properly discretized in a finite number of cells which can be modeled as an equivalent three dimensional network of lumped components, in order to characterize its electromagnetic behavior at wavelengths much longer than the relevant average linear size of the constitutive cells. Therefore, any mixture and the corresponding tissue can be characterized in terms of its effective conductance at extremely low frequency, with respect to a reference set of electrodes (ports of the equivalent network). When the above procedure is implemented for evaluating any of the aforesaid conductances, a resulting nonlinear characteristic should be expected. In reality, it may happen that the effect of the constitutive nonlinearities and the related asymmetries are smeared out by the randomness of the interconnections of the lumped components, leading at a macroscopic level to an isotropic constant equivalent conductance, i.e., to an isotropic constant equivalent conductivity of the mixture. The closed form analysis of a random network of nonlinear (piecewise linear) resistors offers a simple but clear cut example of such a property. This result, if extrapolated to biological media, suggests a new hint for explaining why there is no inconsistency between the typical electric characterization of biological tissues as almost linear macroscopic media, by means of their effective conductivity and permittivity, and the nonlinearities of the biochemical processes occurring in the tissue cells. In fact, the nonlinearities may not be observable by means of macroscopic electrical measurements because of the randomized spatial orientation and location of the processes.
Boys, C A; Robinson, W; Miller, B; Pflugrath, B; Baumgartner, L J; Navarro, A; Brown, R; Deng, Z
2016-05-01
A piecewise regression approach was used to objectively quantify barotrauma injury thresholds in two physoclistous species, Murray cod Maccullochella peelii and silver perch Bidyanus bidyanus, following simulated infrastructure passage in a barometric chamber. The probability of injuries such as swimbladder rupture, exophthalmia and haemorrhage, and emphysema in various organs increased as the ratio between the lowest exposure pressure and the acclimation pressure (ratio of pressure change, R(NE:A) ) reduced. The relationship was typically non-linear and piecewise regression was able to quantify thresholds in R(NE:A) that once exceeded resulted in a substantial increase in barotrauma injury. Thresholds differed among injury types and between species but by applying a multispecies precautionary principle, the maintenance of exposure pressures at river infrastructure above 70% of acclimation pressure (R(NE:A) of 0·7) should protect downstream migrating juveniles of these two physoclistous species sufficiently. These findings have important implications for determining the risk posed by current infrastructures and informing the design and operation of new ones. PMID:26991929
Boys, C A; Robinson, W; Miller, B; Pflugrath, B; Baumgartner, L J; Navarro, A; Brown, R; Deng, Z
2016-05-01
A piecewise regression approach was used to objectively quantify barotrauma injury thresholds in two physoclistous species, Murray cod Maccullochella peelii and silver perch Bidyanus bidyanus, following simulated infrastructure passage in a barometric chamber. The probability of injuries such as swimbladder rupture, exophthalmia and haemorrhage, and emphysema in various organs increased as the ratio between the lowest exposure pressure and the acclimation pressure (ratio of pressure change, R(NE:A) ) reduced. The relationship was typically non-linear and piecewise regression was able to quantify thresholds in R(NE:A) that once exceeded resulted in a substantial increase in barotrauma injury. Thresholds differed among injury types and between species but by applying a multispecies precautionary principle, the maintenance of exposure pressures at river infrastructure above 70% of acclimation pressure (R(NE:A) of 0·7) should protect downstream migrating juveniles of these two physoclistous species sufficiently. These findings have important implications for determining the risk posed by current infrastructures and informing the design and operation of new ones.
Zainudin, Mohd Lutfi; Saaban, Azizan; Bakar, Mohd Nazari Abu
2015-12-11
The solar radiation values have been composed by automatic weather station using the device that namely pyranometer. The device is functions to records all the radiation values that have been dispersed, and these data are very useful for it experimental works and solar device’s development. In addition, for modeling and designing on solar radiation system application is needed for complete data observation. Unfortunately, lack for obtained the complete solar radiation data frequently occur due to several technical problems, which mainly contributed by monitoring device. Into encountering this matter, estimation missing values in an effort to substitute absent values with imputed data. This paper aimed to evaluate several piecewise interpolation techniques likes linear, splines, cubic, and nearest neighbor into dealing missing values in hourly solar radiation data. Then, proposed an extendable work into investigating the potential used of cubic Bezier technique and cubic Said-ball method as estimator tools. As result, methods for cubic Bezier and Said-ball perform the best compare to another piecewise imputation technique.
Movie approximation technique for the implementation of fast bandwidth-smoothing algorithms
NASA Astrophysics Data System (ADS)
Feng, Wu-chi; Lam, Chi C.; Liu, Ming
1997-12-01
Bandwidth smoothing algorithms can effectively reduce the network resource requirements for the delivery of compressed video streams. For stored video, a large number of bandwidth smoothing algorithms have been introduced that are optimal under certain constraints but require access to all the frame size data in order to achieve their optimal properties. This constraint, however, can be both resource and computationally expensive, especially for moderately priced set-top-boxes. In this paper, we introduce a movie approximation technique for the representation of the frame sizes of a video, reducing the complexity of the bandwidth smoothing algorithms and the amount of frame data that must be transmitted prior to the start of playback. Our results show that the proposed approximation technique can accurately approximate the frame data with a small number of piece-wise linear segments without affecting the performance measures that the bandwidth soothing algorithms are attempting to achieve by more than 1%. In addition, we show that implementations of this technique can speed up execution times by 100 to 400 times, allowing the bandwidth plan calculation times to be reduced to tens of milliseconds. Evaluation using a compressed full-length motion-JPEG video is provided.
Linear equality constraints in the general linear mixed model.
Edwards, L J; Stewart, P W; Muller, K E; Helms, R W
2001-12-01
Scientists may wish to analyze correlated outcome data with constraints among the responses. For example, piecewise linear regression in a longitudinal data analysis can require use of a general linear mixed model combined with linear parameter constraints. Although well developed for standard univariate models, there are no general results that allow a data analyst to specify a mixed model equation in conjunction with a set of constraints on the parameters. We resolve the difficulty by precisely describing conditions that allow specifying linear parameter constraints that insure the validity of estimates and tests in a general linear mixed model. The recommended approach requires only straightforward and noniterative calculations to implement. We illustrate the convenience and advantages of the methods with a comparison of cognitive developmental patterns in a study of individuals from infancy to early adulthood for children from low-income families.
Wide-Angle Shift-Map PE for a Piecewise Linear Terrain
NASA Astrophysics Data System (ADS)
Holm, Peter D.
2009-03-01
A wide-angle parabolic wave equation solution, using shift-map and finite-difference techniques, is presented. The corresponding split-step Fourier solution is well known. The solution using finite-difference technique, where the standard parabolic wave equation is modified into the so-called Claerbout equation allowing propagation angles up to 45° from the paraxial direction, is also well known. Here, we present an extension to that solution in which the shift-map technique is incorporated into the finite-difference scheme allowing a varying terrain to be considered. The result is a solution that corresponds to the well known split-step solution, which is believed to perform well for terrain slopes up to 10°-15°, and discontinuous slope changes on the order of 15°-20°. This solution is of first order with respect to the terrain slope. The resulting difference scheme is for solving a tridiagonal system. When using the finite-difference technique, it is also possible to find a second order solution with respect to the terrain slope. The resulting difference scheme in this case is for solving a pentadiagonal system. This new solution performs well for slopes up to about 15°-20° and discontinuous slope changes up to about 30°-40°, which is an improvement.
Rate-distortion optimized tree-structured compression algorithms for piecewise polynomial images.
Shukla, Rahul; Dragotti, Pier Luigi; Do, Minh N; Vetterli, Martin
2005-03-01
This paper presents novel coding algorithms based on tree-structured segmentation, which achieve the correct asymptotic rate-distortion (R-D) behavior for a simple class of signals, known as piecewise polynomials, by using an R-D based prune and join scheme. For the one-dimensional case, our scheme is based on binary-tree segmentation of the signal. This scheme approximates the signal segments using polynomial models and utilizes an R-D optimal bit allocation strategy among the different signal segments. The scheme further encodes similar neighbors jointly to achieve the correct exponentially decaying R-D behavior (D(R) - c(o)2(-c1R)), thus improving over classic wavelet schemes. We also prove that the computational complexity of the scheme is of O(N log N). We then show the extension of this scheme to the two-dimensional case using a quadtree. This quadtree-coding scheme also achieves an exponentially decaying R-D behavior, for the polygonal image model composed of a white polygon-shaped object against a uniform black background, with low computational cost of O(N log N). Again, the key is an R-D optimized prune and join strategy. Finally, we conclude with numerical results, which show that the proposed quadtree-coding scheme outperforms JPEG2000 by about 1 dB for real images, like cameraman, at low rates of around 0.15 bpp. PMID:15762332
Discrete extrinsic curvatures and approximation of surfaces by polar polyhedra
NASA Astrophysics Data System (ADS)
Garanzha, V. A.
2010-01-01
Duality principle for approximation of geometrical objects (also known as Eu-doxus exhaustion method) was extended and perfected by Archimedes in his famous tractate “Measurement of circle”. The main idea of the approximation method by Archimedes is to construct a sequence of pairs of inscribed and circumscribed polygons (polyhedra) which approximate curvilinear convex body. This sequence allows to approximate length of curve, as well as area and volume of the bodies and to obtain error estimates for approximation. In this work it is shown that a sequence of pairs of locally polar polyhedra allows to construct piecewise-affine approximation to spherical Gauss map, to construct convergent point-wise approximations to mean and Gauss curvature, as well as to obtain natural discretizations of bending energies. The Suggested approach can be applied to nonconvex surfaces and in the case of multiple dimensions.
Archibald, Richard K; Deiterding, Ralf; Hauck, Cory D; Jakeman, John D; Xiu, Dongbin
2012-01-01
We have develop a fast method that can capture piecewise smooth functions in high dimensions with high order and low computational cost. This method can be used for both approximation and error estimation of stochastic simulations where the computations can either be guided or come from a legacy database.
Unfolding homoclinic connections formed by corner intersections in piecewise-smooth maps
NASA Astrophysics Data System (ADS)
Simpson, D. J. W.
2016-07-01
The stable and unstable manifolds of an invariant set of a piecewise-smooth map are themselves piecewise-smooth. Consequently, as parameters of a piecewise-smooth map are varied, an invariant set can develop a homoclinic connection when its stable manifold intersects a non-differentiable point of its unstable manifold (or vice-versa). This is a codimension-one bifurcation analogous to a homoclinic tangency of a smooth map, referred to here as a homoclinic corner. This paper presents an unfolding of generic homoclinic corners for saddle fixed points of planar piecewise-smooth continuous maps. It is shown that a sequence of border-collision bifurcations limits to a homoclinic corner and that all nearby periodic solutions are unstable.
NASA Astrophysics Data System (ADS)
Denisov, A. M.; Zakharov, E. V.; Kalinin, A. V.; Kalinin, V. V.
2010-07-01
A numerical method is proposed for solving an inverse electrocardiography problem for a medium with a piecewise constant electrical conductivity. The method is based on the method of boundary integral equations and Tikhonov regularization.
Construction of a Class of Four-Dimensional Piecewise Affine Systems with Homoclinic Orbits
NASA Astrophysics Data System (ADS)
Wu, Tiantian; Yang, Xiao-Song
2016-06-01
Based on mathematical analysis, this paper provides a methodology to ensure the existence of homoclinic orbits in a class of four-dimensional piecewise affine systems. In addition, an example is provided to illustrate the effectiveness of the method.
Hirata, Yoshito; Aihara, Kazuyuki
2012-06-01
We introduce a low-dimensional description for a high-dimensional system, which is a piecewise affine model whose state space is divided by permutations. We show that the proposed model tends to predict wind speeds and photovoltaic outputs for the time scales from seconds to 100 s better than by global affine models. In addition, computations using the piecewise affine model are much faster than those of usual nonlinear models such as radial basis function models.
Vision servoing of robot systems using piecewise continuous controllers and observers
NASA Astrophysics Data System (ADS)
Wang, H. P.; Vasseur, C.; Christov, N.; Koncar, V.
2012-11-01
This paper deals with the visual servoing of X-Y robot systems using low cost CCD camera. The proposed approach is based on the theory of piecewise continuous systems which are a particular class of hybrid systems with autonomous switching and controlled impulses. Visual trajectory tracking systems comprising piecewise continuous controllers and observers, are developed. Real-time results are given to illustrate the effectiveness of the proposed visual control system.
Robust Morse decompositions of piecewise constant vector fields.
Szymczak, Andrzej; Zhang, Eugene
2012-06-01
In this paper, we introduce a new approach to computing a Morse decomposition of a vector field on a triangulated manifold surface. The basic idea is to convert the input vector field to a piecewise constant (PC) vector field, whose trajectories can be computed using simple geometric rules. To overcome the intrinsic difficulty in PC vector fields (in particular, discontinuity along mesh edges), we borrow results from the theory of differential inclusions. The input vector field and its PC variant have similar Morse decompositions. We introduce a robust and efficient algorithm to compute Morse decompositions of a PC vector field. Our approach provides subtriangle precision for Morse sets. In addition, we describe a Morse set classification framework which we use to color code the Morse sets in order to enhance the visualization. We demonstrate the benefits of our approach with three well-known simulation data sets, for which our method has produced Morse decompositions that are similar to or finer than those obtained using existing techniques, and is over an order of magnitude faster. PMID:21747131
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.
NASA Technical Reports Server (NTRS)
Gottlieb, David; Shu, Chi-Wang
1993-01-01
The investigation of overcoming Gibbs phenomenon was continued, i.e., obtaining exponential accuracy at all points including at the discontinuities themselves, from the knowledge of a spectral partial sum of a discontinuous but piecewise analytic function. It was shown that if we are given the first N expansion coefficients of an L(sub 2) function f(x) in terms of either the trigonometrical polynomials or the Chebyshev or Legendre polynomials, an exponentially convergent approximation to the point values of f(x) in any sub-interval in which it is analytic can be constructed.
An approximation technique for jet impingement flow
Najafi, Mahmoud; Fincher, Donald; Rahni, Taeibi; Javadi, KH.; Massah, H.
2015-03-10
The analytical approximate solution of a non-linear jet impingement flow model will be demonstrated. We will show that this is an improvement over the series approximation obtained via the Adomian decomposition method, which is itself, a powerful method for analysing non-linear differential equations. The results of these approximations will be compared to the Runge-Kutta approximation in order to demonstrate their validity.
Melnikov Method for a Class of Planar Hybrid Piecewise-Smooth Systems
NASA Astrophysics Data System (ADS)
Li, Shuangbao; Ma, Wensai; Zhang, Wei; Hao, Yuxin
In this paper, we extend the well-known Melnikov method for smooth systems to a class of periodic perturbed planar hybrid piecewise-smooth systems. In this class, the switching manifold is a straight line which divides the plane into two zones, and the dynamics in each zone is governed by a smooth system. When a trajectory reaches the separation line, then a reset map is applied instantaneously before entering the trajectory in the other zone. We assume that the unperturbed system is a piecewise Hamiltonian system which possesses a piecewise-smooth homoclinic solution transversally crossing the switching manifold. Then, we study the persistence of the homoclinic orbit under a nonautonomous periodic perturbation and the reset map. To achieve this objective, we obtain the Melnikov function to measure the distance of the perturbed stable and unstable manifolds and present the theorem for homoclinic bifurcations for the class of planar hybrid piecewise-smooth systems. Furthermore, we employ the obtained Melnikov function to detect the chaotic boundaries for a concrete planar hybrid piecewise-smooth system.
Straten, Marcel van; Venema, Henk W.; Streekstra, Geert J.; Majoie, Charles B.L.M.; Heeten, Gerard J. den; Grimbergen, Cornelis A.
2004-10-01
In maximum intensity projection (MIP) images of CT angiography (CTA) scans, the arteries are often obscured by bone. A bone removal method is presented that uses an additional, nonenhanced scan to create a mask of the bone by thresholding and dilation. After registration of the CTA scan and the additional scan, the bone in the CTA scan is masked. As the cervical area contains bones that can move with respect to each other, these bones are separated first using a watershed algorithm, and then registered individually. A phantom study was performed to evaluate and quantify the tradeoff between the removal of the bone and the preservation of the arteries contiguous to the bone. The influence of algorithm parameters and scan parameters was studied. The method was clinically evaluated with data sets of 35 patients. Best results were obtained with a threshold of 150 HU and a dilation of 8 in-plane voxels and two out-of-plane voxels. The mean width of the soft tissue layer, which is also masked, was approximately 1 mm. The mAs value of the nonenhanced scan could be reduced from 250 mAs to 65 mAs without a loss of quality. In 32 cases the bones were registered correctly and removed completely. In three cases the bone separation was not completely successful, and consequently the bone was not completely removed. The piecewise matched mask bone elimination method proved to be able to obtain MIP images of the cervical arteries free from overprojecting bone in a fully automatic way and with only a slight increase of radiation dose.
Optimizing the Zeldovich approximation
NASA Technical Reports Server (NTRS)
Melott, Adrian L.; Pellman, Todd F.; Shandarin, Sergei F.
1994-01-01
We have recently learned that the Zeldovich approximation can be successfully used for a far wider range of gravitational instability scenarios than formerly proposed; we study here how to extend this range. In previous work (Coles, Melott and Shandarin 1993, hereafter CMS) we studied the accuracy of several analytic approximations to gravitational clustering in the mildly nonlinear regime. We found that what we called the 'truncated Zeldovich approximation' (TZA) was better than any other (except in one case the ordinary Zeldovich approximation) over a wide range from linear to mildly nonlinear (sigma approximately 3) regimes. TZA was specified by setting Fourier amplitudes equal to zero for all wavenumbers greater than k(sub nl), where k(sub nl) marks the transition to the nonlinear regime. Here, we study the cross correlation of generalized TZA with a group of n-body simulations for three shapes of window function: sharp k-truncation (as in CMS), a tophat in coordinate space, or a Gaussian. We also study the variation in the crosscorrelation as a function of initial truncation scale within each type. We find that k-truncation, which was so much better than other things tried in CMS, is the worst of these three window shapes. We find that a Gaussian window e(exp(-k(exp 2)/2k(exp 2, sub G))) applied to the initial Fourier amplitudes is the best choice. It produces a greatly improved crosscorrelation in those cases which most needed improvement, e.g. those with more small-scale power in the initial conditions. The optimum choice of kG for the Gaussian window is (a somewhat spectrum-dependent) 1 to 1.5 times k(sub nl). Although all three windows produce similar power spectra and density distribution functions after application of the Zeldovich approximation, the agreement of the phases of the Fourier components with the n-body simulation is better for the Gaussian window. We therefore ascribe the success of the best-choice Gaussian window to its superior treatment
Calculation of the biological effective dose for piecewise defined dose-rate fits
Hobbs, Robert F.; Sgouros, George
2009-03-15
An algorithmic solution to the biological effective dose (BED) calculation from the Lea-Catcheside formula for a piecewise defined function is presented. Data from patients treated for metastatic thyroid cancer were used to illustrate the solution. The Lea-Catcheside formula for the G-factor of the BED is integrated numerically using a large number of small trapezoidal fits to each integral. The algorithmically calculated BED is compatible with an analytic calculation for a similarly valued exponentially fitted dose-rate plot and is the only resolution for piecewise defined dose-rate functions.
What Can Tobit-Piecewise Regression Tell Us about the Determinants of Household Educational Debt?
ERIC Educational Resources Information Center
Thipbharos, Titirut
2014-01-01
Educational debt as part of household debt remains a problem for Thailand. The significant factors of household characteristics with regard to educational debt are shown by constructing a Tobit-piecewise regression for three different clusters, namely poor, middle and affluent households in Thailand. It was found that household debt is likely to…
ERIC Educational Resources Information Center
Jaggars, Shanna Smith; Xu, Di
2016-01-01
Policymakers have become increasingly concerned with measuring--and holding colleges accountable for--students' labor market outcomes. In this article we introduce a piecewise growth curve approach to analyzing community college students' labor market outcomes, and we discuss how this approach differs from two popular econometric approaches:…
A Piecewise Modeling Approach for Sensitivity Studies - Tests with WRF Model
NASA Astrophysics Data System (ADS)
Shao, Aimei; Qiu, Chongjian
2016-04-01
The model-based numerical sensitivity experiment is the basis for the attribution studies of the climate change and weather/climate anomalies. In order to reduce systematic errors and prevent model drift caused by accumulated model errors in the long-term simulations, a piecewise modeling approach that subdivides long integrations into a sequence of short ones is adopted for the sensitivity studies. For the control run with realistic external forcings to produce the reference state, it is similar to the common model reinitialization method in which the simulated state is updated periodically with available analysis data or observations at the end of each segment. For the perturbation run with some modified external forcings to produce perturbed states, no analysis data or observation is available. In this case, we update the simulated perturbed states with the sum of the analysis and the difference fields between two simulated states. An idealized twin sensitivity experiment was conducted with the Weather Research and Forecasting model (WRF) to evaluate the advantages of the piecewise approach over the conventional continuous modeling approach. The sensitivity experiment is designed to assess the effect of anomalous snow depth over the Tibetan Plateau in the preceding winter on East Asian summer climate, which consists of one conventional continuous run and eight piecewise runs with different segment length and error size in the analysis data. The results show that the piecewise approach can improve the credibility of the modeled sensitivity experiment and all eight piecewise runs perform significantly better than the continuous run. Compared with the continuous run, the mean absolute errors in summer 2-m temperature difference field and precipitation difference field over East Asia can be reduced by 36.9% and 16.3% by using the piecewise approach, respectively. The spatial correlation coefficient between the true and simulated state increases from 0.26 (for
Approximation by hinge functions
Faber, V.
1997-05-01
Breiman has defined {open_quotes}hinge functions{close_quotes} for use as basis functions in least squares approximations to data. A hinge function is the max (or min) function of two linear functions. In this paper, the author assumes the existence of smooth function f(x) and a set of samples of the form (x, f(x)) drawn from a probability distribution {rho}(x). The author hopes to find the best fitting hinge function h(x) in the least squares sense. There are two problems with this plan. First, Breiman has suggested an algorithm to perform this fit. The author shows that this algorithm is not robust and also shows how to create examples on which the algorithm diverges. Second, if the author tries to use the data to minimize the fit in the usual discrete least squares sense, the functional that must be minimized is continuous in the variables, but has a derivative which jumps at the data. This paper takes a different approach. This approach is an example of a method that the author has developed called {open_quotes}Monte Carlo Regression{close_quotes}. (A paper on the general theory is in preparation.) The author shall show that since the function f is continuous, the analytic form of the least squares equation is continuously differentiable. A local minimum is solved for by using Newton`s method, where the entries of the Hessian are estimated directly from the data by Monte Carlo. The algorithm has the desirable properties that it is quadratically convergent from any starting guess sufficiently close to a solution and that each iteration requires only a linear system solve.
Wolf, Paul A; Jørgensen, Jakob S; Schmidt, Taly G; Sidky, Emil Y
2013-01-01
A sparsity-exploiting algorithm intended for few-view Single Photon Emission Computed Tomography (SPECT) reconstruction is proposed and characterized. The algorithm models the object as piecewise constant subject to a blurring operation. To validate that the algorithm closely approximates the true object in the noiseless case, projection data were generated from an object assuming this model and using the system matrix. Monte Carlo simulations were performed to provide more realistic data of a phantom with varying smoothness across the field of view and a cardiac phantom. Reconstructions were performed across a sweep of two primary design parameters. The results demonstrate that the algorithm recovers the object in a noiseless simulation case. While the algorithm assumes a specific blurring model, the results suggest that the algorithm may provide high reconstruction accuracy even when the object does not match the assumed blurring model. Generally, increased values of the blurring parameter and Total Variation (TV) weighting parameters reduced streaking artifacts, while decreasing spatial resolution. The proposed algorithm demonstrated higher correlation with respect to the true phantom compared to Maximum Likelihood Expectation Maximization (MLEM) reconstructions. Images reconstructed with the proposed algorithm demonstrated reduced streaking artifacts when reconstructing from few views compared to MLEM. The proposed algorithm introduced patchy artifacts in some reconstructed images, depending on the noise level and the selected algorithm parameters. Overall, the results demonstrate preliminary feasibility of a sparsity-exploiting reconstruction algorithm which may be beneficial for few-view SPECT. PMID:23892823
NASA Astrophysics Data System (ADS)
Wolf, Paul A.; Jørgensen, Jakob S.; Schmidt, Taly G.; Sidky, Emil Y.
2013-08-01
A sparsity-exploiting algorithm intended for few-view single photon emission computed tomography (SPECT) reconstruction is proposed and characterized. The algorithm models the object as piecewise constant subject to a blurring operation. To validate that the algorithm closely approximates the true object in the noiseless case, projection data were generated from an object assuming this model and using the system matrix. Monte Carlo simulations were performed to provide more realistic data of a phantom with varying smoothness across the field of view and a cardiac phantom. Reconstructions were performed across a sweep of two primary design parameters. The results demonstrate that the algorithm recovers the object in a noiseless simulation case. While the algorithm assumes a specific blurring model, the results suggest that the algorithm may provide high reconstruction accuracy even when the object does not match the assumed blurring model. Generally, increased values of the blurring parameter and total variation weighting parameters reduced streaking artifacts, while decreasing spatial resolution. The proposed algorithm demonstrated higher correlation with respect to the true phantom compared to maximum-likelihood expectation maximization (MLEM) reconstructions. Images reconstructed with the proposed algorithm demonstrated reduced streaking artifacts when reconstructing from few views compared to MLEM. The proposed algorithm introduced patchy artifacts in some reconstructed images, depending on the noise level and the selected algorithm parameters. Overall, the results demonstrate preliminary feasibility of a sparsity-exploiting reconstruction algorithm which may be beneficial for few-view SPECT.
NASA Astrophysics Data System (ADS)
Guo, Yongfeng; Shen, Yajun; Tan, Jianguo
2016-09-01
The phenomenon of stochastic resonance (SR) in a piecewise nonlinear model driven by a periodic signal and correlated noises for the cases of a multiplicative non-Gaussian noise and an additive Gaussian white noise is investigated. Applying the path integral approach, the unified colored noise approximation and the two-state model theory, the analytical expression of the signal-to-noise ratio (SNR) is derived. It is found that conventional stochastic resonance exists in this system. From numerical computations we obtain that: (i) As a function of the non-Gaussian noise intensity, the SNR is increased when the non-Gaussian noise deviation parameter q is increased. (ii) As a function of the Gaussian noise intensity, the SNR is decreased when q is increased. This demonstrates that the effect of the non-Gaussian noise on SNR is different from that of the Gaussian noise in this system. Moreover, we further discuss the effect of the correlation time of the non-Gaussian noise, cross-correlation strength, the amplitude and frequency of the periodic signal on SR.
NASA Astrophysics Data System (ADS)
Wu, Tiantian; Yang, Xiao-Song
2016-05-01
Based on mathematical analysis, this paper provides a methodology to ensure the existence of heteroclinic cycles in a class of four-dimensional piecewise affine systems. In addition, examples are provided to illustrate the effectiveness of the method.
Probabilistic approach for predicting periodic orbits in piecewise affine differential models.
Chaves, Madalena; Farcot, Etienne; Gouzé, Jean-Luc
2013-06-01
Piecewise affine models provide a qualitative description of the dynamics of a system, and are often used to study genetic regulatory networks. The state space of a piecewise affine system is partitioned into hyperrectangles, which can be represented as nodes in a directed graph, so that the system's trajectories follow a path in a transition graph. This paper proposes and compares two definitions of probability of transition between two nodes A and B of the graph, based on the volume of the initial conditions on the hyperrectangle A whose trajectories cross to B. The parameters of the system can thus be compared to the observed transitions between two hyperrectangles. This property may become useful to identify sets of parameters for which the system yields a desired periodic orbit with a high probability, or to predict the most likely periodic orbit given a set of parameters, as illustrated by a gene regulatory system composed of two intertwined negative loops.
NASA Astrophysics Data System (ADS)
Akhmet, M. U.
2007-12-01
In this paper we introduce a new type of differential equations with piecewise constant argument (EPCAG), more general than EPCA [K.L. Cooke, J. Wiener, Retarded differential equations with piecewise constant delays, J. Math. Anal. Appl. 99 (1984) 265-297; J. Wiener, Generalized Solutions of Functional Differential Equations, World Scientific, Singapore, 1993]. The Reduction Principle [V.A. Pliss, The reduction principle in the theory of the stability of motion, Izv. Akad. Nauk SSSR Ser. Mat. 27 (1964) 1297-1324 (in Russian); V.A. Pliss, Integral Sets of Periodic Systems of Differential Equations, Nauka, Moskow, 1977 (in Russian)] is proved for EPCAG. The structure of the set of solutions is specified. We establish also the existence of global integral manifolds of quasilinear EPCAG in the so-called critical case and investigate the stability of the zero solution.
Li, Bing Nan; Chui, Chee Kong; Ong, Sim Heng; Numano, Tomokazu; Washio, Toshikatsu; Homma, Kazuhiro; Chang, Stephen; Venkatesh, Sudhakar; Kobayashi, Etsuko
2012-04-01
Magnetic resonance elastography (MRE) is designed for imaging the mechanical properties of soft tissues. However, the interpretation of shear modulus distribution is often confusing and cumbersome. For reliable evaluation, a common practice is to specify the regions of interest and consider regional elasticity. Such an experience-dependent protocol is susceptible to intrapersonal and interpersonal variability. In this study we propose to remodel shear modulus distribution with piecewise constant level sets by referring to the corresponding magnitude image. Optimal segmentation and registration are achieved by a new hybrid level set model comprised of alternating global and local region competitions. Experimental results on the simulated MRE data sets show that the mean error of elasticity reconstruction is 11.33% for local frequency estimation and 18.87% for algebraic inversion of differential equation. Piecewise constant level set modeling is effective to improve the quality of shear modulus distribution, and facilitates MRE analysis and interpretation.
On piecewise interpolation techniques for estimating solar radiation missing values in Kedah
Saaban, Azizan; Zainudin, Lutfi; Bakar, Mohd Nazari Abu
2014-12-04
This paper discusses the use of piecewise interpolation method based on cubic Ball and Bézier curves representation to estimate the missing value of solar radiation in Kedah. An hourly solar radiation dataset is collected at Alor Setar Meteorology Station that is taken from Malaysian Meteorology Deparment. The piecewise cubic Ball and Bézier functions that interpolate the data points are defined on each hourly intervals of solar radiation measurement and is obtained by prescribing first order derivatives at the starts and ends of the intervals. We compare the performance of our proposed method with existing methods using Root Mean Squared Error (RMSE) and Coefficient of Detemination (CoD) which is based on missing values simulation datasets. The results show that our method is outperformed the other previous methods.
Low-complexity piecewise-affine virtual sensors: theory and design
NASA Astrophysics Data System (ADS)
Rubagotti, Matteo; Poggi, Tomaso; Oliveri, Alberto; Pascucci, Carlo Alberto; Bemporad, Alberto; Storace, Marco
2014-03-01
This paper is focused on the theoretical development and the hardware implementation of low-complexity piecewise-affine direct virtual sensors for the estimation of unmeasured variables of interest of nonlinear systems. The direct virtual sensor is designed directly from measured inputs and outputs of the system and does not require a dynamical model. The proposed approach allows one to design estimators which mitigate the effect of the so-called 'curse of dimensionality' of simplicial piecewise-affine functions, and can be therefore applied to relatively high-order systems, enjoying convergence and optimality properties. An automatic toolchain is also presented to generate the VHDL code describing the digital circuit implementing the virtual sensor, starting from the set of measured input and output data. The proposed methodology is applied to generate an FPGA implementation of the virtual sensor for the estimation of vehicle lateral velocity, using a hardware-in-the-loop setting.
Normal form and limit cycle bifurcation of piecewise smooth differential systems with a center
NASA Astrophysics Data System (ADS)
Wei, Lijun; Zhang, Xiang
2016-07-01
In this paper we prove that any Σ-center (either nondegenerate or degenerate) of a planar piecewise Cr smooth vector field Z is topologically equivalent to that of Z0: (x ˙ , y ˙) = (- 1 , 2 x) for y ≥ 0, (x ˙ , y ˙) = (1 , 2 x) for y ≤ 0, and that the homeomorphism between Z and Z0 is Cr smoothness when restricted to each side of the switching line except at the center p. We illustrate by examples that there are degenerate Σ-centers whose flows are conjugate to that of Z0, and also there exist nondegenerate Σ-centers whose flows cannot be conjugate to that of Z0. Finally applying the normal form Z0 together with the piecewise smooth equivalence, we study the number of limit cycles which can be bifurcated from the Σ-center of Z.
A piecewise modeling approach for climate sensitivity studies: Tests with a shallow-water model
NASA Astrophysics Data System (ADS)
Shao, Aimei; Qiu, Chongjian; Niu, Guo-Yue
2015-10-01
In model-based climate sensitivity studies, model errors may grow during continuous long-term integrations in both the "reference" and "perturbed" states and hence the climate sensitivity (defined as the difference between the two states). To reduce the errors, we propose a piecewise modeling approach that splits the continuous long-term simulation into subintervals of sequential short-term simulations, and updates the modeled states through re-initialization at the end of each subinterval. In the re-initialization processes, this approach updates the reference state with analysis data and updates the perturbed states with the sum of analysis data and the difference between the perturbed and the reference states, thereby improving the credibility of the modeled climate sensitivity. We conducted a series of experiments with a shallow-water model to evaluate the advantages of the piecewise approach over the conventional continuous modeling approach. We then investigated the impacts of analysis data error and subinterval length used in the piecewise approach on the simulations of the reference and perturbed states as well as the resulting climate sensitivity. The experiments show that the piecewise approach reduces the errors produced by the conventional continuous modeling approach, more effectively when the analysis data error becomes smaller and the subinterval length is shorter. In addition, we employed a nudging assimilation technique to solve possible spin-up problems caused by re-initializations by using analysis data that contain inconsistent errors between mass and velocity. The nudging technique can effectively diminish the spin-up problem, resulting in a higher modeling skill.
Zeng, Gengsheng L; Gullberg, Grant T
2013-03-01
It has been widely believed that the non-negativity and the piecewise-constant constraints guarantee a unique solution to an interior tomographic problem. This letter points out that if the number of views is finite, the non-negativity and the piecewise-constant constraints do not guarantee a unique solution to an interior tomographic problem and the reconstruction could be biased by a constant polygon.
Dolgin, Madlena; Einziger, Pinchas D
2010-05-01
Image reconstruction in electrical impedance tomography is, generally, an ill-posed nonlinear inverse problem. Regularization methods are widely used to ensure a stable solution. Herein, we present a case study, which uses a novel electrical impedance tomography method for reconstruction of layered biological tissues with piecewise continuous plane-stratified profiles. The algorithm implements the recently proposed reconstruction scheme for piecewise constant conductivity profiles, utilizing Legendre expansion in conjunction with improved Prony method. It is shown that the proposed algorithm is capable of successfully reconstructing piecewise continuous conductivity profiles with moderate slop. This reconstruction procedure, which calculates both the locations and the conductivities, repetitively provides inhomogeneous depth discretization, i.e., the depths grid is not equispaced. Incorporation of this specific inhomogeneous grid in the widely used mean least square reconstruction procedure results in a stable and accurate reconstruction, whereas, the commonly selected equispaced depth grid leads to unstable reconstruction. This observation establishes the main result of our investigation, highlighting the impact of physical phenomenon (the image series expansion) on electrical impedance tomography, leading to a physically motivated stabilization of the inverse problem, i.e., an inhomogeneous depth discretization renders an inherent regularization of the mean least square algorithm. The effectiveness and the significance of inhomogeneous discretization in electrical impedance tomography reconstruction procedure is further demonstrated and verified via numerical simulations.
NASA Technical Reports Server (NTRS)
2006-01-01
[figure removed for brevity, see original site] Context image for PIA03667 Linear Clouds
These clouds are located near the edge of the south polar region. The cloud tops are the puffy white features in the bottom half of the image.
Image information: VIS instrument. Latitude -80.1N, Longitude 52.1E. 17 meter/pixel resolution.
Note: this THEMIS visual image has not been radiometrically nor geometrically calibrated for this preliminary release. An empirical correction has been performed to remove instrumental effects. A linear shift has been applied in the cross-track and down-track direction to approximate spacecraft and planetary motion. Fully calibrated and geometrically projected images will be released through the Planetary Data System in accordance with Project policies at a later time.
NASA's Jet Propulsion Laboratory manages the 2001 Mars Odyssey mission for NASA's Office of Space Science, Washington, D.C. The Thermal Emission Imaging System (THEMIS) was developed by Arizona State University, Tempe, in collaboration with Raytheon Santa Barbara Remote Sensing. The THEMIS investigation is led by Dr. Philip Christensen at Arizona State University. Lockheed Martin Astronautics, Denver, is the prime contractor for the Odyssey project, and developed and built the orbiter. Mission operations are conducted jointly from Lockheed Martin and from JPL, a division of the California Institute of Technology in Pasadena.
Rasin, A.
1994-04-01
We discuss the idea of approximate flavor symmetries. Relations between approximate flavor symmetries and natural flavor conservation and democracy models is explored. Implications for neutrino physics are also discussed.
Identification of pulmonary fissures using a piecewise plane fitting algorithm.
Gu, Suicheng; Wilson, David; Wang, Zhimin; Bigbee, William L; Siegfried, Jill; Gur, David; Pu, Jiantao
2012-10-01
We describe an automated computerized scheme to identify pulmonary fissures depicted in chest computed tomography (CT) examinations from a novel perspective. Whereas CT images can be regarded as a cloud of points, the underlying idea is to search for surface-like structures in the three-dimensional (3D) Euclidean space by using an efficient plane fitting algorithm. The proposed plane fitting operation is performed in a number of small spherical lung sub-volumes to detect small planar patches. Using a simple clustering criterion based on their spatial coherence and surface area, the identified planar patches, assumed to represent fissures, are classified into different types of fissures, namely left oblique, right oblique and right horizontal fissures. The performance of the developed scheme was assessed by comparing with a manually created "reference standard" and the results obtained by a previously developed approach on a dataset of 30 lung CT examinations. The experiments show that the average discrepancy is around 1.0mm in comparison with the reference standard, while the corresponding maximum discrepancy is 20.5mm. In addition, 94% of the fissure voxels identified by the computerized scheme are within 3mm of the fissures in the reference standard. As compared to a previously developed approach, we also found that the newly developed scheme had a smaller discrepancy with the standard reference. In efficiency, it takes approximately 8 min to identify the fissures in a chest CT examination on a typical PC. The developed scheme demonstrates a reasonable performance in terms of accuracy, robustness, and computational efficiency.
NASA Technical Reports Server (NTRS)
Gottlieb, David; Shu, Chi-Wang
1994-01-01
We continue our investigation of overcoming Gibbs phenomenon, i.e., to obtain exponential accuracy at all points (including at the discontinuities themselves), from the knowledge of a spectral partial sum of a discontinuous but piecewise analytic function. We show that if we are given the first N Gegenbauer expansion coefficients, based on the Gegenbauer polynomials C(sub k)(sup mu)(x) with the weight function (1 - x(exp 2))(exp mu - 1/2) for any constant mu is greater than or equal to 0, of an L(sub 1) function f(x), we can construct an exponentially convergent approximation to the point values of f(x) in any subinterval in which the function is analytic. The proof covers the cases of Chebyshev or Legendre partial sums, which are most common in applications.
NASA Astrophysics Data System (ADS)
Niiniluoto, Ilkka
2014-03-01
Approximation of laws is an important theme in the philosophy of science. If we can make sense of the idea that two scientific laws are "close" to each other, then we can also analyze such methodological notions as approximate explanation of laws, approximate reduction of theories, approximate empirical success of theories, and approximate truth of laws. Proposals for measuring the distance between quantitative scientific laws were given in Niiniluoto (1982, 1987). In this paper, these definitions are reconsidered as a response to the interesting critical remarks by Liu (1999).
Wavelet Approximation in Data Assimilation
NASA Technical Reports Server (NTRS)
Tangborn, Andrew; Atlas, Robert (Technical Monitor)
2002-01-01
Estimation of the state of the atmosphere with the Kalman filter remains a distant goal because of high computational cost of evolving the error covariance for both linear and nonlinear systems. Wavelet approximation is presented here as a possible solution that efficiently compresses both global and local covariance information. We demonstrate the compression characteristics on the the error correlation field from a global two-dimensional chemical constituent assimilation, and implement an adaptive wavelet approximation scheme on the assimilation of the one-dimensional Burger's equation. In the former problem, we show that 99%, of the error correlation can be represented by just 3% of the wavelet coefficients, with good representation of localized features. In the Burger's equation assimilation, the discrete linearized equations (tangent linear model) and analysis covariance are projected onto a wavelet basis and truncated to just 6%, of the coefficients. A nearly optimal forecast is achieved and we show that errors due to truncation of the dynamics are no greater than the errors due to covariance truncation.
Linearization of the T-matrix solution for quasi-homogeneous scatterers.
Valagiannopoulos, Constantine A; Tsitsas, Nikolaos L
2009-04-01
Interesting applications arising in optical and chemical engineering, environmental science, and biology motivate the investigation of electromagnetic wave scattering problems by radially inhomogeneous obstacles. Our main purpose is the investigation of plane-wave scattering by quasi-homogeneous obstacles, that is, obstacles with wavenumbers not exhibiting large variations from a specific average value k . The analysis is presented separately for a slab (1D), a cylindrical (2D), and a spherical (3D) scatterer. First, we consider a step approximation of the wavenumber and express the field coefficients by applying a T-matrix method for the corresponding piecewise homogeneous scatterer. Then, by performing an appropriate Taylor expansion, we express the field coefficients as linear combinations of the distances of the wavenumber samples from k . The combinations' weights are called layer-factors, because each one describes the contribution of a specific layer in the scattered field. Furthermore, it is shown that the far-field pattern of the quasi-homogeneous scatterer is decomposed into that of the respective homogeneous scatterer plus the perturbation far-field pattern, depending on the wavenumber's deviations from k . Several numerical results are presented concerning the comparison of the far-field patterns computed by the proposed technique and the T-matrix method, as well as investigations of the perturbation far-field pattern and the layer-factors. Linear, sinusoidal, Lunenburg type, and triangular wavenumber profiles are analyzed.
Sparse pseudospectral approximation method
NASA Astrophysics Data System (ADS)
Constantine, Paul G.; Eldred, Michael S.; Phipps, Eric T.
2012-07-01
Multivariate global polynomial approximations - such as polynomial chaos or stochastic collocation methods - are now in widespread use for sensitivity analysis and uncertainty quantification. The pseudospectral variety of these methods uses a numerical integration rule to approximate the Fourier-type coefficients of a truncated expansion in orthogonal polynomials. For problems in more than two or three dimensions, a sparse grid numerical integration rule offers accuracy with a smaller node set compared to tensor product approximation. However, when using a sparse rule to approximately integrate these coefficients, one often finds unacceptable errors in the coefficients associated with higher degree polynomials. By reexamining Smolyak's algorithm and exploiting the connections between interpolation and projection in tensor product spaces, we construct a sparse pseudospectral approximation method that accurately reproduces the coefficients of basis functions that naturally correspond to the sparse grid integration rule. The compelling numerical results show that this is the proper way to use sparse grid integration rules for pseudospectral approximation.
Approximations for photoelectron scattering
NASA Astrophysics Data System (ADS)
Fritzsche, V.
1989-04-01
The errors of several approximations in the theoretical approach of photoelectron scattering are systematically studied, in tungsten, for electron energies ranging from 10 to 1000 eV. The large inaccuracies of the plane-wave approximation (PWA) are substantially reduced by means of effective scattering amplitudes in the modified small-scattering-centre approximation (MSSCA). The reduced angular momentum expansion (RAME) is so accurate that it allows reliable calculations of multiple-scattering contributions for all the energies considered.
NASA Astrophysics Data System (ADS)
Liu, Fang; Lin, Lin; Vigil-Fowler, Derek; Lischner, Johannes; Kemper, Alexander F.; Sharifzadeh, Sahar; da Jornada, Felipe H.; Deslippe, Jack; Yang, Chao; Neaton, Jeffrey B.; Louie, Steven G.
2015-04-01
We present a numerical integration scheme for evaluating the convolution of a Green's function with a screened Coulomb potential on the real axis in the GW approximation of the self energy. Our scheme takes the zero broadening limit in Green's function first, replaces the numerator of the integrand with a piecewise polynomial approximation, and performs principal value integration on subintervals analytically. We give the error bound of our numerical integration scheme and show by numerical examples that it is more reliable and accurate than the standard quadrature rules such as the composite trapezoidal rule. We also discuss the benefit of using different self energy expressions to perform the numerical convolution at different frequencies.
Generalized Gradient Approximation Made Simple
Perdew, J.P.; Burke, K.; Ernzerhof, M.
1996-10-01
Generalized gradient approximations (GGA{close_quote}s) for the exchange-correlation energy improve upon the local spin density (LSD) description of atoms, molecules, and solids. We present a simple derivation of a simple GGA, in which all parameters (other than those in LSD) are fundamental constants. Only general features of the detailed construction underlying the Perdew-Wang 1991 (PW91) GGA are invoked. Improvements over PW91 include an accurate description of the linear response of the uniform electron gas, correct behavior under uniform scaling, and a smoother potential. {copyright} {ital 1996 The American Physical Society.}
NASA Technical Reports Server (NTRS)
Voorhies, Coerte V.
1993-01-01
The problem of estimating a steady fluid velocity field near the top of Earth's core which induces the secular variation (SV) indicated by models of the observed geomagnetic field is examined in the source-free mantle/frozen-flux core (SFI/VFFC) approximation. This inverse problem is non-linear because solutions of the forward problem are deterministically chaotic. The SFM/FFC approximation is inexact, and neither the models nor the observations they represent are either complete or perfect. A method is developed for solving the non-linear inverse motional induction problem posed by the hypothesis of (piecewise, statistically) steady core surface flow and the supposition of a complete initial geomagnetic condition. The method features iterative solution of the weighted, linearized least-squares problem and admits optional biases favoring surficially geostrophic flow and/or spatially simple flow. Two types of weights are advanced radial field weights for fitting the evolution of the broad-scale portion of the radial field component near Earth's surface implied by the models, and generalized weights for fitting the evolution of the broad-scale portion of the scalar potential specified by the models.
NASA Technical Reports Server (NTRS)
Dutta, Soumitra
1988-01-01
A model for approximate spatial reasoning using fuzzy logic to represent the uncertainty in the environment is presented. Algorithms are developed which can be used to reason about spatial information expressed in the form of approximate linguistic descriptions similar to the kind of spatial information processed by humans. Particular attention is given to static spatial reasoning.
Regularity of absolutely continuous invariant measures for piecewise expanding unimodal maps
NASA Astrophysics Data System (ADS)
Contreras, Fabián; Dolgopyat, Dmitry
2016-09-01
Let f:[0,1]\\to [0,1] be a piecewise expanding unimodal map of class C k+1, with k≥slant 1 , and μ =ρ \\text{d}x the (unique) SRB measure associated to it. We study the regularity of ρ. In particular, points N where ρ is not differentiable has zero Hausdorff dimension, but is uncountable if the critical orbit of f is dense. This improves on a work of Szewc (1984). We also obtain results about higher orders of differentiability of ρ in the sense of Whitney.
ON THE PIECEWISE PARABOLIC METHOD FOR COMPRESSIBLE FLOW WITH STELLAR EQUATIONS OF STATE
Zingale, Michael; Katz, Max P.
2015-02-01
The piecewise parabolic method and related schemes are widely used to model stellar flows. Several different methods for extending the validity of these methods to a general equation of state (EOS) have been proposed over time, but direct comparisons among one-another and exact solutions with stellar EOSs are not widely available. We introduce some simple test problems with exact solutions run with a popular stellar EOS and test how two existing codes with different approaches to incorporating general gases perform. The source code for generating the exact solutions is made available.
Steering by transient destabilization in piecewise-holonomic models of legged locomotion
NASA Astrophysics Data System (ADS)
Proctor, J.; Holmes, P.
2008-08-01
We study turning strategies in low-dimensional models of legged locomotion in the horizontal plane. Since the constraints due to foot placement switch from stride to stride, these models are piecewise-holonomic, and this can cause stride-to-stride changes in angular momentum and in the ratio of rotational to translational kinetic energy. Using phase plane analyses and parameter studies based on experimental observations of insects, we investigate how these changes can be harnessed to produce rapid turns, and compare the results with dynamical cockroach data. Qualitative similarities between the model and insect data suggest general strategies that could be implemented in legged robots.
Sensitivity analysis of aeroelastic response of a wing using piecewise pressure representation
NASA Astrophysics Data System (ADS)
Eldred, Lloyd B.; Kapania, Rakesh K.; Barthelemy, Jean-Francois M.
1993-04-01
A sensitivity analysis scheme of the static aeroelastic response of a wing is developed, by incorporating a piecewise panel-based pressure representation into an existing wing aeroelastic model to improve the model's fidelity, including the sensitivity of the wing static aeroelastic response with respect to various shape parameters. The new formulation is quite general and accepts any aerodynamics and structural analysis capability. A program is developed which combines the local sensitivities, such as the sensitivity of the stiffness matrix or the aerodynamic kernel matrix, into global sensitivity derivatives.
Piecewise oblique boundary treatment for the elastic-plastic wave equation on a cartesian grid
NASA Astrophysics Data System (ADS)
Giese, Guido
2009-11-01
Numerical schemes for hyperbolic conservation laws in 2-D on a Cartesian grid usually have the advantage of being easy to implement and showing good computational performances, without allowing the simulation of “real-world” problems on arbitrarily shaped domains. In this paper a numerical treatment of boundary conditions for the elastic-plastic wave equation is developed, which allows the simulation of problems on an arbitrarily shaped physical domain surrounded by a piece-wise smooth boundary curve, but using a PDE solver on a rectangular Cartesian grid with the afore-mentioned advantages.
Arithmetic coding as a non-linear dynamical system
NASA Astrophysics Data System (ADS)
Nagaraj, Nithin; Vaidya, Prabhakar G.; Bhat, Kishor G.
2009-04-01
In order to perform source coding (data compression), we treat messages emitted by independent and identically distributed sources as imprecise measurements (symbolic sequence) of a chaotic, ergodic, Lebesgue measure preserving, non-linear dynamical system known as Generalized Luröth Series (GLS). GLS achieves Shannon's entropy bound and turns out to be a generalization of arithmetic coding, a popular source coding algorithm, used in international compression standards such as JPEG2000 and H.264. We further generalize GLS to piecewise non-linear maps (Skewed-nGLS). We motivate the use of Skewed-nGLS as a framework for joint source coding and encryption.
Approximate kernel competitive learning.
Wu, Jian-Sheng; Zheng, Wei-Shi; Lai, Jian-Huang
2015-03-01
Kernel competitive learning has been successfully used to achieve robust clustering. However, kernel competitive learning (KCL) is not scalable for large scale data processing, because (1) it has to calculate and store the full kernel matrix that is too large to be calculated and kept in the memory and (2) it cannot be computed in parallel. In this paper we develop a framework of approximate kernel competitive learning for processing large scale dataset. The proposed framework consists of two parts. First, it derives an approximate kernel competitive learning (AKCL), which learns kernel competitive learning in a subspace via sampling. We provide solid theoretical analysis on why the proposed approximation modelling would work for kernel competitive learning, and furthermore, we show that the computational complexity of AKCL is largely reduced. Second, we propose a pseudo-parallelled approximate kernel competitive learning (PAKCL) based on a set-based kernel competitive learning strategy, which overcomes the obstacle of using parallel programming in kernel competitive learning and significantly accelerates the approximate kernel competitive learning for large scale clustering. The empirical evaluation on publicly available datasets shows that the proposed AKCL and PAKCL can perform comparably as KCL, with a large reduction on computational cost. Also, the proposed methods achieve more effective clustering performance in terms of clustering precision against related approximate clustering approaches.
Approximate kernel competitive learning.
Wu, Jian-Sheng; Zheng, Wei-Shi; Lai, Jian-Huang
2015-03-01
Kernel competitive learning has been successfully used to achieve robust clustering. However, kernel competitive learning (KCL) is not scalable for large scale data processing, because (1) it has to calculate and store the full kernel matrix that is too large to be calculated and kept in the memory and (2) it cannot be computed in parallel. In this paper we develop a framework of approximate kernel competitive learning for processing large scale dataset. The proposed framework consists of two parts. First, it derives an approximate kernel competitive learning (AKCL), which learns kernel competitive learning in a subspace via sampling. We provide solid theoretical analysis on why the proposed approximation modelling would work for kernel competitive learning, and furthermore, we show that the computational complexity of AKCL is largely reduced. Second, we propose a pseudo-parallelled approximate kernel competitive learning (PAKCL) based on a set-based kernel competitive learning strategy, which overcomes the obstacle of using parallel programming in kernel competitive learning and significantly accelerates the approximate kernel competitive learning for large scale clustering. The empirical evaluation on publicly available datasets shows that the proposed AKCL and PAKCL can perform comparably as KCL, with a large reduction on computational cost. Also, the proposed methods achieve more effective clustering performance in terms of clustering precision against related approximate clustering approaches. PMID:25528318
Boundary-equilibrium bifurcations in piecewise-smooth slow-fast systems.
Kowalczyk, P; Glendinning, P
2011-06-01
In this paper we study the qualitative dynamics of piecewise-smooth slow-fast systems (singularly perturbed systems) which are everywhere continuous. We consider phase space topology of systems with one-dimensional slow dynamics and one-dimensional fast dynamics. The slow manifold of the reduced system is formed by a piecewise-continuous curve, and the differentiability is lost across the switching surface. In the full system the slow manifold is no longer continuous, and there is an O(ɛ) discontinuity across the switching manifold, but the discontinuity cannot qualitatively alter system dynamics. Revealed phase space topology is used to unfold qualitative dynamics of planar slow-fast systems with an equilibrium point on the switching surface. In this case the local dynamics corresponds to so-called boundary-equilibrium bifurcations, and four qualitative phase portraits are uncovered. Our results are then used to investigate the dynamics of a box model of a thermohaline circulation, and the presence of a boundary-equilibrium bifurcation of a fold type is shown.
Three-Dimensional Piecewise-Continuous Class-Shape Transformation of Wings
NASA Technical Reports Server (NTRS)
Olson, Erik D.
2015-01-01
Class-Shape Transformation (CST) is a popular method for creating analytical representations of the surface coordinates of various components of aerospace vehicles. A wide variety of two- and three-dimensional shapes can be represented analytically using only a modest number of parameters, and the surface representation is smooth and continuous to as fine a degree as desired. This paper expands upon the original two-dimensional representation of airfoils to develop a generalized three-dimensional CST parametrization scheme that is suitable for a wider range of aircraft wings than previous formulations, including wings with significant non-planar shapes such as blended winglets and box wings. The method uses individual functions for the spanwise variation of airfoil shape, chord, thickness, twist, and reference axis coordinates to build up the complete wing shape. An alternative formulation parameterizes the slopes of the reference axis coordinates in order to relate the spanwise variation to the tangents of the sweep and dihedral angles. Also discussed are methods for fitting existing wing surface coordinates, including the use of piecewise equations to handle discontinuities, and mathematical formulations of geometric continuity constraints. A subsonic transport wing model is used as an example problem to illustrate the application of the methodology and to quantify the effects of piecewise representation and curvature constraints.
Boundary-equilibrium bifurcations in piecewise-smooth slow-fast systems.
Kowalczyk, P; Glendinning, P
2011-06-01
In this paper we study the qualitative dynamics of piecewise-smooth slow-fast systems (singularly perturbed systems) which are everywhere continuous. We consider phase space topology of systems with one-dimensional slow dynamics and one-dimensional fast dynamics. The slow manifold of the reduced system is formed by a piecewise-continuous curve, and the differentiability is lost across the switching surface. In the full system the slow manifold is no longer continuous, and there is an O(ɛ) discontinuity across the switching manifold, but the discontinuity cannot qualitatively alter system dynamics. Revealed phase space topology is used to unfold qualitative dynamics of planar slow-fast systems with an equilibrium point on the switching surface. In this case the local dynamics corresponds to so-called boundary-equilibrium bifurcations, and four qualitative phase portraits are uncovered. Our results are then used to investigate the dynamics of a box model of a thermohaline circulation, and the presence of a boundary-equilibrium bifurcation of a fold type is shown. PMID:21721768
NASA Technical Reports Server (NTRS)
Shen, Ji Yao; Abu-Saba, Elias G.; Mcginley, William M.; Sharpe, Lonnie, Jr.; Taylor, Lawrence W., Jr.
1992-01-01
Distributed parameter modeling offers a viable alternative to the finite element approach for modeling large flexible space structures. The introduction of the transfer matrix method into the continuum modeling process provides a very useful tool to facilitate the distributed parameter model applied to some more complex configurations. A uniform Timoshenko beam model for the estimation of the dynamic properties of beam-like structures has given comparable results. But many aeronautical and aerospace structures are comprised of non-uniform sections or sectional properties, such as aircraft wings and satellite antennas. This paper proposes a piecewise continuous Timoshenko beam model which is used for the dynamic analysis of tapered beam-like structures. A tapered beam is divided into several segments of uniform beam elements. Instead of arbitrarily assumed shape functions used in finite element analysis, the closed-form solution of the Timoshenko beam equation is used. Application of the transfer matrix method relates all the elements as a whole. By corresponding boundary conditions and compatible conditions a characteristic equation for the global tapered beam has been developed, from which natural frequencies can be derived. A computer simulation is shown in this paper, and compared with the results obtained from the finite element analysis. While piecewise continuous Timoshenko beam model decreases the number of elements significantly; comparable results to the finite element method are obtained.
Group lassoing change-points in piecewise-constant AR processes
NASA Astrophysics Data System (ADS)
Angelosante, Daniele; Giannakis, Georgios B.
2012-12-01
Regularizing the least-squares criterion with the total number of coefficient changes, it is possible to estimate time-varying (TV) autoregressive (AR) models with piecewise-constant coefficients. Such models emerge in various applications including speech segmentation, biomedical signal processing, and geophysics. To cope with the inherent lack of continuity and the high computational burden when dealing with high-dimensional data sets, this article introduces a convex regularization approach enabling efficient and continuous estimation of TV-AR models. To this end, the problem is cast as a sparse regression one with grouped variables, and is solved by resorting to the group least-absolute shrinkage and selection operator (Lasso). The fresh look advocated here permeates benefits from advances in variable selection and compressive sampling to signal segmentation. An efficient block-coordinate descent algorithm is developed to implement the novel segmentation method. Issues regarding regularization and uniqueness of the solution are also discussed. Finally, an alternative segmentation technique is introduced to improve the detection of change instants. Numerical tests using synthetic and real data corroborate the merits of the developed segmentation techniques in identifying piecewise-constant TV-AR models.
GRMHD Simulations of Binary Neutron Star Mergers with Piecewise Polytropic Equations of State
NASA Astrophysics Data System (ADS)
Giacomazzo, Bruno
2015-04-01
We present new results of fully general relativistic magnetohydrodynamic (GRMHD) simulations of binary neutron star (BNS) mergers performed with the Whisky code. Our new simulations consider both equal and unequal-mass systems and describe the NS matter via piecewise polytropic equations of state (EOSs). BNS mergers are powerful sources of gravitational waves (GWs) that can be detected by ground based detectors, such as advanced Virgo and LIGO, and they are also thought to be behind the central engine powering short gamma-ray bursts. In our simulations we therefore focus both on the GW emission and on the dynamics of matter and magnetic fields, both in the case a black hole is promptly formed and in the case of the formation of a long-lived magnetized NS. Since the EOS has an important role in both GW emission and matter dynamics, our simulations employ piecewise polytropic EOSs composed by seven pieces, four for the low-density regions (including the crust) and three for the core, in order to more accurately match physically motivated EOSs. Thermal effects are also included in order to more properly describe the post-merger dynamics.
NASA Astrophysics Data System (ADS)
Yuan, Rong
2005-03-01
We study the spectrum containment of almost periodic solution to neutral delay differential equations with piecewise constant argument (EPCA, for short). We find an important property, which is different from that given by Cartwright for ordinary differential equations (ODE). Some known (periodic solution) results would be expanded. As a corollary, it is shown that EPCA with periodic perturbations possess a quasi-periodic solution and no periodic solution. This new phenomenon is due to the piecewise constant argument and illustrates a crucial difference between ODE and EPCA.
Sidorin, Anatoly
2010-01-05
In linear accelerators the particles are accelerated by either electrostatic fields or oscillating Radio Frequency (RF) fields. Accordingly the linear accelerators are divided in three large groups: electrostatic, induction and RF accelerators. Overview of the different types of accelerators is given. Stability of longitudinal and transverse motion in the RF linear accelerators is briefly discussed. The methods of beam focusing in linacs are described.
Congruence Approximations for Entrophy Endowed Hyperbolic Systems
NASA Technical Reports Server (NTRS)
Barth, Timothy J.; Saini, Subhash (Technical Monitor)
1998-01-01
Building upon the standard symmetrization theory for hyperbolic systems of conservation laws, congruence properties of the symmetrized system are explored. These congruence properties suggest variants of several stabilized numerical discretization procedures for hyperbolic equations (upwind finite-volume, Galerkin least-squares, discontinuous Galerkin) that benefit computationally from congruence approximation. Specifically, it becomes straightforward to construct the spatial discretization and Jacobian linearization for these schemes (given a small amount of derivative information) for possible use in Newton's method, discrete optimization, homotopy algorithms, etc. Some examples will be given for the compressible Euler equations and the nonrelativistic MHD equations using linear and quadratic spatial approximation.
Approximate Solutions Of Equations Of Steady Diffusion
NASA Technical Reports Server (NTRS)
Edmonds, Larry D.
1992-01-01
Rigorous analysis yields reliable criteria for "best-fit" functions. Improved "curve-fitting" method yields approximate solutions to differential equations of steady-state diffusion. Method applies to problems in which rates of diffusion depend linearly or nonlinearly on concentrations of diffusants, approximate solutions analytic or numerical, and boundary conditions of Dirichlet type, of Neumann type, or mixture of both types. Applied to equations for diffusion of charge carriers in semiconductors in which mobilities and lifetimes of charge carriers depend on concentrations.
Inverse Modeling Via Linearized Functional Minimization
NASA Astrophysics Data System (ADS)
Barajas-Solano, D. A.; Wohlberg, B.; Vesselinov, V. V.; Tartakovsky, D. M.
2014-12-01
We present a novel parameter estimation methodology for transient models of geophysical systems with uncertain, spatially distributed, heterogeneous and piece-wise continuous parameters.The methodology employs a bayesian approach to propose an inverse modeling problem for the spatial configuration of the model parameters.The likelihood of the configuration is formulated using sparse measurements of both model parameters and transient states.We propose using total variation regularization (TV) as the prior reflecting the heterogeneous, piece-wise continuity assumption on the parameter distribution.The maximum a posteriori (MAP) estimator of the parameter configuration is then computed by minimizing the negative bayesian log-posterior using a linearized functional minimization approach. The computation of the MAP estimator is a large-dimensional nonlinear minimization problem with two sources of nonlinearity: (1) the TV operator, and (2) the nonlinear relation between states and parameters provided by the model's governing equations.We propose a a hybrid linearized functional minimization (LFM) algorithm in two stages to efficiently treat both sources of nonlinearity.The relation between states and parameters is linearized, resulting in a linear minimization sub-problem equipped with the TV operator; this sub-problem is then minimized using the Alternating Direction Method of Multipliers (ADMM). The methodology is illustrated with a transient saturated groundwater flow application in a synthetic domain, stimulated by external point-wise loadings representing aquifer pumping, together with an array of discrete measurements of hydraulic conductivity and transient measurements of hydraulic head.We show that our inversion strategy is able to recover the overall large-scale features of the parameter configuration, and that the reconstruction is improved by the addition of transient information of the state variable.
Approximations of nonlinear systems having outputs
NASA Technical Reports Server (NTRS)
Hunt, L. R.; Su, R.
1985-01-01
For a nonlinear system with output derivative x = f(x) and y = h(x), two types of linearizations about a point x(0) in state space are considered. One is the usual Taylor series approximation, and the other is defined by linearizing the appropriate Lie derivatives of the output with respect to f about x(0). The latter is called the obvservation model and appears to be quite natural for observation. It is noted that there is a coordinate system in which these two kinds of linearizations agree. In this coordinate system, a technique to construct an observer is introduced.
ERIC Educational Resources Information Center
Wolff, Hans
This paper deals with a stochastic process for the approximation of the root of a regression equation. This process was first suggested by Robbins and Monro. The main result here is a necessary and sufficient condition on the iteration coefficients for convergence of the process (convergence with probability one and convergence in the quadratic…
NASA Astrophysics Data System (ADS)
Huang, Siendong
2009-11-01
The nonlocality of quantum states on a bipartite system \\mathcal {A+B} is tested by comparing probabilistic outcomes of two local observables of different subsystems. For a fixed observable A of the subsystem \\mathcal {A,} its optimal approximate double A' of the other system \\mathcal {B} is defined such that the probabilistic outcomes of A' are almost similar to those of the fixed observable A. The case of σ-finite standard von Neumann algebras is considered and the optimal approximate double A' of an observable A is explicitly determined. The connection between optimal approximate doubles and quantum correlations is explained. Inspired by quantum states with perfect correlation, like Einstein-Podolsky-Rosen states and Bohm states, the nonlocality power of an observable A for general quantum states is defined as the similarity that the outcomes of A look like the properties of the subsystem \\mathcal {B} corresponding to A'. As an application of optimal approximate doubles, maximal Bell correlation of a pure entangled state on \\mathcal {B}(\\mathbb {C}^{2})\\otimes \\mathcal {B}(\\mathbb {C}^{2}) is found explicitly.
Approximating Integrals Using Probability
ERIC Educational Resources Information Center
Maruszewski, Richard F., Jr.; Caudle, Kyle A.
2005-01-01
As part of a discussion on Monte Carlo methods, which outlines how to use probability expectations to approximate the value of a definite integral. The purpose of this paper is to elaborate on this technique and then to show several examples using visual basic as a programming tool. It is an interesting method because it combines two branches of…
New generalized gradient approximation functionals
NASA Astrophysics Data System (ADS)
Boese, A. Daniel; Doltsinis, Nikos L.; Handy, Nicholas C.; Sprik, Michiel
2000-01-01
New generalized gradient approximation (GGA) functionals are reported, using the expansion form of A. D. Becke, J. Chem. Phys. 107, 8554 (1997), with 15 linear parameters. Our original such GGA functional, called HCTH, was determined through a least squares refinement to data of 93 systems. Here, the data are extended to 120 systems and 147 systems, introducing electron and proton affinities, and weakly bound dimers to give the new functionals HCTH/120 and HCTH/147. HCTH/120 has already been shown to give high quality predictions for weakly bound systems. The functionals are applied in a comparative study of the addition reaction of water to formaldehyde and sulfur trioxide, respectively. Furthermore, the performance of the HCTH/120 functional in Car-Parrinello molecular dynamics simulations of liquid water is encouraging.
Aggregate formation in a model fluid with microscopic piecewise-continuous competing interactions
NASA Astrophysics Data System (ADS)
Cigala, G.; Costa, D.; Bomont, J.-M.; Caccamo, C.
2015-09-01
We use the hypernetted chain theory - complemented in a few selected cases by Monte Carlo simulations to assess its accuracy - to study a model fluid composed by hard spheres surrounded by an attractive square-well, followed by a linearly decreasing repulsive ramp. The proposed competing interaction allows us to study individually the influence of all the defining parameters on the formation of aggregates out of the homogeneous fluid, as signalled by the development of a local peak at low wavevectors in the static structure factor. We document an accurate linear scaling of the temperature whereupon aggregates develop (at constant density) as a function of both the square-well width and the height of the repulsive barrier, over a wide range of such properties; a much drastic dependence is instead observed on the repulsion range, well described by an exponential increase. We show that typical correlation distances among aggregates, in early conditions for their development, fall approximately at a constant fraction of the width of the repulsive barrier. We analyse different possibilities in which attraction or repulsion prevails in the overall balance of microscopic interaction: we document, as expected, that for sufficiently weak repulsion barriers, or sufficiently large attractive wells, the structure factor shows a trend to increase in the limit of vanishing wavevectors, heralding a standard liquid-vapour phase separation at low temperatures. At the opposite end, the development of aggregates seems to be hardly favoured, for the present model, if attraction is completely neglected.
NASA Astrophysics Data System (ADS)
Yamamoto, Akira; Yokoya, Kaoru
2015-02-01
An overview of linear collider programs is given. The history and technical challenges are described and the pioneering electron-positron linear collider, the SLC, is first introduced. For future energy frontier linear collider projects, the International Linear Collider (ILC) and the Compact Linear Collider (CLIC) are introduced and their technical features are discussed. The ILC is based on superconducting RF technology and the CLIC is based on two-beam acceleration technology. The ILC collaboration completed the Technical Design Report in 2013, and has come to the stage of "Design to Reality." The CLIC collaboration published the Conceptual Design Report in 2012, and the key technology demonstration is in progress. The prospects for further advanced acceleration technology are briefly discussed for possible long-term future linear colliders.
NASA Astrophysics Data System (ADS)
Yamamoto, Akira; Yokoya, Kaoru
An overview of linear collider programs is given. The history and technical challenges are described and the pioneering electron-positron linear collider, the SLC, is first introduced. For future energy frontier linear collider projects, the International Linear Collider (ILC) and the Compact Linear Collider (CLIC) are introduced and their technical features are discussed. The ILC is based on superconducting RF technology and the CLIC is based on two-beam acceleration technology. The ILC collaboration completed the Technical Design Report in 2013, and has come to the stage of "Design to Reality." The CLIC collaboration published the Conceptual Design Report in 2012, and the key technology demonstration is in progress. The prospects for further advanced acceleration technology are briefly discussed for possible long-term future linear colliders.
ERIC Educational Resources Information Center
Jaggars, Shanna Smith; Xu, Di
2015-01-01
Policymakers have become increasingly concerned with measuring--and holding colleges accountable for--students' labor market outcomes. In this paper we introduce a piecewise growth curve approach to analyzing community college students' labor market outcomes, and we discuss how this approach differs from Mincerian and fixed-effects approaches. Our…
ERIC Educational Resources Information Center
Hindman, Annemarie H.; Cromley, Jennifer G.; Skibbe, Lori E.; Miller, Alison L.
2011-01-01
This article reviews the mechanics of conventional and piecewise growth models to demonstrate the unique affordances of each technique for examining the nature and predictors of children's early literacy learning during the transition from preschool through first grade. Using the nationally representative Family and Child Experiences Survey…
Abgrall, Rémi; Congedo, Pietro Marco
2013-02-15
This paper deals with the formulation of a semi-intrusive (SI) method allowing the computation of statistics of linear and non linear PDEs solutions. This method shows to be very efficient to deal with probability density function of whatsoever form, long-term integration and discontinuities in stochastic space. Given a stochastic PDE where randomness is defined on Ω, starting from (i) a description of the solution in term of a space variables, (ii) a numerical scheme defined for any event ω∈Ω and (iii) a (family) of random variables that may be correlated, the solution is numerically described by its conditional expectancies of point values or cell averages and its evaluation constructed from the deterministic scheme. One of the tools is a tessellation of the random space as in finite volume methods for the space variables. Then, using these conditional expectancies and the geometrical description of the tessellation, a piecewise polynomial approximation in the random variables is computed using a reconstruction method that is standard for high order finite volume space, except that the measure is no longer the standard Lebesgue measure but the probability measure. This reconstruction is then used to formulate a scheme on the numerical approximation of the solution from the deterministic scheme. This new approach is said semi-intrusive because it requires only a limited amount of modification in a deterministic solver to quantify uncertainty on the state when the solver includes uncertain variables. The effectiveness of this method is illustrated for a modified version of Kraichnan–Orszag three-mode problem where a discontinuous pdf is associated to the stochastic variable, and for a nozzle flow with shocks. The results have been analyzed in terms of accuracy and probability measure flexibility. Finally, the importance of the probabilistic reconstruction in the stochastic space is shown up on an example where the exact solution is computable, the viscous
A Dynamical Analysis of a Piecewise Smooth Pest Control SI Model
NASA Astrophysics Data System (ADS)
Liu, Bing; Liu, Wanbo; Tao, Fennmei; Kang, Baolin; Cong, Jiguang
In this paper, we propose a piecewise smooth SI pest control system to model the process of spraying pesticides and releasing infectious pests. We assume that the pest population consists of susceptible pests and infectious pests, and that the disease spreads horizontally between pests. We take the susceptible pest as the control index on whether to implement chemical control and biological control strategies. Based on the theory of Filippov system, the sliding-mode domain and conditions for the existence of real equilibria, virtual equilibria, pseudo-equilibrium and boundary equilibria are given. Further, we show the global stability of real equilibria (or boundary equilibria) and pseudo-equilibrium. Our results can provide theoretical guidance for the problem of pest control.
Piecewise Smooth Dynamical Systems Theory: The Case of the Missing Boundary Equilibrium Bifurcations
NASA Astrophysics Data System (ADS)
Hogan, S. J.; Homer, M. E.; Jeffrey, M. R.; Szalai, R.
2016-10-01
We present two codimension-one bifurcations that occur when an equilibrium collides with a discontinuity in a piecewise smooth dynamical system. These simple cases appear to have escaped recent classifications. We present them here to highlight some of the powerful results from Filippov's book Differential Equations with Discontinuous Righthand Sides (Kluwer, 1988). Filippov classified the so-called boundary equilibrium collisions without providing their unfolding. We show the complete unfolding here, for the first time, in the particularly interesting case of a node changing its stability as it collides with a discontinuity. We provide a prototypical model that can be used to generate all codimension-one boundary equilibrium collisions, and summarize the elements of Filippov's work that are important in achieving a full classification.
A New Wavelength Calibration Method for LAMOST Based on Piecewise Fitting
NASA Astrophysics Data System (ADS)
Ye, Gen-hong; Ye, Zhong-fu; Zhu, Jia
2014-04-01
The traditional methods of wavelength calibration for the LAM-OST (Large Area Multi-Object Fiber Spectroscopic Telescope) usually use a fifth-order polynomial to perform fitting and calibration in a broad wavelength range. Obviously, it is unable to reflect very well the local dispersion relations by using only one polynomial to fit the whole waveband. In order to reflect accurately the dispersion characteristics of local wavebands, a new wavelength calibration method based on piecewise fitting is proposed. In this method, the entire wavelength range is divided into several sub-bands according to certain principles, and a proper polynomial is used to perform fitting and calibration for each sub-band separately. Compared with the traditional methods of wave-length calibration, the experimental results show that this new method can get a more precise description of the dispersion characteristics of local wavebands. Therefore, the accuracy of wavelength calibration in the whole waveband will be further improved.
Abdelhedi, Abdessamad; Boutat, Driss; Sbita, Lassaad; Tami, Ramdane; Liu, Da-Yan
2016-01-01
Susceptible Exposed Infectious and Recovered epidemic model endowed with a treatment function (SEIR-T model) is a well-known model used to reproduce the behavior of an epidemic, where the susceptible population and the exposed population need to be estimated to predict and control the propagation of a contagious disease. This paper focuses on the nonlinear observer design for a class of nonlinear piecewise systems including SEIR-T models. For this purpose, two changes of coordinates are provided to transform the considered systems into an extended nonlinear observer normal form, on which a high gain observer can be applied. Then, the proposed method is applied to a SEIR-T model. Finally, simulation results are given to show its efficiency. PMID:26606994
Heteroclinic bifurcation in a class of planar piecewise smooth systems with multiple zones
NASA Astrophysics Data System (ADS)
Shen, Jun; Du, Zhengdong
2016-06-01
We discuss heteroclinic bifurcation in a class of periodically excited planar piecewise smooth systems with discontinuities on finitely many smooth curves intersecting at the origin. Assume that the unperturbed system has a hyperbolic saddle in each subregion, and those saddles are connected by a heteroclinic cycle that crosses every switching curve transversally exactly once. We present a method of Melnikov type to derive sufficient conditions under which the perturbed stable and unstable manifolds intersect transversally. Such transversal intersections imply that the corresponding Poincaré map has a transverse heteroclinic cycle. As applications, we present examples with 2 and 4 switching curves respectively. Our numerical simulations suggest that such transversal intersections result in the appearance of chaotic motions in those example systems.
Zhang, Li; Li, Liang; Shen, Le; Xing, Yuxiang
2013-01-01
Dual energy CT has the ability to give more information about the test object by reconstructing the attenuation factors under different energies. These images under different energies share identical structures but different attenuation factors. By referring to the fully sampled low-energy image, we show that it is possible to greatly reduce the sampling rate of the high-energy image in order to lower dose. To compensate the attenuation factor difference between the two modalities, we use piecewise polynomial fitting to fit the low-energy image to the high-energy image. During the reconstruction, the result is constrained by its distance to the fitted image, and the structural information thus can be preserved. An ASD-POCS-based optimization schedule is proposed to solve the problem, and numerical simulations are taken to verify the algorithm. PMID:24282443
New aspects of the Casimir energy theory for a piecewise uniform string
Brevik, I.; Elizalde, E. )
1994-05-15
The Casimir energy for the transverse oscillations of a piecewise uniform closed string is calculated. The string consists of two parts (I and II) each having in general different tension and mass density but adjusted in such a way that the velocity of sound always equals the velocity of light. This model was introduced by Brevik and Nielsen, and the present paper contains new developments of the theory, in particular, a very simple regularization of the energy density. Using the technique introduced by van Kampen, Nijboer, and Schram, the Casimir energy is written as a contour integral, from which the energy can be readily calculated, for arbitrary length [ital s]=[ital L][sub II]/[ital L][sub I] and tension [ital x]=[ital T][sub I]/[ital T][sub II] ratios. Also, the finite temperature version of the theory is constructed.
The stiffness variation of a micro-ring driven by a traveling piecewise-electrode.
Li, Yingjie; Yu, Tao; Hu, Yuh-Chung
2014-01-01
In the practice of electrostatically actuated micro devices; the electrostatic force is implemented by sequentially actuated piecewise-electrodes which result in a traveling distributed electrostatic force. However; such force was modeled as a traveling concentrated electrostatic force in literatures. This article; for the first time; presents an analytical study on the stiffness variation of microstructures driven by a traveling piecewise electrode. The analytical model is based on the theory of shallow shell and uniform electrical field. The traveling electrode not only applies electrostatic force on the circular-ring but also alters its dynamical characteristics via the negative electrostatic stiffness. It is known that; when a structure is subjected to a traveling constant force; its natural mode will be resonated as the traveling speed approaches certain critical speeds; and each natural mode refers to exactly one critical speed. However; for the case of a traveling electrostatic force; the number of critical speeds is more than that of the natural modes. This is due to the fact that the traveling electrostatic force makes the resonant frequencies of the forward and backward traveling waves of the circular-ring different. Furthermore; the resonance and stability can be independently controlled by the length of the traveling electrode; though the driving voltage and traveling speed of the electrostatic force alter the dynamics and stabilities of microstructures. This paper extends the fundamental insights into the electromechanical behavior of microstructures driven by electrostatic forces as well as the future development of MEMS/NEMS devices with electrostatic actuation and sensing. PMID:25230308
The Stiffness Variation of a Micro-Ring Driven by a Traveling Piecewise-Electrode
Li, Yingjie; Yu, Tao; Hu, Yuh-Chung
2014-01-01
In the practice of electrostatically actuated micro devices; the electrostatic force is implemented by sequentially actuated piecewise-electrodes which result in a traveling distributed electrostatic force. However; such force was modeled as a traveling concentrated electrostatic force in literatures. This article; for the first time; presents an analytical study on the stiffness variation of microstructures driven by a traveling piecewise electrode. The analytical model is based on the theory of shallow shell and uniform electrical field. The traveling electrode not only applies electrostatic force on the circular-ring but also alters its dynamical characteristics via the negative electrostatic stiffness. It is known that; when a structure is subjected to a traveling constant force; its natural mode will be resonated as the traveling speed approaches certain critical speeds; and each natural mode refers to exactly one critical speed. However; for the case of a traveling electrostatic force; the number of critical speeds is more than that of the natural modes. This is due to the fact that the traveling electrostatic force makes the resonant frequencies of the forward and backward traveling waves of the circular-ring different. Furthermore; the resonance and stability can be independently controlled by the length of the traveling electrode; though the driving voltage and traveling speed of the electrostatic force alter the dynamics and stabilities of microstructures. This paper extends the fundamental insights into the electromechanical behavior of microstructures driven by electrostatic forces as well as the future development of MEMS/NEMS devices with electrostatic actuation and sensing. PMID:25230308
ERIC Educational Resources Information Center
Walkiewicz, T. A.; Newby, N. D., Jr.
1972-01-01
A discussion of linear collisions between two or three objects is related to a junior-level course in analytical mechanics. The theoretical discussion uses a geometrical approach that treats elastic and inelastic collisions from a unified point of view. Experiments with a linear air track are described. (Author/TS)
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.
Chalasani, P.; Saias, I.; Jha, S.
1996-04-08
As increasingly large volumes of sophisticated options (called derivative securities) are traded in world financial markets, determining a fair price for these options has become an important and difficult computational problem. Many valuation codes use the binomial pricing model, in which the stock price is driven by a random walk. In this model, the value of an n-period option on a stock is the expected time-discounted value of the future cash flow on an n-period stock price path. Path-dependent options are particularly difficult to value since the future cash flow depends on the entire stock price path rather than on just the final stock price. Currently such options are approximately priced by Monte carlo methods with error bounds that hold only with high probability and which are reduced by increasing the number of simulation runs. In this paper the authors show that pricing an arbitrary path-dependent option is {number_sign}-P hard. They show that certain types f path-dependent options can be valued exactly in polynomial time. Asian options are path-dependent options that are particularly hard to price, and for these they design deterministic polynomial-time approximate algorithms. They show that the value of a perpetual American put option (which can be computed in constant time) is in many cases a good approximation to the value of an otherwise identical n-period American put option. In contrast to Monte Carlo methods, the algorithms have guaranteed error bounds that are polynormally small (and in some cases exponentially small) in the maturity n. For the error analysis they derive large-deviation results for random walks that may be of independent interest.
Meta-Regression Approximations to Reduce Publication Selection Bias
ERIC Educational Resources Information Center
Stanley, T. D.; Doucouliagos, Hristos
2014-01-01
Publication selection bias is a serious challenge to the integrity of all empirical sciences. We derive meta-regression approximations to reduce this bias. Our approach employs Taylor polynomial approximations to the conditional mean of a truncated distribution. A quadratic approximation without a linear term, precision-effect estimate with…
Predictability of Acoustic Propagation in Shallow Water Using Parabolic Approximation Models
NASA Astrophysics Data System (ADS)
Cederberg, Robert John
for which sound speed is comprised of piecewise linear segments. In addition, bottom sound speed range dependence was modeled with discrete as well as continuous changes between profiles for variations of the geophysical parameters in this area and no substantial difference was observed. Comparisons of the experimental results with model predictions illustrate how parameter uncertainties may have significant effects.
Superconducting linear actuator
NASA Technical Reports Server (NTRS)
Johnson, Bruce; Hockney, Richard
1993-01-01
Special actuators are needed to control the orientation of large structures in space-based precision pointing systems. Electromagnetic actuators that presently exist are too large in size and their bandwidth is too low. Hydraulic fluid actuation also presents problems for many space-based applications. Hydraulic oil can escape in space and contaminate the environment around the spacecraft. A research study was performed that selected an electrically-powered linear actuator that can be used to control the orientation of a large pointed structure. This research surveyed available products, analyzed the capabilities of conventional linear actuators, and designed a first-cut candidate superconducting linear actuator. The study first examined theoretical capabilities of electrical actuators and determined their problems with respect to the application and then determined if any presently available actuators or any modifications to available actuator designs would meet the required performance. The best actuator was then selected based on available design, modified design, or new design for this application. The last task was to proceed with a conceptual design. No commercially-available linear actuator or modification capable of meeting the specifications was found. A conventional moving-coil dc linear actuator would meet the specification, but the back-iron for this actuator would weigh approximately 12,000 lbs. A superconducting field coil, however, eliminates the need for back iron, resulting in an actuator weight of approximately 1000 lbs.
Christofilos, N.C.; Polk, I.J.
1959-02-17
Improvements in linear particle accelerators are described. A drift tube system for a linear ion accelerator reduces gap capacity between adjacent drift tube ends. This is accomplished by reducing the ratio of the diameter of the drift tube to the diameter of the resonant cavity. Concentration of magnetic field intensity at the longitudinal midpoint of the external sunface of each drift tube is reduced by increasing the external drift tube diameter at the longitudinal center region.
Roy, Swapnoneel; Thakur, Ashok Kumar
2008-01-01
Genome rearrangements have been modelled by a variety of primitives such as reversals, transpositions, block moves and block interchanges. We consider such a genome rearrangement primitive Strip Exchanges. Given a permutation, the challenge is to sort it by using minimum number of strip exchanges. A strip exchanging move interchanges the positions of two chosen strips so that they merge with other strips. The strip exchange problem is to sort a permutation using minimum number of strip exchanges. We present here the first non-trivial 2-approximation algorithm to this problem. We also observe that sorting by strip-exchanges is fixed-parameter-tractable. Lastly we discuss the application of strip exchanges in a different area Optical Character Recognition (OCR) with an example.
Hierarchical Approximate Bayesian Computation
Turner, Brandon M.; Van Zandt, Trisha
2013-01-01
Approximate Bayesian computation (ABC) is a powerful technique for estimating the posterior distribution of a model’s parameters. It is especially important when the model to be fit has no explicit likelihood function, which happens for computational (or simulation-based) models such as those that are popular in cognitive neuroscience and other areas in psychology. However, ABC is usually applied only to models with few parameters. Extending ABC to hierarchical models has been difficult because high-dimensional hierarchical models add computational complexity that conventional ABC cannot accommodate. In this paper we summarize some current approaches for performing hierarchical ABC and introduce a new algorithm called Gibbs ABC. This new algorithm incorporates well-known Bayesian techniques to improve the accuracy and efficiency of the ABC approach for estimation of hierarchical models. We then use the Gibbs ABC algorithm to estimate the parameters of two models of signal detection, one with and one without a tractable likelihood function. PMID:24297436
The Linear KdV Equation with an Interface
NASA Astrophysics Data System (ADS)
Deconinck, Bernard; Sheils, Natalie E.; Smith, David A.
2016-10-01
The interface problem for the linear Korteweg-de Vries (KdV) equation in one-dimensional piecewise homogeneous domains is examined by constructing an explicit solution in each domain. The location of the interface is known and a number of compatibility conditions at the boundary are imposed. We provide an explicit characterization of sufficient interface conditions for the construction of a solution using Fokas's Unified Transform Method. The problem and the method considered here extend that of earlier papers to problems with more than two spatial derivatives.
The Linear KdV Equation with an Interface
NASA Astrophysics Data System (ADS)
Deconinck, Bernard; Sheils, Natalie E.; Smith, David A.
2016-07-01
The interface problem for the linear Korteweg-de Vries (KdV) equation in one-dimensional piecewise homogeneous domains is examined by constructing an explicit solution in each domain. The location of the interface is known and a number of compatibility conditions at the boundary are imposed. We provide an explicit characterization of sufficient interface conditions for the construction of a solution using Fokas's Unified Transform Method. The problem and the method considered here extend that of earlier papers to problems with more than two spatial derivatives.
Linearly Adjustable International Portfolios
NASA Astrophysics Data System (ADS)
Fonseca, R. J.; Kuhn, D.; Rustem, B.
2010-09-01
We present an approach to multi-stage international portfolio optimization based on the imposition of a linear structure on the recourse decisions. Multiperiod decision problems are traditionally formulated as stochastic programs. Scenario tree based solutions however can become intractable as the number of stages increases. By restricting the space of decision policies to linear rules, we obtain a conservative tractable approximation to the original problem. Local asset prices and foreign exchange rates are modelled separately, which allows for a direct measure of their impact on the final portfolio value.
NASA Astrophysics Data System (ADS)
Sultan, Cornel
2010-10-01
The design of vector second-order linear systems for accurate proportional damping approximation is addressed. For this purpose an error system is defined using the difference between the generalized coordinates of the non-proportionally damped system and its proportionally damped approximation in modal space. The accuracy of the approximation is characterized using the energy gain of the error system and the design problem is formulated as selecting parameters of the non-proportionally damped system to ensure that this gain is sufficiently small. An efficient algorithm that combines linear matrix inequalities and simultaneous perturbation stochastic approximation is developed to solve the problem and examples of its application to tensegrity structures design are presented.
NASA Astrophysics Data System (ADS)
Chernov, A. V.
2013-12-01
Approximating finite-dimensional mathematical programming problems are studied that arise from piecewise constant discretization of controls in the optimization of distributed systems of a fairly broad class. The smoothness of the approximating problems is established. Gradient formulas are derived that make use of the analytical solution of the original control system and its adjoint, thus providing an opportunity for algorithmic separation of numerical optimization and the task of solving a controlled initial-boundary value problem. The approximating problems are proved to converge to the original optimization problem with respect to the functional as the discretization is refined. The application of the approach to optimization problems is illustrated by solving the semilinear wave equation controlled by applying an integral criterion. The results of numerical experiments are analyzed.
Li, Yutian; Zhu, Jianxin
2015-05-01
In this paper we consider the problem of computing the eigen-modes for the varying refractive-index profile in an open waveguide. We first approximate the refractive-index by a piecewise polynomial of degree two, and the corresponding Sturm-Liouville problem (eigenvalue problem) of the Helmholtz operator in each layer can be solved analytically by the Kummer functions. Then, analytical approximate dispersion equations are established for both TE and TM cases. Furthermore, the approximate dispersion equations converge fast to the exact ones for the continuous refractive-index function as the maximum value of the subinterval sizes tends to zero. Suitable numerical methods, such as Müller's method or the chord secant method, may be applied to the dispersion relations to compute the eigenmodes. Numerical simulations show that our method is very practical and efficient for computing eigenmodes. PMID:25969285
Interior region-of-interest reconstruction using a small, nearly piecewise constant subregion
Taguchi, Katsuyuki; Xu Jingyan; Srivastava, Somesh; Tsui, Benjamin M. W.; Cammin, Jochen; Tang Qiulin
2011-03-15
Purpose: To develop a method to reconstruct an interior region-of-interest (ROI) image with sufficient accuracy that uses differentiated backprojection (DBP) projection onto convex sets (POCS) [H. Kudo et al., ''Tiny a priori knowledge solves the interior problem in computed tomography'', Phys. Med. Biol. 53, 2207-2231 (2008)] and a tiny knowledge that there exists a nearly piecewise constant subregion. Methods: The proposed method first employs filtered backprojection to reconstruct an image on which a tiny region P with a small variation in the pixel values is identified inside the ROI. Total variation minimization [H. Yu and G. Wang, ''Compressed sensing based interior tomography'', Phys. Med. Biol. 54, 2791-2805 (2009); W. Han et al., ''A general total variation minimization theorem for compressed sensing based interior tomography'', Int. J. Biomed. Imaging 2009, Article 125871 (2009)] is then employed to obtain pixel values in the subregion P, which serve as a priori knowledge in the next step. Finally, DBP-POCS is performed to reconstruct f(x,y) inside the ROI. Clinical data and the reconstructed image obtained by an x-ray computed tomography system (SOMATOM Definition; Siemens Healthcare) were used to validate the proposed method. The detector covers an object with a diameter of {approx}500 mm. The projection data were truncated either moderately to limit the detector coverage to diameter 350 mm of the object or severely to cover diameter 199 mm. Images were reconstructed using the proposed method. Results: The proposed method provided ROI images with correct pixel values in all areas except near the edge of the ROI. The coefficient of variation, i.e., the root mean square error divided by the mean pixel values, was less than 2.0% or 4.5% with the moderate or severe truncation cases, respectively, except near the boundary of the ROI. Conclusions: The proposed method allows for reconstructing interior ROI images with sufficient accuracy with a tiny knowledge
Approximate Bayesian multibody tracking.
Lanz, Oswald
2006-09-01
Visual tracking of multiple targets is a challenging problem, especially when efficiency is an issue. Occlusions, if not properly handled, are a major source of failure. Solutions supporting principled occlusion reasoning have been proposed but are yet unpractical for online applications. This paper presents a new solution which effectively manages the trade-off between reliable modeling and computational efficiency. The Hybrid Joint-Separable (HJS) filter is derived from a joint Bayesian formulation of the problem, and shown to be efficient while optimal in terms of compact belief representation. Computational efficiency is achieved by employing a Markov random field approximation to joint dynamics and an incremental algorithm for posterior update with an appearance likelihood that implements a physically-based model of the occlusion process. A particle filter implementation is proposed which achieves accurate tracking during partial occlusions, while in cases of complete occlusion, tracking hypotheses are bound to estimated occlusion volumes. Experiments show that the proposed algorithm is efficient, robust, and able to resolve long-term occlusions between targets with identical appearance. PMID:16929730
Vonica-Gligor, Andreea Loredana; Tomuţă, Ioan; Leucuţa, Sorin E
2016-06-01
The aim of this work was to develop a pulsatile release system with metoprolol for chronotherapeutical use by coating swellable mini-tablets with Eudragit RS. To study the influence of the formulation factors (amount of coating polymer, plasticizer percentage in film coating and swelling agent percentage in mini-tablets), a Box-Behnken design of experiment (DoE) was used. To evaluate the influence of the studied factors on the sigmoid shape of the dissolution profile, piecewise function parameters were used as the responses of DoE. The results show that higher concentrations of coating polymer and higher concentrations of plasticizer polymer led to a thicker and more elastic polymeric film, which led to a delay in drug release. Using the parameters of the piecewise function as DoE responses, an optimum formulation with a sigmoid shape dissolution profile and a 2.5-h lag time followed by rapid drug release were obtained. PMID:27279062
NASA Astrophysics Data System (ADS)
Lubkin, Elihu
2002-04-01
In 1993,(E. & T. Lubkin, Int.J.Theor.Phys. 32), 993 (1993) we gave exact mean trace
Piecewise-Constant-Model-Based Interior Tomography Applied to Dentin Tubules
He, Peng; Wei, Biao; Wang, Steve; Stock, Stuart R.; Yu, Hengyong; Wang, Ge
2013-01-01
Dentin is a hierarchically structured biomineralized composite material, and dentin’s tubules are difficult to study in situ. Nano-CT provides the requisite resolution, but the field of view typically contains only a few tubules. Using a plate-like specimen allows reconstruction of a volume containing specific tubules from a number of truncated projections typically collected over an angular range of about 140°, which is practically accessible. Classical computed tomography (CT) theory cannot exactly reconstruct an object only from truncated projections, needless to say a limited angular range. Recently, interior tomography was developed to reconstruct a region-of-interest (ROI) from truncated data in amore » theoretically exact fashion via the total variation (TV) minimization under the condition that the ROI is piecewise constant. In this paper, we employ a TV minimization interior tomography algorithm to reconstruct interior microstructures in dentin from truncated projections over a limited angular range. Compared to the filtered backprojection (FBP) reconstruction, our reconstruction method reduces noise and suppresses artifacts. Volume rendering confirms the merits of our method in terms of preserving the interior microstructure of the dentin specimen.« less
Development and evaluation of the piecewise Prony method for evoked potential analysis.
Garoosi, V; Jansen, B H
2000-12-01
A new method is presented to decompose nonstationary signals into a summation of oscillatory components with time varying frequency, amplitude, and phase characteristics. This method, referred to as piecewise Prony method (PPM), is an improvement over the classical Prony method, which can only deal with signals containing components with fixed frequency, amplitude and phase, and monotonically increasing or decreasing rate of change. PPM allows the study of the temporal profile of post-stimulus signal changes in single-trial evoked potentials (EPs), which can lead to new insights in EP generation. We have evaluated this method on simulated data to test its limitations and capabilities, and also on single-trial EPs. The simulation experiments showed that the PPM can detect amplitude changes as small as 10%, rate changes as small as 10%, and 0.15 Hz of frequency changes. The capabilities of the PPM were demonstrated using single electroencephalogram/EP trials of flash visual EPs recorded from one normal subject. The trial-by-trial results confirmed that the stimulation drastically attenuates the alpha activity shortly after stimulus presentation, with the alpha activity returning about 0.5 s later. The PPM results also provided evidence that delta activity undergoes phase alignment following stimulus presentation.
Verification and Validation of a Chemical Reaction Solver Coupled to the Piecewise Parabolic Method
NASA Astrophysics Data System (ADS)
Attal, Nitesh; Ramaprabhu, Praveen; Hossain, Jahed; Karkhanis, Varad; Roy, Sukesh; Gord, James; Uddin, Mesbah
2012-11-01
We present a detailed chemical kinetics reaction solver coupled to the Piecewise Parabolic Method (PPM) embedded in the widely used astrophysical FLASH code. The FLASH code solves the compressible Euler equations with a directionally split, PPM with Adaptive Mesh Refinement (AMR). The reaction network is solved using a library of coupled ODE solvers, specialized for handling stiff systems of equations. Finally, the diffusion of heat, mass, and momentum is handled either through an update of the fluxes of each quantity, or by directly solving a diffusion equation for each. The resulting product is capable of handling a variety of physics such as gas-phase chemical kinetics, diffusive transport of mass, momentum, and heat, shocks, sharp interfaces, multi-species mixtures, and thermal radiation. We will present results from verification and validation of the above capabilities through comparison with analytical solutions, and published numerical and experimental data. Our validation cases include advection of reacting fronts in 1-D and 2D, laminar premixed flames in a Bunsen burner configuration, and shock-driven combustion. We acknowledge funding from Spectral Energies LLC.
Laplace approximation in measurement error models.
Battauz, Michela
2011-05-01
Likelihood analysis for regression models with measurement errors in explanatory variables typically involves integrals that do not have a closed-form solution. In this case, numerical methods such as Gaussian quadrature are generally employed. However, when the dimension of the integral is large, these methods become computationally demanding or even unfeasible. This paper proposes the use of the Laplace approximation to deal with measurement error problems when the likelihood function involves high-dimensional integrals. The cases considered are generalized linear models with multiple covariates measured with error and generalized linear mixed models with measurement error in the covariates. The asymptotic order of the approximation and the asymptotic properties of the Laplace-based estimator for these models are derived. The method is illustrated using simulations and real-data analysis.
Successive linear optimization approach to the dynamic traffic assignment problem
Ho, J.K.
1980-11-01
A dynamic model for the optimal control of traffic flow over a network is considered. The model, which treats congestion explicitly in the flow equations, gives rise to nonlinear, nonconvex mathematical programming problems. It has been shown for a piecewise linear version of this model that a global optimum is contained in the set of optimal solutions of a certain linear program. A sufficient condition for optimality which implies that a global optimum can be obtained by successively optimizing at most N + 1 objective functions for the linear program, where N is the number of time periods in the planning horizon is presented. Computational results are reported to indicate the efficiency of this approach.
Merge Frame Design for Video Stream Switching Using Piecewise Constant Functions
NASA Astrophysics Data System (ADS)
Dai, Wei; Cheung, Gene; Cheung, Ngai-Man; Ortega, Antonio; Au, Oscar C.
2016-08-01
The ability to efficiently switch from one pre-encoded video stream to another (e.g., for bitrate adaptation or view switching) is important for many interactive streaming applications. Recently, stream-switching mechanisms based on distributed source coding (DSC) have been proposed. In order to reduce the overall transmission rate, these approaches provide a "merge" mechanism, where information is sent to the decoder such that the exact same frame can be reconstructed given that any one of a known set of side information (SI) frames is available at the decoder (e.g., each SI frame may correspond to a different stream from which we are switching). However, the use of bit-plane coding and channel coding in many DSC approaches leads to complex coding and decoding. In this paper, we propose an alternative approach for merging multiple SI frames, using a piecewise constant (PWC) function as the merge operator. In our approach, for each block to be reconstructed, a series of parameters of these PWC merge functions are transmitted in order to guarantee identical reconstruction given the known side information blocks. We consider two different scenarios. In the first case, a target frame is first given, and then merge parameters are chosen so that this frame can be reconstructed exactly at the decoder. In contrast, in the second scenario, the reconstructed frame and merge parameters are jointly optimized to meet a rate-distortion criteria. Experiments show that for both scenarios, our proposed merge techniques can outperform both a recent approach based on DSC and the SP-frame approach in H.264, in terms of compression efficiency and decoder complexity.
A piecewise model of virus-immune system with two thresholds.
Tang, Biao; Xiao, Yanni; Wu, Jianhong
2016-08-01
The combined antiretroviral therapy with interleukin (IL)-2 treatment may not be enough to preclude exceptionally high growth of HIV virus nor rebuilt the HIV-specific CD4 or CD8 T-cell proliferative immune response for management of HIV infected patients. Whether extra inclusion of immune therapy can induce the HIV-specific immune response and control HIV replication remains challenging. Here a piecewise virus-immune model with two thresholds is proposed to represent the HIV-1 RNA and effector cell-guided therapy strategies. We first analyze the dynamics of the virus-immune system with effector cell-guided immune therapy only and prove that there exists a critical level of the intensity of immune therapy determining whether the HIV-1 RAN virus loads can be controlled below a relative low level. Our analysis of the global dynamics of the proposed model shows that the pseudo-equilibrium can be globally stable or locally bistable with order 1 periodic solution or bistable with the virus-free periodic solution under various appropriate conditions. This indicates that HIV viral loads can either be eradicated or stabilize at a previously given level or go to infinity (corresponding to the effector cells oscillating), depending on the threshold levels and the initial HIV virus loads and effector cell counts. Comparing with the single threshold therapy strategy we obtain that with two thresholds therapy strategies either virus can be eradicated or the controllable region, where HIV viral loads can be maintained below a certain value, can be enlarged.
Cubic spline approximation techniques for parameter estimation in distributed systems
NASA Technical Reports Server (NTRS)
Banks, H. T.; Crowley, J. M.; Kunisch, K.
1983-01-01
Approximation schemes employing cubic splines in the context of a linear semigroup framework are developed for both parabolic and hyperbolic second-order partial differential equation parameter estimation problems. Convergence results are established for problems with linear and nonlinear systems, and a summary of numerical experiments with the techniques proposed is given.
Colgate, S.A.
1958-05-27
An improvement is presented in linear accelerators for charged particles with respect to the stable focusing of the particle beam. The improvement consists of providing a radial electric field transverse to the accelerating electric fields and angularly introducing the beam of particles in the field. The results of the foregoing is to achieve a beam which spirals about the axis of the acceleration path. The combination of the electric fields and angular motion of the particles cooperate to provide a stable and focused particle beam.
State space approximation for general fractional order dynamic systems
NASA Astrophysics Data System (ADS)
Liang, Shu; Peng, Cheng; Liao, Zeng; Wang, Yong
2014-10-01
Approximations for general fractional order dynamic systems are of much theoretical and practical interest. In this paper, a new approximate method for fractional order integrator is proposed. The poles of the approximate model are unrelated to the order of integrator. This feature shows benefits on extending the algorithm to the systems containing various fractional orders. Then a unified approximate method is derived for general fractional order linear or nonlinear dynamic systems via combining the proposed new method with the distributed frequency model approach. Numerical examples are given to show the wide applicability of our method and to illustrate the acceptable accuracy for approximations as well.
Reply to Steele & Ferrer: Modeling Oscillation, Approximately or Exactly?
ERIC Educational Resources Information Center
Oud, Johan H. L.; Folmer, Henk
2011-01-01
This article addresses modeling oscillation in continuous time. It criticizes Steele and Ferrer's article "Latent Differential Equation Modeling of Self-Regulatory and Coregulatory Affective Processes" (2011), particularly the approximate estimation procedure applied. This procedure is the latent version of the local linear approximation procedure…
Impact of using linear optimization models in dose planning for HDR brachytherapy
Holm, Aasa; Larsson, Torbjoern; Carlsson Tedgren, Aasa
2012-02-15
Purpose: Dose plans generated with optimization models hitherto used in high-dose-rate (HDR) brachytherapy have shown a tendency to yield longer dwell times than manually optimized plans. Concern has been raised for the corresponding undesired hot spots, and various methods to mitigate these have been developed. The hypotheses upon this work is based are (a) that one cause for the long dwell times is the use of objective functions comprising simple linear penalties and (b) that alternative penalties, as these are piecewise linear, would lead to reduced length of individual dwell times. Methods: The characteristics of the linear penalties and the piecewise linear penalties are analyzed mathematically. Experimental comparisons between the two types of penalties are carried out retrospectively for a set of prostate cancer patients. Results: When the two types of penalties are compared, significant changes can be seen in the dwell times, while most dose-volume parameters do not differ significantly. On average, total dwell times were reduced by 4.2%, with a reduction of maximum dwell times by 25%, when the alternative penalties were used. Conclusions: The use of linear penalties in optimization models for HDR brachytherapy is one cause for the undesired long dwell times that arise in mathematically optimized plans. By introducing alternative penalties, a significant reduction in dwell times can be achieved for HDR brachytherapy dose plans. Although various measures for mitigating the long dwell times are already available, the observation that linear penalties contribute to their appearance is of fundamental interest.
CMB-lensing beyond the Born approximation
NASA Astrophysics Data System (ADS)
Marozzi, Giovanni; Fanizza, Giuseppe; Di Dio, Enea; Durrer, Ruth
2016-09-01
We investigate the weak lensing corrections to the cosmic microwave background temperature anisotropies considering effects beyond the Born approximation. To this aim, we use the small deflection angle approximation, to connect the lensed and unlensed power spectra, via expressions for the deflection angles up to third order in the gravitational potential. While the small deflection angle approximation has the drawback to be reliable only for multipoles l lesssim 2500, it allows us to consistently take into account the non-Gaussian nature of cosmological perturbation theory beyond the linear level. The contribution to the lensed temperature power spectrum coming from the non-Gaussian nature of the deflection angle at higher order is a new effect which has not been taken into account in the literature so far. It turns out to be the leading contribution among the post-Born lensing corrections. On the other hand, the effect is smaller than corrections coming from non-linearities in the matter power spectrum, and its imprint on CMB lensing is too small to be seen in present experiments.
Approximate Controllability of Fractional Neutral Stochastic System with Infinite Delay
NASA Astrophysics Data System (ADS)
Sakthivel, R.; Ganesh, R.; Suganya, S.
2012-12-01
The concept of controllability plays an important role in analysis and design of linear and nonlinear control systems. Further, fractional differential equations have wide applications in engineering and science. In this paper, the approximate controllability of neutral stochastic fractional integro-differential equation with infinite delay in a Hilbert space is studied. By using Krasnoselskii's fixed point theorem with stochastic analysis theory, we derive a new set of sufficient conditions for the approximate controllability of nonlinear fractional stochastic system under the assumption that the corresponding linear system is approximately controllable. Finally, an example is provided to illustrate the obtained theory.
Watts, Benjamin
2014-09-22
An algorithm is presented for the calculation of the Kramers-Kronig transform of a spectrum via a piecewise Laurent polynomial method. This algorithm is demonstrated to be highly accurate, while also being computationally efficient. The algorithm places no requirements on data point spacing and is capable of integrating across the full spectrum (i.e. from zero to infinity). Further, we present a computer application designed to aid in calculating the Kramers-Kronig transform on near-edge experimental X-ray absorption spectra (extended with atomic scattering factor data) in order to produce the dispersive part of the X-ray refractive index, including near-edge features.
Woodward, P. R.
2003-03-26
This report summarizes the results of the project entitled, ''Piecewise-Parabolic Methods for Parallel Computation with Applications to Unstable Fluid Flow in 2 and 3 Dimensions'' This project covers a span of many years, beginning in early 1987. It has provided over that considerable period the core funding to my research activities in scientific computation at the University of Minnesota. It has supported numerical algorithm development, application of those algorithms to fundamental fluid dynamics problems in order to demonstrate their effectiveness, and the development of scientific visualization software and systems to extract scientific understanding from those applications.
Watts, Benjamin
2014-09-22
An algorithm is presented for the calculation of the Kramers-Kronig transform of a spectrum via a piecewise Laurent polynomial method. This algorithm is demonstrated to be highly accurate, while also being computationally efficient. The algorithm places no requirements on data point spacing and is capable of integrating across the full spectrum (i.e. from zero to infinity). Further, we present a computer application designed to aid in calculating the Kramers-Kronig transform on near-edge experimental X-ray absorption spectra (extended with atomic scattering factor data) in order to produce the dispersive part of the X-ray refractive index, including near-edge features. PMID:25321829
Djuric, P.M.; Fwu, Jong-Kae; Jovanovic, S.; Lynn, K.
1996-03-01
A new approach for processing of piecewise-constant signals is proposed. It is based on modeling the observed data as a sum of a random signal and noise. The random signal has a Gibbs distribution, and the noise is Gaussian. A MAP criterion in derived for joint estimation of the number of signal levels and reconstruction of signal. The criterion comprises of three terms, one corresponding to the likelihood of the data and two to penalties. One penalty term penalizes for unnecessary transitions, and the other, for unnecessary levels. The method has been tested on synthesized data and applied to single ion channel recording.
Linearization algorithms for line transfer
Scott, H.A.
1990-11-06
Complete linearization is a very powerful technique for solving multi-line transfer problems that can be used efficiently with a variety of transfer formalisms. The linearization algorithm we describe is computationally very similar to ETLA, but allows an effective treatment of strongly-interacting lines. This algorithm has been implemented (in several codes) with two different transfer formalisms in all three one-dimensional geometries. We also describe a variation of the algorithm that handles saturable laser transport. Finally, we present a combination of linearization with a local approximate operator formalism, which has been implemented in two dimensions and is being developed in three dimensions. 11 refs.
Phenomenological applications of rational approximants
NASA Astrophysics Data System (ADS)
Gonzàlez-Solís, Sergi; Masjuan, Pere
2016-08-01
We illustrate the powerfulness of Padé approximants (PAs) as a summation method and explore one of their extensions, the so-called quadratic approximant (QAs), to access both space- and (low-energy) time-like (TL) regions. As an introductory and pedagogical exercise, the function 1 zln(1 + z) is approximated by both kind of approximants. Then, PAs are applied to predict pseudoscalar meson Dalitz decays and to extract Vub from the semileptonic B → πℓνℓ decays. Finally, the π vector form factor in the TL region is explored using QAs.
Analysing organic transistors based on interface approximation
Akiyama, Yuto; Mori, Takehiko
2014-01-15
Temperature-dependent characteristics of organic transistors are analysed thoroughly using interface approximation. In contrast to amorphous silicon transistors, it is characteristic of organic transistors that the accumulation layer is concentrated on the first monolayer, and it is appropriate to consider interface charge rather than band bending. On the basis of this model, observed characteristics of hexamethylenetetrathiafulvalene (HMTTF) and dibenzotetrathiafulvalene (DBTTF) transistors with various surface treatments are analysed, and the trap distribution is extracted. In turn, starting from a simple exponential distribution, we can reproduce the temperature-dependent transistor characteristics as well as the gate voltage dependence of the activation energy, so we can investigate various aspects of organic transistors self-consistently under the interface approximation. Small deviation from such an ideal transistor operation is discussed assuming the presence of an energetically discrete trap level, which leads to a hump in the transfer characteristics. The contact resistance is estimated by measuring the transfer characteristics up to the linear region.
Approximate inverse preconditioners for general sparse matrices
Chow, E.; Saad, Y.
1994-12-31
Preconditioned Krylov subspace methods are often very efficient in solving sparse linear matrices that arise from the discretization of elliptic partial differential equations. However, for general sparse indifinite matrices, the usual ILU preconditioners fail, often because of the fact that the resulting factors L and U give rise to unstable forward and backward sweeps. In such cases, alternative preconditioners based on approximate inverses may be attractive. We are currently developing a number of such preconditioners based on iterating on each column to get the approximate inverse. For this approach to be efficient, the iteration must be done in sparse mode, i.e., we must use sparse-matrix by sparse-vector type operatoins. We will discuss a few options and compare their performance on standard problems from the Harwell-Boeing collection.
LINEAR - DERIVATION AND DEFINITION OF A LINEAR AIRCRAFT MODEL
NASA Technical Reports Server (NTRS)
Duke, E. L.
1994-01-01
interest, or a full non-linear aerodynamic model as used in simulations. LINEAR is written in FORTRAN and has been implemented on a DEC VAX computer operating under VMS with a virtual memory requirement of approximately 296K of 8 bit bytes. Both an interactive and batch version are included. LINEAR was developed in 1988.
Parameter Choices for Approximation by Harmonic Splines
NASA Astrophysics Data System (ADS)
Gutting, Martin
2016-04-01
The approximation by harmonic trial functions allows the construction of the solution of boundary value problems in geoscience, e.g., in terms of harmonic splines. Due to their localizing properties regional modeling or the improvement of a global model in a part of the Earth's surface is possible with splines. Fast multipole methods have been developed for some cases of the occurring kernels to obtain a fast matrix-vector multiplication. The main idea of the fast multipole algorithm consists of a hierarchical decomposition of the computational domain into cubes and a kernel approximation for the more distant points. This reduces the numerical effort of the matrix-vector multiplication from quadratic to linear in reference to the number of points for a prescribed accuracy of the kernel approximation. The application of the fast multipole method to spline approximation which also allows the treatment of noisy data requires the choice of a smoothing parameter. We investigate different methods to (ideally automatically) choose this parameter with and without prior knowledge of the noise level. Thereby, the performance of these methods is considered for different types of noise in a large simulation study. Applications to gravitational field modeling are presented as well as the extension to boundary value problems where the boundary is the known surface of the Earth itself.
Approximation and compression with sparse orthonormal transforms.
Sezer, Osman Gokhan; Guleryuz, Onur G; Altunbasak, Yucel
2015-08-01
We propose a new transform design method that targets the generation of compression-optimized transforms for next-generation multimedia applications. The fundamental idea behind transform compression is to exploit regularity within signals such that redundancy is minimized subject to a fidelity cost. Multimedia signals, in particular images and video, are well known to contain a diverse set of localized structures, leading to many different types of regularity and to nonstationary signal statistics. The proposed method designs sparse orthonormal transforms (SOTs) that automatically exploit regularity over different signal structures and provides an adaptation method that determines the best representation over localized regions. Unlike earlier work that is motivated by linear approximation constructs and model-based designs that are limited to specific types of signal regularity, our work uses general nonlinear approximation ideas and a data-driven setup to significantly broaden its reach. We show that our SOT designs provide a safe and principled extension of the Karhunen-Loeve transform (KLT) by reducing to the KLT on Gaussian processes and by automatically exploiting non-Gaussian statistics to significantly improve over the KLT on more general processes. We provide an algebraic optimization framework that generates optimized designs for any desired transform structure (multiresolution, block, lapped, and so on) with significantly better n -term approximation performance. For each structure, we propose a new prototype codec and test over a database of images. Simulation results show consistent increase in compression and approximation performance compared with conventional methods. PMID:25823033
Achievements and Problems in Diophantine Approximation Theory
NASA Astrophysics Data System (ADS)
Sprindzhuk, V. G.
1980-08-01
ContentsIntroduction I. Metrical theory of approximation on manifolds § 1. The basic problem § 2. Brief survey of results § 3. The principal conjecture II. Metrical theory of transcendental numbers § 1. Mahler's classification of numbers § 2. Metrical characterization of numbers with a given type of approximation § 3. Further problems III. Approximation of algebraic numbers by rationals § 1. Simultaneous approximations § 2. The inclusion of p-adic metrics § 3. Effective improvements of Liouville's inequality IV. Estimates of linear forms in logarithms of algebraic numbers § 1. The basic method § 2. Survey of results § 3. Estimates in the p-adic metric V. Diophantine equations § 1. Ternary exponential equations § 2. The Thue and Thue-Mahler equations § 3. Equations of hyperelliptic type § 4. Algebraic-exponential equations VI. The arithmetic structure of polynomials and the class number § 1. The greatest prime divisor of a polynomial in one variable § 2. The greatest prime divisor of a polynomial in two variables § 3. Square-free divisors of polynomials and the class number § 4. The general problem of the size of the class number Conclusion References
Linearized motion estimation for articulated planes.
Datta, Ankur; Sheikh, Yaser; Kanade, Takeo
2011-04-01
In this paper, we describe the explicit application of articulation constraints for estimating the motion of a system of articulated planes. We relate articulations to the relative homography between planes and show that these articulations translate into linearized equality constraints on a linear least-squares system, which can be solved efficiently using a Karush-Kuhn-Tucker system. The articulation constraints can be applied for both gradient-based and feature-based motion estimation algorithms and to illustrate this, we describe a gradient-based motion estimation algorithm for an affine camera and a feature-based motion estimation algorithm for a projective camera that explicitly enforces articulation constraints. We show that explicit application of articulation constraints leads to numerically stable estimates of motion. The simultaneous computation of motion estimates for all of the articulated planes in a scene allows us to handle scene areas where there is limited texture information and areas that leave the field of view. Our results demonstrate the wide applicability of the algorithm in a variety of challenging real-world cases such as human body tracking, motion estimation of rigid, piecewise planar scenes, and motion estimation of triangulated meshes.
Abou-Jaoudé, Wassim; Chaves, Madalena; Gouzé, Jean-Luc
2014-12-01
A class of piecewise affine differential (PWA) models, initially proposed by Glass and Kauffman (in J Theor Biol 39:103-129, 1973), has been widely used for the modelling and the analysis of biological switch-like systems, such as genetic or neural networks. Its mathematical tractability facilitates the qualitative analysis of dynamical behaviors, in particular periodic phenomena which are of prime importance in biology. Notably, a discrete qualitative description of the dynamics, called the transition graph, can be directly associated to this class of PWA systems. Here we present a study of periodic behaviours (i.e. limit cycles) in a class of two-dimensional piecewise affine biological models. Using concavity and continuity properties of Poincaré maps, we derive structural principles linking the topology of the transition graph to the existence, number and stability of limit cycles. These results notably extend previous works on the investigation of structural principles to the case of unequal and regulated decay rates for the 2-dimensional case. Some numerical examples corresponding to minimal models of biological oscillators are treated to illustrate the use of these structural principles.
Mamey, Mary Rose; Barbosa-Leiker, Celestina; McPherson, Sterling; Burns, G Leonard; Parks, Craig; Roll, John
2015-12-01
Researchers often want to examine 2 comorbid conditions simultaneously. One strategy to do so is through the use of parallel latent growth curve modeling (LGCM). This statistical technique allows for the simultaneous evaluation of 2 disorders to determine the explanations and predictors of change over time. Additionally, a piecewise model can help identify whether there are more than 2 growth processes within each disorder (e.g., during a clinical trial). A parallel piecewise LGCM was applied to self-reported attention-deficit/hyperactivity disorder (ADHD) and self-reported substance use symptoms in 303 adolescents enrolled in cognitive-behavioral therapy treatment for a substance use disorder and receiving either oral-methylphenidate or placebo for ADHD across 16 weeks. Assessing these 2 disorders concurrently allowed us to determine whether elevated levels of 1 disorder predicted elevated levels or increased risk of the other disorder. First, a piecewise growth model measured ADHD and substance use separately. Next, a parallel piecewise LGCM was used to estimate the regressions across disorders to determine whether higher scores at baseline of the disorders (i.e., ADHD or substance use disorder) predicted rates of change in the related disorder. Finally, treatment was added to the model to predict change. While the analyses revealed no significant relationships across disorders, this study explains and applies a parallel piecewise growth model to examine the developmental processes of comorbid conditions over the course of a clinical trial. Strengths of piecewise and parallel LGCMs for other addictions researchers interested in examining dual processes over time are discussed.
Mamey, Mary Rose; Barbosa-Leiker, Celestina; McPherson, Sterling; Burns, G Leonard; Parks, Craig; Roll, John
2015-12-01
Researchers often want to examine 2 comorbid conditions simultaneously. One strategy to do so is through the use of parallel latent growth curve modeling (LGCM). This statistical technique allows for the simultaneous evaluation of 2 disorders to determine the explanations and predictors of change over time. Additionally, a piecewise model can help identify whether there are more than 2 growth processes within each disorder (e.g., during a clinical trial). A parallel piecewise LGCM was applied to self-reported attention-deficit/hyperactivity disorder (ADHD) and self-reported substance use symptoms in 303 adolescents enrolled in cognitive-behavioral therapy treatment for a substance use disorder and receiving either oral-methylphenidate or placebo for ADHD across 16 weeks. Assessing these 2 disorders concurrently allowed us to determine whether elevated levels of 1 disorder predicted elevated levels or increased risk of the other disorder. First, a piecewise growth model measured ADHD and substance use separately. Next, a parallel piecewise LGCM was used to estimate the regressions across disorders to determine whether higher scores at baseline of the disorders (i.e., ADHD or substance use disorder) predicted rates of change in the related disorder. Finally, treatment was added to the model to predict change. While the analyses revealed no significant relationships across disorders, this study explains and applies a parallel piecewise growth model to examine the developmental processes of comorbid conditions over the course of a clinical trial. Strengths of piecewise and parallel LGCMs for other addictions researchers interested in examining dual processes over time are discussed. PMID:26389639
Approximate circuits for increased reliability
Hamlet, Jason R.; Mayo, Jackson R.
2015-12-22
Embodiments of the invention describe a Boolean circuit having a voter circuit and a plurality of approximate circuits each based, at least in part, on a reference circuit. The approximate circuits are each to generate one or more output signals based on values of received input signals. The voter circuit is to receive the one or more output signals generated by each of the approximate circuits, and is to output one or more signals corresponding to a majority value of the received signals. At least some of the approximate circuits are to generate an output value different than the reference circuit for one or more input signal values; however, for each possible input signal value, the majority values of the one or more output signals generated by the approximate circuits and received by the voter circuit correspond to output signal result values of the reference circuit.
Approximate circuits for increased reliability
Hamlet, Jason R.; Mayo, Jackson R.
2015-08-18
Embodiments of the invention describe a Boolean circuit having a voter circuit and a plurality of approximate circuits each based, at least in part, on a reference circuit. The approximate circuits are each to generate one or more output signals based on values of received input signals. The voter circuit is to receive the one or more output signals generated by each of the approximate circuits, and is to output one or more signals corresponding to a majority value of the received signals. At least some of the approximate circuits are to generate an output value different than the reference circuit for one or more input signal values; however, for each possible input signal value, the majority values of the one or more output signals generated by the approximate circuits and received by the voter circuit correspond to output signal result values of the reference circuit.
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.
Canards and curvature: nonsmooth approximation by pinching
NASA Astrophysics Data System (ADS)
Desroches, M.; Jeffrey, M. R.
2011-05-01
In multiple time-scale (singularly perturbed) dynamical systems, canards are counterintuitive solutions that evolve along both attracting and repelling invariant manifolds. In two dimensions, canards result in periodic oscillations whose amplitude and period grow in a highly nonlinear way: they are slowly varying with respect to a control parameter, except for an exponentially small range of values where they grow extremely rapidly. This sudden growth, called a canard explosion, has been encountered in many applications ranging from chemistry to neuronal dynamics, aerospace engineering and ecology. Canards were initially studied using nonstandard analysis, and later the same results were proved by standard techniques such as matched asymptotics, invariant manifold theory and parameter blow-up. More recently, canard-like behaviour has been linked to surfaces of discontinuity in piecewise-smooth dynamical systems. This paper provides a new perspective on the canard phenomenon by showing that the nonstandard analysis of canard explosions can be recast into the framework of piecewise-smooth dynamical systems. An exponential coordinate scaling is applied to a singularly perturbed system of ordinary differential equations. The scaling acts as a lens that resolves dynamics across all time-scales. The changes of local curvature that are responsible for canard explosions are then analysed. Regions where different time-scales dominate are separated by hypersurfaces, and these are pinched together to obtain a piecewise-smooth system, in which curvature changes manifest as discontinuity-induced bifurcations. The method is used to classify canards in arbitrary dimensions, and to derive the parameter values over which canards form either small cycles (canards without head) or large cycles (canards with head).
Linearized Kernel Dictionary Learning
NASA Astrophysics Data System (ADS)
Golts, Alona; Elad, Michael
2016-06-01
In this paper we present a new approach of incorporating kernels into dictionary learning. The kernel K-SVD algorithm (KKSVD), which has been introduced recently, shows an improvement in classification performance, with relation to its linear counterpart K-SVD. However, this algorithm requires the storage and handling of a very large kernel matrix, which leads to high computational cost, while also limiting its use to setups with small number of training examples. We address these problems by combining two ideas: first we approximate the kernel matrix using a cleverly sampled subset of its columns using the Nystr\\"{o}m method; secondly, as we wish to avoid using this matrix altogether, we decompose it by SVD to form new "virtual samples," on which any linear dictionary learning can be employed. Our method, termed "Linearized Kernel Dictionary Learning" (LKDL) can be seamlessly applied as a pre-processing stage on top of any efficient off-the-shelf dictionary learning scheme, effectively "kernelizing" it. We demonstrate the effectiveness of our method on several tasks of both supervised and unsupervised classification and show the efficiency of the proposed scheme, its easy integration and performance boosting properties.
On some applications of diophantine approximations
Chudnovsky, G. V.
1984-01-01
Siegel's results [Siegel, C. L. (1929) Abh. Preuss. Akad. Wiss. Phys.-Math. Kl. 1] on the transcendence and algebraic independence of values of E-functions are refined to obtain the best possible bound for the measures of irrationality and linear independence of values of arbitrary E-functions at rational points. Our results show that values of E-functions at rational points have measures of diophantine approximations typical to “almost all” numbers. In particular, any such number has the “2 + ε” exponent of irrationality: ǀΘ - p/qǀ > ǀqǀ-2-ε for relatively prime rational integers p,q, with q ≥ q0 (Θ, ε). These results answer some problems posed by Lang. The methods used here are based on the introduction of graded Padé approximations to systems of functions satisfying linear differential equations with rational function coefficients. The constructions and proofs of this paper were used in the functional (nonarithmetic case) in a previous paper [Chudnovsky, D. V. & Chudnovsky, G. V. (1983) Proc. Natl. Acad. Sci. USA 80, 5158-5162]. PMID:16593441
Online segmentation of time series based on polynomial least-squares approximations.
Fuchs, Erich; Gruber, Thiemo; Nitschke, Jiri; Sick, Bernhard
2010-12-01
The paper presents SwiftSeg, a novel technique for online time series segmentation and piecewise polynomial representation. The segmentation approach is based on a least-squares approximation of time series in sliding and/or growing time windows utilizing a basis of orthogonal polynomials. This allows the definition of fast update steps for the approximating polynomial, where the computational effort depends only on the degree of the approximating polynomial and not on the length of the time window. The coefficients of the orthogonal expansion of the approximating polynomial-obtained by means of the update steps-can be interpreted as optimal (in the least-squares sense) estimators for average, slope, curvature, change of curvature, etc., of the signal in the time window considered. These coefficients, as well as the approximation error, may be used in a very intuitive way to define segmentation criteria. The properties of SwiftSeg are evaluated by means of some artificial and real benchmark time series. It is compared to three different offline and online techniques to assess its accuracy and runtime. It is shown that SwiftSeg-which is suitable for many data streaming applications-offers high accuracy at very low computational costs. PMID:20975120
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.
Mathematical algorithms for approximate reasoning
NASA Technical Reports Server (NTRS)
Murphy, John H.; Chay, Seung C.; Downs, Mary M.
1988-01-01
Most state of the art expert system environments contain a single and often ad hoc strategy for approximate reasoning. Some environments provide facilities to program the approximate reasoning algorithms. However, the next generation of expert systems should have an environment which contain a choice of several mathematical algorithms for approximate reasoning. To meet the need for validatable and verifiable coding, the expert system environment must no longer depend upon ad hoc reasoning techniques but instead must include mathematically rigorous techniques for approximate reasoning. Popular approximate reasoning techniques are reviewed, including: certainty factors, belief measures, Bayesian probabilities, fuzzy logic, and Shafer-Dempster techniques for reasoning. A group of mathematically rigorous algorithms for approximate reasoning are focused on that could form the basis of a next generation expert system environment. These algorithms are based upon the axioms of set theory and probability theory. To separate these algorithms for approximate reasoning various conditions of mutual exclusivity and independence are imposed upon the assertions. Approximate reasoning algorithms presented include: reasoning with statistically independent assertions, reasoning with mutually exclusive assertions, reasoning with assertions that exhibit minimum overlay within the state space, reasoning with assertions that exhibit maximum overlay within the state space (i.e. fuzzy logic), pessimistic reasoning (i.e. worst case analysis), optimistic reasoning (i.e. best case analysis), and reasoning with assertions with absolutely no knowledge of the possible dependency among the assertions. A robust environment for expert system construction should include the two modes of inference: modus ponens and modus tollens. Modus ponens inference is based upon reasoning towards the conclusion in a statement of logical implication, whereas modus tollens inference is based upon reasoning away
Approximating random quantum optimization problems
NASA Astrophysics Data System (ADS)
Hsu, B.; Laumann, C. R.; Läuchli, A. M.; Moessner, R.; Sondhi, S. L.
2013-06-01
We report a cluster of results regarding the difficulty of finding approximate ground states to typical instances of the quantum satisfiability problem k-body quantum satisfiability (k-QSAT) on large random graphs. As an approximation strategy, we optimize the solution space over “classical” product states, which in turn introduces a novel autonomous classical optimization problem, PSAT, over a space of continuous degrees of freedom rather than discrete bits. Our central results are (i) the derivation of a set of bounds and approximations in various limits of the problem, several of which we believe may be amenable to a rigorous treatment; (ii) a demonstration that an approximation based on a greedy algorithm borrowed from the study of frustrated magnetism performs well over a wide range in parameter space, and its performance reflects the structure of the solution space of random k-QSAT. Simulated annealing exhibits metastability in similar “hard” regions of parameter space; and (iii) a generalization of belief propagation algorithms introduced for classical problems to the case of continuous spins. This yields both approximate solutions, as well as insights into the free energy “landscape” of the approximation problem, including a so-called dynamical transition near the satisfiability threshold. Taken together, these results allow us to elucidate the phase diagram of random k-QSAT in a two-dimensional energy-density-clause-density space.
Burgers approximation for two-dimensional flow past an ellipse
NASA Technical Reports Server (NTRS)
Dorrepaal, J. M.
1982-01-01
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 solution. 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 approximation 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 approximation. In the special case of flow past a circular cylinder, Burgers approximation predicts a boundary layer whose thickness is roughly proportional to R-1/2.
Approximate Bisimulation-Based Reduction of Power System Dynamic Models
Stankovic, AM; Dukic, SD; Saric, AT
2015-05-01
In this paper we propose approximate bisimulation relations and functions for reduction of power system dynamic models in differential- algebraic (descriptor) form. The full-size dynamic model is obtained by linearization of the nonlinear transient stability model. We generalize theoretical results on approximate bisimulation relations and bisimulation functions, originally derived for a class of constrained linear systems, to linear systems in descriptor form. An algorithm for transient stability assessment is proposed and used to determine whether the power system is able to maintain the synchronism after a large disturbance. Two benchmark power systems are used to illustrate the proposed algorithm and to evaluate the applicability of approximate bisimulation relations and bisimulation functions for reduction of the power system dynamic models.
Sparse Multinomial Logistic Regression via Approximate Message Passing
NASA Astrophysics Data System (ADS)
Byrne, Evan; Schniter, Philip
2016-11-01
For the problem of multi-class linear classification and feature selection, we propose approximate message passing approaches to sparse multinomial logistic regression (MLR). First, we propose two algorithms based on the Hybrid Generalized Approximate Message Passing (HyGAMP) framework: one finds the maximum a posteriori (MAP) linear classifier and the other finds an approximation of the test-error-rate minimizing linear classifier. Then we design computationally simplified variants of these two algorithms. Next, we detail methods to tune the hyperparameters of their assumed statistical models using Stein's unbiased risk estimate (SURE) and expectation-maximization (EM), respectively. Finally, using both synthetic and real-world datasets, we demonstrate improved error-rate and runtime performance relative to existing state-of-the-art approaches to sparse MLR.
Fractional Electron Loss in Approximate DFT and Hartree-Fock Theory.
Peach, Michael J G; Teale, Andrew M; Helgaker, Trygve; Tozer, David J
2015-11-10
Plots of electronic energy vs electron number, determined using approximate density functional theory (DFT) and Hartree-Fock theory, are typically piecewise convex and piecewise concave, respectively. The curves also commonly exhibit a minimum and maximum, respectively, in the neutral → anion segment, which lead to positive DFT anion HOMO energies and positive Hartree-Fock neutral LUMO energies. These minima/maxima are a consequence of using basis sets that are local to the system, preventing fractional electron loss. Ground-state curves are presented that illustrate the idealized behavior that would occur if the basis set were to be modified to enable fractional electron loss without changing the description in the vicinity of the system. The key feature is that the energy cannot increase when the electron number increases, so the slope cannot be anywhere positive, meaning frontier orbital energies cannot be positive. For the convex (DFT) case, the idealized curve is flat beyond a critical electron number such that any additional fraction of an electron added to the system is unbound. The anion HOMO energy is zero. For the concave (Hartree-Fock) case, the idealized curve is flat up to some critical electron number, beyond which it curves down to the anion energy. A minimum fraction of an electron is required before any binding occurs, but beyond that, the full fraction abruptly binds. The neutral LUMO energy is zero. Approximate DFT and Hartree-Fock results are presented for the F → F(-) segment, and results approaching the idealized behavior are recovered for highly diffuse basis sets. It is noted that if a DFT calculation using a highly diffuse basis set yields a negative LUMO energy then a fraction of an electron must bind and the electron affinity must be positive, irrespective of whether an electron binds experimentally. This is illustrated by calculations on Ne → Ne(-).
Waveform feature extraction based on tauberian approximation.
De Figueiredo, R J; Hu, C L
1982-02-01
A technique is presented for feature extraction of a waveform y based on its Tauberian approximation, that is, on the approximation of y by a linear combination of appropriately delayed versions of a single basis function x, i.e., y(t) = ¿M i = 1 aix(t - ¿i), where the coefficients ai and the delays ¿i are adjustable parameters. Considerations in the choice or design of the basis function x are given. The parameters ai and ¿i, i=1, . . . , M, are retrieved by application of a suitably adapted version of Prony's method to the Fourier transform of the above approximation of y. A subset of the parameters ai and ¿i, i = 1, . . . , M, is used to construct the feature vector, the value of which can be used in a classification algorithm. Application of this technique to the classification of wide bandwidth radar return signatures is presented. Computer simulations proved successful and are also discussed.
Generalized Quasilinear Approximation: Application to Zonal Jets.
Marston, J B; Chini, G P; Tobias, S M
2016-05-27
Quasilinear theory is often utilized to approximate the dynamics of fluids exhibiting significant interactions between mean flows and eddies. We present a generalization of quasilinear theory to include dynamic mode interactions on the large scales. This generalized quasilinear (GQL) approximation is achieved by separating the state variables into large and small zonal scales via a spectral filter rather than by a decomposition into a formal mean and fluctuations. Nonlinear interactions involving only small zonal scales are then removed. The approximation is conservative and allows for scattering of energy between small-scale modes via the large scale (through nonlocal spectral interactions). We evaluate GQL for the paradigmatic problems of the driving of large-scale jets on a spherical surface and on the beta plane and show that it is accurate even for a small number of large-scale modes. As GQL is formally linear in the small zonal scales, it allows for the closure of the system and can be utilized in direct statistical simulation schemes that have proved an attractive alternative to direct numerical simulation for many geophysical and astrophysical problems. PMID:27284660
Model reduction using new optimal Routh approximant technique
NASA Technical Reports Server (NTRS)
Hwang, Chyi; Guo, Tong-Yi; Sheih, Leang-San
1992-01-01
An optimal Routh approximant of a single-input single-output dynamic system is a reduced-order transfer function of which the denominator is obtained by the Routh approximation method while the numerator is determined by minimizing a time-response integral-squared-error (ISE) criterion. In this paper, a new elegant approach is presented for obtaining the optimal Routh approximants for linear time-invariant continuous-time systems. The approach is based on the Routh canonical expansion, which is a finite-term orthogonal series of rational basis functions, and minimization of the ISE criterion. A procedure for combining the above approach with the bilinear transformation is also presented in order to obtain the optimal bilinear Routh approximants of linear time-invariant discrete-time systems. The proposed technique is simple in formulation and is amenable to practical implementation.
Optimal feedback control infinite dimensional parabolic evolution systems: Approximation techniques
NASA Technical Reports Server (NTRS)
Banks, H. T.; Wang, C.
1989-01-01
A general approximation framework is discussed for computation of optimal feedback controls in linear quadratic regular problems for nonautonomous parabolic distributed parameter systems. This is done in the context of a theoretical framework using general evolution systems in infinite dimensional Hilbert spaces. Conditions are discussed for preservation under approximation of stabilizability and detectability hypotheses on the infinite dimensional system. The special case of periodic systems is also treated.
NASA Technical Reports Server (NTRS)
Lee, Jong-Won; Harris, Charles E.
1988-01-01
A mathematical model based on the Euler-Bernoulli beam theory is proposed for predicting the effective Young's moduli of piece-wise isotropic composite laminates with wavy patterns in the main load-carrying layers. Strains in corrugated layers, in-phase layers, and out-of-phase layers are predicted for various geometries and material configurations by assuming matrix layers as elastic foundations. Experimental results obtained from corrugated aluminum specimens and aluminum/epoxy specimens with in-phase and out-of-phase wavy patterns coincide very well with the predictions. The work represents a preliminary effort toward further generalization of the model for two-dimensional anisotropic laminates containing wavy patterns in the main load-carrying layers.
An improved LMI-based approach for stability of piecewise affine time-delay systems with uncertainty
NASA Astrophysics Data System (ADS)
Duan, Shiming; Ni, Jun; Galip Ulsoy, A.
2012-09-01
The stability problem for uncertain piecewise affine (PWA) time-delay systems is investigated in this article. It is assumed that there exists a known constant time delay in the system and the uncertainly is norm-bounded. Sufficient conditions for the stability of nominal systems and the stability of systems subject to uncertainty are derived using the Lyapunov-Krasovskii functional with a triple integration term. This approach handles switching based on the delayed states (in addition to the states) for a PWA time-delay system, considers structured as well as unstructured uncertainty and reduces the conservativeness of previous approaches. The effectiveness of the proposed approach is demonstrated by comparing with the existing methods through numerical examples.
Samet Y. Kadioglu
2011-12-01
We present a computational gas dynamics method based on the Spectral Deferred Corrections (SDC) time integration technique and the Piecewise Parabolic Method (PPM) finite volume method. The PPM framework is used to define edge averaged quantities which are then used to evaluate numerical flux functions. The SDC technique is used to integrate solution in time. This kind of approach was first taken by Anita et al in [17]. However, [17] is problematic when it is implemented to certain shock problems. Here we propose significant improvements to [17]. The method is fourth order (both in space and time) for smooth flows, and provides highly resolved discontinuous solutions. We tested the method by solving variety of problems. Results indicate that the fourth order of accuracy in both space and time has been achieved when the flow is smooth. Results also demonstrate the shock capturing ability of the method.
NASA Astrophysics Data System (ADS)
Krauze, W.; Makowski, P.; Kujawińska, M.
2015-06-01
Standard tomographic algorithms applied to optical limited-angle tomography result in the reconstructions that have highly anisotropic resolution and thus special algorithms are developed. State of the art approaches utilize the Total Variation (TV) minimization technique. These methods give very good results but are applicable to piecewise constant structures only. In this paper, we propose a novel algorithm for 3D limited-angle tomography - Total Variation Iterative Constraint method (TVIC) which enhances the applicability of the TV regularization to non-piecewise constant samples, like biological cells. This approach consists of two parts. First, the TV minimization is used as a strong regularizer to create a sharp-edged image converted to a 3D binary mask which is then iteratively applied in the tomographic reconstruction as a constraint in the object domain. In the present work we test the method on a synthetic object designed to mimic basic structures of a living cell. For simplicity, the test reconstructions were performed within the straight-line propagation model (SIRT3D solver from the ASTRA Tomography Toolbox), but the strategy is general enough to supplement any algorithm for tomographic reconstruction that supports arbitrary geometries of plane-wave projection acquisition. This includes optical diffraction tomography solvers. The obtained reconstructions present resolution uniformity and general shape accuracy expected from the TV regularization based solvers, but keeping the smooth internal structures of the object at the same time. Comparison between three different patterns of object illumination arrangement show very small impact of the projection acquisition geometry on the image quality.
Adaptive approximation models in optimization
Voronin, A.N.
1995-05-01
The paper proposes a method for optimization of functions of several variables that substantially reduces the number of objective function evaluations compared to traditional methods. The method is based on the property of iterative refinement of approximation models of the optimand function in approximation domains that contract to the extremum point. It does not require subjective specification of the starting point, step length, or other parameters of the search procedure. The method is designed for efficient optimization of unimodal functions of several (not more than 10-15) variables and can be applied to find the global extremum of polymodal functions and also for optimization of scalarized forms of vector objective functions.
NASA Astrophysics Data System (ADS)
Wang, Ronghao; Xing, Jianchun; Li, Juelong; Xiang, Zhengrong
2016-10-01
This paper studies the problem of stabilising a sampled-data switched linear system by quantised feedback asynchronously switched controllers. The idea of a quantised feedback asynchronously switched control strategy originates in earlier work reflecting actual system characteristic of switching and quantising, respectively. A quantised scheme is designed depending on switching time using dynamic quantiser. When sampling time, system switching time and controller switching time are all not uniform, the proposed switching controllers guarantee the system to be finite-time stable by a piecewise Lyapunov function and the average dwell-time method. Simulation examples are provided to show the effectiveness of the developed results.
NASA Astrophysics Data System (ADS)
Tian, Wenyi; Yuan, Xiaoming
2016-11-01
Linear inverse problems with total variation regularization can be reformulated as saddle-point problems; the primal and dual variables of such a saddle-point reformulation can be discretized in piecewise affine and constant finite element spaces, respectively. Thus, the well-developed primal-dual approach (a.k.a. the inexact Uzawa method) is conceptually applicable to such a regularized and discretized model. When the primal-dual approach is applied, the resulting subproblems may be highly nontrivial and it is necessary to discuss how to tackle them and thus make the primal-dual approach implementable. In this paper, we suggest linearizing the data-fidelity quadratic term of the hard subproblems so as to obtain easier ones. A linearized primal-dual method is thus proposed. Inspired by the fact that the linearized primal-dual method can be explained as an application of the proximal point algorithm, a relaxed version of the linearized primal-dual method, which can often accelerate the convergence numerically with the same order of computation, is also proposed. The global convergence and worst-case convergence rate measured by the iteration complexity are established for the new algorithms. Their efficiency is verified by some numerical results.
Pythagorean Approximations and Continued Fractions
ERIC Educational Resources Information Center
Peralta, Javier
2008-01-01
In this article, we will show that the Pythagorean approximations of [the square root of] 2 coincide with those achieved in the 16th century by means of continued fractions. Assuming this fact and the known relation that connects the Fibonacci sequence with the golden section, we shall establish a procedure to obtain sequences of rational numbers…
Error Bounds for Interpolative Approximations.
ERIC Educational Resources Information Center
Gal-Ezer, J.; Zwas, G.
1990-01-01
Elementary error estimation in the approximation of functions by polynomials as a computational assignment, error-bounding functions and error bounds, and the choice of interpolation points are discussed. Precalculus and computer instruction are used on some of the calculations. (KR)
Linearized Functional Minimization for Inverse Modeling
Wohlberg, Brendt; Tartakovsky, Daniel M.; Dentz, Marco
2012-06-21
Heterogeneous aquifers typically consist of multiple lithofacies, whose spatial arrangement significantly affects flow and transport. The estimation of these lithofacies is complicated by the scarcity of data and by the lack of a clear correlation between identifiable geologic indicators and attributes. We introduce a new inverse-modeling approach to estimate both the spatial extent of hydrofacies and their properties from sparse measurements of hydraulic conductivity and hydraulic head. Our approach is to minimize a functional defined on the vectors of values of hydraulic conductivity and hydraulic head fields defined on regular grids at a user-determined resolution. This functional is constructed to (i) enforce the relationship between conductivity and heads provided by the groundwater flow equation, (ii) penalize deviations of the reconstructed fields from measurements where they are available, and (iii) penalize reconstructed fields that are not piece-wise smooth. We develop an iterative solver for this functional that exploits a local linearization of the mapping from conductivity to head. This approach provides a computationally efficient algorithm that rapidly converges to a solution. A series of numerical experiments demonstrates the robustness of our approach.
Chemical Laws, Idealization and Approximation
NASA Astrophysics Data System (ADS)
Tobin, Emma
2013-07-01
This paper examines the notion of laws in chemistry. Vihalemm ( Found Chem 5(1):7-22, 2003) argues that the laws of chemistry are fundamentally the same as the laws of physics they are all ceteris paribus laws which are true "in ideal conditions". In contrast, Scerri (2000) contends that the laws of chemistry are fundamentally different to the laws of physics, because they involve approximations. Christie ( Stud Hist Philos Sci 25:613-629, 1994) and Christie and Christie ( Of minds and molecules. Oxford University Press, New York, pp. 34-50, 2000) agree that the laws of chemistry are operationally different to the laws of physics, but claim that the distinction between exact and approximate laws is too simplistic to taxonomise them. Approximations in chemistry involve diverse kinds of activity and often what counts as a scientific law in chemistry is dictated by the context of its use in scientific practice. This paper addresses the question of what makes chemical laws distinctive independently of the separate question as to how they are related to the laws of physics. From an analysis of some candidate ceteris paribus laws in chemistry, this paper argues that there are two distinct kinds of ceteris paribus laws in chemistry; idealized and approximate chemical laws. Thus, while Christie ( Stud Hist Philos Sci 25:613-629, 1994) and Christie and Christie ( Of minds and molecules. Oxford University Press, New York, pp. 34--50, 2000) are correct to point out that the candidate generalisations in chemistry are diverse and heterogeneous, a distinction between idealizations and approximations can nevertheless be used to successfully taxonomise them.
Taylor Approximations to Logistic IRT Models and Their Use in Adaptive Testing.
ERIC Educational Resources Information Center
Veerkamp, Wim J. J.
2000-01-01
Showed how Taylor approximation can be used to generate a linear approximation to a logistic item characteristic curve and a linear ability estimator. Demonstrated how, for a specific simulation, this could result in the special case of a Robbins-Monro item selection procedure for adaptive testing. (SLD)
Animal models and integrated nested Laplace approximations.
Holand, Anna Marie; Steinsland, Ingelin; Martino, Sara; Jensen, Henrik
2013-08-07
Animal models are generalized linear mixed models used in evolutionary biology and animal breeding to identify the genetic part of traits. Integrated Nested Laplace Approximation (INLA) is a methodology for making fast, nonsampling-based Bayesian inference for hierarchical Gaussian Markov models. In this article, we demonstrate that the INLA methodology can be used for many versions of Bayesian animal models. We analyze animal models for both synthetic case studies and house sparrow (Passer domesticus) population case studies with Gaussian, binomial, and Poisson likelihoods using INLA. Inference results are compared with results using Markov Chain Monte Carlo methods. For model choice we use difference in deviance information criteria (DIC). We suggest and show how to evaluate differences in DIC by comparing them with sampling results from simulation studies. We also introduce an R package, AnimalINLA, for easy and fast inference for Bayesian Animal models using INLA.
Approximate polynomial preconditioning applied to biharmonic equations on vector supercomputers
NASA Technical Reports Server (NTRS)
Wong, Yau Shu; Jiang, Hong
1987-01-01
Applying a finite difference approximation to a biharmonic equation results in a very ill-conditioned system of equations. This paper examines the conjugate gradient method used in conjunction with the generalized and approximate polynomial preconditionings for solving such linear systems. An approximate polynomial preconditioning is introduced, and is shown to be more efficient than the generalized polynomial preconditionings. This new technique provides a simple but effective preconditioning polynomial, which is based on another coefficient matrix rather than the original matrix operator as commonly used.
New approximation for the effective energy of nonlinear conducting composites
NASA Astrophysics Data System (ADS)
Gibiansky, Leonid; Torquato, Salvatore
1998-07-01
Approximations for the effective energy and, thus, effective conductivity of nonlinear, isotropic conducting dispersions are developed. This is accomplished by using the Ponte Castaneda variational principles [Philos. Trans. R. Soc. London Ser. A 340, 1321 (1992)] and the Torquato approximation [J. Appl. Phys. 58, 3790 (1985)] of the effective conductivity of corresponding linear composites. The results are obtained for dispersions with superconducting or insulating inclusions, and, more generally, for phases with a power-law energy. It is shown that the new approximations lie within the best available rigorous upper and lower bounds on the effective energy.
Airborne Linear Array Image Geometric Rectification Method Based on Unequal Segmentation
NASA Astrophysics Data System (ADS)
Li, J. M.; Li, C. R.; Zhou, M.; Hu, J.; Yang, C. M.
2016-06-01
As the linear array sensor such as multispectral and hyperspectral sensor has great potential in disaster monitoring and geological survey, the quality of the image geometric rectification should be guaranteed. Different from the geometric rectification of airborne planar array images or multi linear array images, exterior orientation elements need to be determined for each scan line of single linear array images. Internal distortion persists after applying GPS/IMU data directly to geometrical rectification. Straight lines may be curving and jagged. Straight line feature -based geometrical rectification algorithm was applied to solve this problem, whereby the exterior orientation elements were fitted by piecewise polynomial and evaluated with the straight line feature as constraint. However, atmospheric turbulence during the flight is unstable, equal piecewise can hardly provide good fitting, resulting in limited precision improvement of geometric rectification or, in a worse case, the iteration cannot converge. To solve this problem, drawing on dynamic programming ideas, unequal segmentation of line feature-based geometric rectification method is developed. The angle elements fitting error is minimized to determine the optimum boundary. Then the exterior orientation elements of each segment are fitted and evaluated with the straight line feature as constraint. The result indicates that the algorithm is effective in improving the precision of geometric rectification.
Testing the frozen flow approximation
NASA Technical Reports Server (NTRS)
Lucchin, Francesco; Matarrese, Sabino; Melott, Adrian L.; Moscardini, Lauro
1993-01-01
We investigate the accuracy of the frozen-flow approximation (FFA), recently proposed by Matarrese, et al. (1992), for following the nonlinear evolution of cosmological density fluctuations under gravitational instability. We compare a number of statistics between results of the FFA and n-body simulations, including those used by Melott, Pellman & Shandarin (1993) to test the Zel'dovich approximation. The FFA performs reasonably well in a statistical sense, e.g. in reproducing the counts-in-cell distribution, at small scales, but it does poorly in the crosscorrelation with n-body which means it is generally not moving mass to the right place, especially in models with high small-scale power.
Calculating Resonance Positions and Widths Using the Siegert Approximation Method
ERIC Educational Resources Information Center
Rapedius, Kevin
2011-01-01
Here, we present complex resonance states (or Siegert states) that describe the tunnelling decay of a trapped quantum particle from an intuitive point of view that naturally leads to the easily applicable Siegert approximation method. This can be used for analytical and numerical calculations of complex resonances of both the linear and nonlinear…
Approximate line shapes for hydrogen
NASA Technical Reports Server (NTRS)
Sutton, K.
1978-01-01
Two independent methods are presented for calculating radiative transport within hydrogen lines. In Method 1, a simple equation is proposed for calculating the line shape. In Method 2, the line shape is assumed to be a dispersion profile and an equation is presented for calculating the half half-width. The results obtained for the line shapes and curves of growth by the two approximate methods are compared with similar results using the detailed line shapes by Vidal et al.
Approximate reasoning using terminological models
NASA Technical Reports Server (NTRS)
Yen, John; Vaidya, Nitin
1992-01-01
Term Subsumption Systems (TSS) form a knowledge-representation scheme in AI that can express the defining characteristics of concepts through a formal language that has a well-defined semantics and incorporates a reasoning mechanism that can deduce whether one concept subsumes another. However, TSS's have very limited ability to deal with the issue of uncertainty in knowledge bases. The objective of this research is to address issues in combining approximate reasoning with term subsumption systems. To do this, we have extended an existing AI architecture (CLASP) that is built on the top of a term subsumption system (LOOM). First, the assertional component of LOOM has been extended for asserting and representing uncertain propositions. Second, we have extended the pattern matcher of CLASP for plausible rule-based inferences. Third, an approximate reasoning model has been added to facilitate various kinds of approximate reasoning. And finally, the issue of inconsistency in truth values due to inheritance is addressed using justification of those values. This architecture enhances the reasoning capabilities of expert systems by providing support for reasoning under uncertainty using knowledge captured in TSS. Also, as definitional knowledge is explicit and separate from heuristic knowledge for plausible inferences, the maintainability of expert systems could be improved.
Computer Experiments for Function Approximations
Chang, A; Izmailov, I; Rizzo, S; Wynter, S; Alexandrov, O; Tong, C
2007-10-15
This research project falls in the domain of response surface methodology, which seeks cost-effective ways to accurately fit an approximate function to experimental data. Modeling and computer simulation are essential tools in modern science and engineering. A computer simulation can be viewed as a function that receives input from a given parameter space and produces an output. Running the simulation repeatedly amounts to an equivalent number of function evaluations, and for complex models, such function evaluations can be very time-consuming. It is then of paramount importance to intelligently choose a relatively small set of sample points in the parameter space at which to evaluate the given function, and then use this information to construct a surrogate function that is close to the original function and takes little time to evaluate. This study was divided into two parts. The first part consisted of comparing four sampling methods and two function approximation methods in terms of efficiency and accuracy for simple test functions. The sampling methods used were Monte Carlo, Quasi-Random LP{sub {tau}}, Maximin Latin Hypercubes, and Orthogonal-Array-Based Latin Hypercubes. The function approximation methods utilized were Multivariate Adaptive Regression Splines (MARS) and Support Vector Machines (SVM). The second part of the study concerned adaptive sampling methods with a focus on creating useful sets of sample points specifically for monotonic functions, functions with a single minimum and functions with a bounded first derivative.
Ultrafast approximation for phylogenetic bootstrap.
Minh, Bui Quang; Nguyen, Minh Anh Thi; von Haeseler, Arndt
2013-05-01
Nonparametric bootstrap has been a widely used tool in phylogenetic analysis to assess the clade support of phylogenetic trees. However, with the rapidly growing amount of data, this task remains a computational bottleneck. Recently, approximation methods such as the RAxML rapid bootstrap (RBS) and the Shimodaira-Hasegawa-like approximate likelihood ratio test have been introduced to speed up the bootstrap. Here, we suggest an ultrafast bootstrap approximation approach (UFBoot) to compute the support of phylogenetic groups in maximum likelihood (ML) based trees. To achieve this, we combine the resampling estimated log-likelihood method with a simple but effective collection scheme of candidate trees. We also propose a stopping rule that assesses the convergence of branch support values to automatically determine when to stop collecting candidate trees. UFBoot achieves a median speed up of 3.1 (range: 0.66-33.3) to 10.2 (range: 1.32-41.4) compared with RAxML RBS for real DNA and amino acid alignments, respectively. Moreover, our extensive simulations show that UFBoot is robust against moderate model violations and the support values obtained appear to be relatively unbiased compared with the conservative standard bootstrap. This provides a more direct interpretation of the bootstrap support. We offer an efficient and easy-to-use software (available at http://www.cibiv.at/software/iqtree) to perform the UFBoot analysis with ML tree inference.
Approximate Counting of Graphical Realizations
2015-01-01
In 1999 Kannan, Tetali and Vempala proposed a MCMC method to uniformly sample all possible realizations of a given graphical degree sequence and conjectured its rapidly mixing nature. Recently their conjecture was proved affirmative for regular graphs (by Cooper, Dyer and Greenhill, 2007), for regular directed graphs (by Greenhill, 2011) and for half-regular bipartite graphs (by Miklós, Erdős and Soukup, 2013). Several heuristics on counting the number of possible realizations exist (via sampling processes), and while they work well in practice, so far no approximation guarantees exist for such an approach. This paper is the first to develop a method for counting realizations with provable approximation guarantee. In fact, we solve a slightly more general problem; besides the graphical degree sequence a small set of forbidden edges is also given. We show that for the general problem (which contains the Greenhill problem and the Miklós, Erdős and Soukup problem as special cases) the derived MCMC process is rapidly mixing. Further, we show that this new problem is self-reducible therefore it provides a fully polynomial randomized approximation scheme (a.k.a. FPRAS) for counting of all realizations. PMID:26161994
Sensitivity analysis and approximation methods for general eigenvalue problems
NASA Technical Reports Server (NTRS)
Murthy, D. V.; Haftka, R. T.
1986-01-01
Optimization of dynamic systems involving complex non-hermitian matrices is often computationally expensive. Major contributors to the computational expense are the sensitivity analysis and reanalysis of a modified design. The present work seeks to alleviate this computational burden by identifying efficient sensitivity analysis and approximate reanalysis methods. For the algebraic eigenvalue problem involving non-hermitian matrices, algorithms for sensitivity analysis and approximate reanalysis are classified, compared and evaluated for efficiency and accuracy. Proper eigenvector normalization is discussed. An improved method for calculating derivatives of eigenvectors is proposed based on a more rational normalization condition and taking advantage of matrix sparsity. Important numerical aspects of this method are also discussed. To alleviate the problem of reanalysis, various approximation methods for eigenvalues are proposed and evaluated. Linear and quadratic approximations are based directly on the Taylor series. Several approximation methods are developed based on the generalized Rayleigh quotient for the eigenvalue problem. Approximation methods based on trace theorem give high accuracy without needing any derivatives. Operation counts for the computation of the approximations are given. General recommendations are made for the selection of appropriate approximation technique as a function of the matrix size, number of design variables, number of eigenvalues of interest and the number of design points at which approximation is sought.
On applications of diophantine approximations
Chudnovsky, G. V.
1984-01-01
This paper is devoted to the study of the arithmetic properties of values of G-functions introduced by Siegel [Siegel, C. L. (1929) Abh. Preuss. Akad. Wiss. Phys.-Math. Kl. 1]. One of the main results is a theorem on the linear independence of values of G-functions at rational points close to the origin. In this theorem, no conditions are imposed on the p-adic convergence of a G-function at a generic point. The theorem finally realizes Siegel's program on G-function values outlined in his paper. PMID:16593530
Approximately Independent Features of Languages
NASA Astrophysics Data System (ADS)
Holman, Eric W.
To facilitate the testing of models for the evolution of languages, the present paper offers a set of linguistic features that are approximately independent of each other. To find these features, the adjusted Rand index (R‧) is used to estimate the degree of pairwise relationship among 130 linguistic features in a large published database. Many of the R‧ values prove to be near zero, as predicted for independent features, and a subset of 47 features is found with an average R‧ of -0.0001. These 47 features are recommended for use in statistical tests that require independent units of analysis.
The structural physical approximation conjecture
NASA Astrophysics Data System (ADS)
Shultz, Fred
2016-01-01
It was conjectured that the structural physical approximation (SPA) of an optimal entanglement witness is separable (or equivalently, that the SPA of an optimal positive map is entanglement breaking). This conjecture was disproved, first for indecomposable maps and more recently for decomposable maps. The arguments in both cases are sketched along with important related results. This review includes background material on topics including entanglement witnesses, optimality, duality of cones, decomposability, and the statement and motivation for the SPA conjecture so that it should be accessible for a broad audience.
Quantum tunneling beyond semiclassical approximation
NASA Astrophysics Data System (ADS)
Banerjee, Rabin; Ranjan Majhi, Bibhas
2008-06-01
Hawking radiation as tunneling by Hamilton-Jacobi method beyond semiclassical approximation is analysed. We compute all quantum corrections in the single particle action revealing that these are proportional to the usual semiclassical contribution. We show that a simple choice of the proportionality constants reproduces the one loop back reaction effect in the spacetime, found by conformal field theory methods, which modifies the Hawking temperature of the black hole. Using the law of black hole mechanics we give the corrections to the Bekenstein-Hawking area law following from the modified Hawking temperature. Some examples are explicitly worked out.
Fermion tunneling beyond semiclassical approximation
NASA Astrophysics Data System (ADS)
Majhi, Bibhas Ranjan
2009-02-01
Applying the Hamilton-Jacobi method beyond the semiclassical approximation prescribed in R. Banerjee and B. R. Majhi, J. High Energy Phys.JHEPFG1029-8479 06 (2008) 09510.1088/1126-6708/2008/06/095 for the scalar particle, Hawking radiation as tunneling of the Dirac particle through an event horizon is analyzed. We show that, as before, all quantum corrections in the single particle action are proportional to the usual semiclassical contribution. We also compute the modifications to the Hawking temperature and Bekenstein-Hawking entropy for the Schwarzschild black hole. Finally, the coefficient of the logarithmic correction to entropy is shown to be related with the trace anomaly.
NASA Astrophysics Data System (ADS)
Chiumenti, M.; Cervera, M.; Agelet de Saracibar, C.; Dialami, N.
2013-05-01
In this work a novel finite element technology based on a three-field mixed formulation is presented. The Variational Multi Scale (VMS) method is used to circumvent the LBB stability condition allowing the use of linear piece-wise interpolations for displacement, stress and pressure fields, respectively. The result is an enhanced stress field approximation which enables for stress-accurate results in nonlinear computational mechanics. The use of an independent nodal variable for the pressure field allows for an adhoc treatment of the incompressibility constraint. This is a mandatory requirement due to the isochoric nature of the plastic strain in metal forming processes. The highly non-linear stress field typically encountered in the Friction Stir Welding (FSW) process is used as an example to show the performance of this new FE technology. The numerical simulation of the FSW process is tackled by means of an Arbitrary-Lagrangian-Eulerian (ALE) formulation. The computational domain is split into three different zones: the work.piece (defined by a rigid visco-plastic behaviour in the Eulerian framework), the pin (within the Lagrangian framework) and finally the stirzone (ALE formulation). A fully coupled thermo-mechanical analysis is introduced showing the heat fluxes generated by the plastic dissipation in the stir-zone (Sheppard rigid-viscoplastic constitutive model) as well as the frictional dissipation at the contact interface (Norton frictional contact model). Finally, tracers have been implemented to show the material flow around the pin allowing a better understanding of the welding mechanism. Numerical results are compared with experimental evidence.
Optimal damping of fast-linear oscillations of stationary satellite with the flywheel
NASA Astrophysics Data System (ADS)
Babadzanjanz, Levon K.; Pototskaya, Irina Yu.; Pupysheva, Yulia Yu.
2016-06-01
The problem of damping of the satellite fast-linear oscillations about its center of mass on yaw and roll channels is considered. The satellite is moving along a stationary orbit and equipped with a hard spun flywheel with a kinematic momentum H. The controlled motion of the satellite can be represented by the linear ODE system with constant coefficients. The admissible control is a piecewise constant function that blanks fast frequency components of the solution of linear equations at the moment T. As "the expenditure" functional we use the integral of the sum of the modules of coordinates of the control along the interval [0, T]. The problem under consideration is to construct an admissible control which minimizes the expenditure. To solve this problem the method is proposed which leads to explicit formulas.
NASA Technical Reports Server (NTRS)
Banks, H. T.; Kunisch, K.
1982-01-01
Approximation results from linear semigroup theory are used to develop a general framework for convergence of approximation schemes in parameter estimation and optimal control problems for nonlinear partial differential equations. These ideas are used to establish theoretical convergence results for parameter identification using modal (eigenfunction) approximation techniques. Results from numerical investigations of these schemes for both hyperbolic and parabolic systems are given.
Plasma Physics Approximations in Ares
Managan, R. A.
2015-01-08
Lee & More derived analytic forms for the transport properties of a plasma. Many hydro-codes use their formulae for electrical and thermal conductivity. The coefficients are complex functions of Fermi-Dirac integrals, F_{n}( μ/θ ), the chemical potential, μ or ζ = ln(1+e^{ μ/θ} ), and the temperature, θ = kT. Since these formulae are expensive to compute, rational function approximations were fit to them. Approximations are also used to find the chemical potential, either μ or ζ . The fits use ζ as the independent variable instead of μ/θ . New fits are provided for A^{α} (ζ ),A^{β} (ζ ), ζ, f(ζ ) = (1 + e^{-μ/θ})F_{1/2}(μ/θ), F_{1/2}'/F_{1/2}, F_{c}^{α}, and F_{c}^{β}. In each case the relative error of the fit is minimized since the functions can vary by many orders of magnitude. The new fits are designed to exactly preserve the limiting values in the non-degenerate and highly degenerate limits or as ζ→ 0 or ∞. The original fits due to Lee & More and George Zimmerman are presented for comparison.
Implementing Linear Algebra Related Algorithms on the TI-92+ Calculator.
ERIC Educational Resources Information Center
Alexopoulos, John; Abraham, Paul
2001-01-01
Demonstrates a less utilized feature of the TI-92+: its natural and powerful programming language. Shows how to implement several linear algebra related algorithms including the Gram-Schmidt process, Least Squares Approximations, Wronskians, Cholesky Decompositions, and Generalized Linear Least Square Approximations with QR Decompositions.…
Approximating metal-insulator transitions
NASA Astrophysics Data System (ADS)
Danieli, Carlo; Rayanov, Kristian; Pavlov, Boris; Martin, Gaven; Flach, Sergej
2015-12-01
We consider quantum wave propagation in one-dimensional quasiperiodic lattices. We propose an iterative construction of quasiperiodic potentials from sequences of potentials with increasing spatial period. At each finite iteration step, the eigenstates reflect the properties of the limiting quasiperiodic potential properties up to a controlled maximum system size. We then observe approximate Metal-Insulator Transitions (MIT) at the finite iteration steps. We also report evidence on mobility edges, which are at variance to the celebrated Aubry-André model. The dynamics near the MIT shows a critical slowing down of the ballistic group velocity in the metallic phase, similar to the divergence of the localization length in the insulating phase.
Interplay of approximate planning strategies.
Huys, Quentin J M; Lally, Níall; Faulkner, Paul; Eshel, Neir; Seifritz, Erich; Gershman, Samuel J; Dayan, Peter; Roiser, Jonathan P
2015-03-10
Humans routinely formulate plans in domains so complex that even the most powerful computers are taxed. To do so, they seem to avail themselves of many strategies and heuristics that efficiently simplify, approximate, and hierarchically decompose hard tasks into simpler subtasks. Theoretical and cognitive research has revealed several such strategies; however, little is known about their establishment, interaction, and efficiency. Here, we use model-based behavioral analysis to provide a detailed examination of the performance of human subjects in a moderately deep planning task. We find that subjects exploit the structure of the domain to establish subgoals in a way that achieves a nearly maximal reduction in the cost of computing values of choices, but then combine partial searches with greedy local steps to solve subtasks, and maladaptively prune the decision trees of subtasks in a reflexive manner upon encountering salient losses. Subjects come idiosyncratically to favor particular sequences of actions to achieve subgoals, creating novel complex actions or "options." PMID:25675480
Indexing the approximate number system.
Inglis, Matthew; Gilmore, Camilla
2014-01-01
Much recent research attention has focused on understanding individual differences in the approximate number system, a cognitive system believed to underlie human mathematical competence. To date researchers have used four main indices of ANS acuity, and have typically assumed that they measure similar properties. Here we report a study which questions this assumption. We demonstrate that the numerical ratio effect has poor test-retest reliability and that it does not relate to either Weber fractions or accuracy on nonsymbolic comparison tasks. Furthermore, we show that Weber fractions follow a strongly skewed distribution and that they have lower test-retest reliability than a simple accuracy measure. We conclude by arguing that in the future researchers interested in indexing individual differences in ANS acuity should use accuracy figures, not Weber fractions or numerical ratio effects. PMID:24361686
IONIS: Approximate atomic photoionization intensities
NASA Astrophysics Data System (ADS)
Heinäsmäki, Sami
2012-02-01
A program to compute relative atomic photoionization cross sections is presented. The code applies the output of the multiconfiguration Dirac-Fock method for atoms in the single active electron scheme, by computing the overlap of the bound electron states in the initial and final states. The contribution from the single-particle ionization matrix elements is assumed to be the same for each final state. This method gives rather accurate relative ionization probabilities provided the single-electron ionization matrix elements do not depend strongly on energy in the region considered. The method is especially suited for open shell atoms where electronic correlation in the ionic states is large. Program summaryProgram title: IONIS Catalogue identifier: AEKK_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEKK_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 1149 No. of bytes in distributed program, including test data, etc.: 12 877 Distribution format: tar.gz Programming language: Fortran 95 Computer: Workstations Operating system: GNU/Linux, Unix Classification: 2.2, 2.5 Nature of problem: Photoionization intensities for atoms. Solution method: The code applies the output of the multiconfiguration Dirac-Fock codes Grasp92 [1] or Grasp2K [2], to compute approximate photoionization intensities. The intensity is computed within the one-electron transition approximation and by assuming that the sum of the single-particle ionization probabilities is the same for all final ionic states. Restrictions: The program gives nonzero intensities for those transitions where only one electron is removed from the initial configuration(s). Shake-type many-electron transitions are not computed. The ionized shell must be closed in the initial state. Running time: Few seconds for a
NASA Astrophysics Data System (ADS)
Alves, Julio Cesar L.; Poppi, Ronei J.
2013-02-01
This paper reports the application of piecewise direct standardization (PDS) for matrix correction in front face fluorescence spectroscopy of solids when different excipients are used in a pharmaceutical preparation based on a mixture of acetylsalicylic acid (ASA), paracetamol (acetaminophen) and caffeine. As verified in earlier studies, the use of different excipients and their ratio can cause a displacement, change in fluorescence intensity or band profile. To overcome this important drawback, a standardization strategy was adopted to convert all the excitation-emission fluorescence spectra into those used for model development. An excitation-emission matrix (EEM) for which excitation and emission wavelengths ranging from 265 to 405 nm and 300 to 480 nm, respectively, was used. Excellent results were obtained using unfolded partial least squares (U-PLS), with RMSEP values of 8.2 mg/g, 10.9 mg/g and 2.7 mg/g for ASA, paracetamol and caffeine, respectively, and with relative errors lesser than 5% for the three analytes.
Wang, Lu; Xu, Lisheng; Zhao, Dazhe; Yao, Yang; Song, Dan
2015-04-01
Because arterial pulse waves contain vital information related to the condition of the cardiovascular system, considerable attention has been devoted to the study of pulse waves in recent years. Accurate acquisition is essential to investigate arterial pulse waves. However, at the stage of developing equipment for acquiring and analyzing arterial pulse waves, specific pulse signals may be unavailable for debugging and evaluating the system under development. To produce test signals that reflect specific physiological conditions, in this paper, an arterial pulse wave generator has been designed and implemented using a field programmable gate array (FPGA), which can produce the desired pulse waves according to the feature points set by users. To reconstruct a periodic pulse wave from the given feature points, a method known as piecewise Gaussian-cosine fitting is also proposed in this paper. Using a test database that contains four types of typical pulse waves with each type containing 25 pulse wave signals, the maximum residual error of each sampling point of the fitted pulse wave in comparison with the real pulse wave is within 8%. In addition, the function for adding baseline drift and three types of noises is integrated into the developed system because the baseline occasionally wanders, and noise needs to be added for testing the performance of the designed circuits and the analysis algorithms. The proposed arterial pulse wave generator can be considered as a special signal generator with a simple structure, low cost and compact size, which can also provide flexible solutions for many other related research purposes.
Approximate Riemann solvers for the Godunov SPH (GSPH)
NASA Astrophysics Data System (ADS)
Puri, Kunal; Ramachandran, Prabhu
2014-08-01
The Godunov Smoothed Particle Hydrodynamics (GSPH) method is coupled with non-iterative, approximate Riemann solvers for solutions to the compressible Euler equations. The use of approximate solvers avoids the expensive solution of the non-linear Riemann problem for every interacting particle pair, as required by GSPH. In addition, we establish an equivalence between the dissipative terms of GSPH and the signal based SPH artificial viscosity, under the restriction of a class of approximate Riemann solvers. This equivalence is used to explain the anomalous “wall heating” experienced by GSPH and we provide some suggestions to overcome it. Numerical tests in one and two dimensions are used to validate the proposed Riemann solvers. A general SPH pairing instability is observed for two-dimensional problems when using unequal mass particles. In general, Ducowicz Roe's and HLLC approximate Riemann solvers are found to be suitable replacements for the iterative Riemann solver in the original GSPH scheme.
Linear stability of shock profiles for systems of conservation laws with semi-linear relaxation
NASA Astrophysics Data System (ADS)
Godillon, Pauline
2001-01-01
The Evans function theory, which has recently been applied to the study of linear stability of viscous shock profiles, is developed below for semi-linear relaxation. We study the linear stability of shock profiles in the Lax, undercompressive and overcompressive cases. The results we obtain are similar to those found for viscous approximations by Gardner and Zumbrun [Commun. Pure Appl. Math. 51 (7) (1998) 797].
Bugeanu, Monica; Di Remigio, Roberto; Mozgawa, Krzysztof; Reine, Simen Sommerfelt; Harbrecht, Helmut; Frediani, Luca
2015-12-21
The simplicity of dielectric continuum models has made them a standard tool in almost any Quantum Chemistry (QC) package. Despite being intuitive from a physical point of view, the actual electrostatic problem at the cavity boundary is challenging: the underlying boundary integral equations depend on singular, long-range operators. The parametrization of the cavity boundary should be molecular-shaped, smooth and differentiable. Even the most advanced implementations, based on the integral equation formulation (IEF) of the polarizable continuum model (PCM), generally lead to working equations which do not guarantee convergence to the exact solution and/or might become numerically unstable in the limit of large refinement of the molecular cavity (small tesserae). This is because they generally make use of a surface parametrization with cusps (interlocking spheres) and employ collocation methods for the discretization (point charges). Wavelets on a smooth cavity are an attractive alternative to consider: for the operators involved, they lead to highly sparse matrices and precise error control. Moreover, by making use of a bilinear basis for the representation of operators and functions on the cavity boundary, all equations can be differentiated to enable the computation of geometrical derivatives. In this contribution, we present our implementation of the IEFPCM with bilinear wavelets on a smooth cavity boundary. The implementation has been carried out in our module PCMSolver and interfaced with LSDalton, demonstrating the accuracy of the method both for the electrostatic solvation energy and for linear response properties. In addition, the implementation in a module makes our framework readily available to any QC software with minimal effort. PMID:26256401
Liu, Jian; Miller, William H.
2006-09-06
The thermal Gaussian approximation (TGA) recently developed by Mandelshtam et al has been demonstrated to be a practical way for approximating the Boltzmann operator exp(-{beta}H) for multidimensional systems. In this paper the TGA is combined with semiclassical (SC) initial value representations (IVRs) for thermal time correlation functions. Specifically, it is used with the linearized SC-IVR (LSC-IVR, equivalent to the classical Wigner model), and the 'forward-backward semiclassical dynamics' (FBSD) approximation developed by Makri et al. Use of the TGA with both of these approximate SC-IVRs allows the oscillatory part of the IVR to be integrated out explicitly, providing an extremely simple result that is readily applicable to large molecular systems. Calculation of the force-force autocorrelation for a strongly anharmonic oscillator demonstrates its accuracy, and of the velocity autocorrelation function (and thus the diffusion coefficient) of liquid neon demonstrates its applicability.
Approximate Schur complement preconditioning of the lowest order nodal discretizations
Moulton, J.D.; Ascher, U.M.; Morel, J.E.
1996-12-31
Particular classes of nodal methods and mixed hybrid finite element methods lead to equivalent, robust and accurate discretizations of 2nd order elliptic PDEs. However, widespread popularity of these discretizations has been hindered by the awkward linear systems which result. The present work exploits this awkwardness, which provides a natural partitioning of the linear system, by defining two optimal preconditioners based on approximate Schur complements. Central to the optimal performance of these preconditioners is their sparsity structure which is compatible with Dendy`s black box multigrid code.
Multidimensional stochastic approximation Monte Carlo.
Zablotskiy, Sergey V; Ivanov, Victor A; Paul, Wolfgang
2016-06-01
Stochastic Approximation Monte Carlo (SAMC) has been established as a mathematically founded powerful flat-histogram Monte Carlo method, used to determine the density of states, g(E), of a model system. We show here how it can be generalized for the determination of multidimensional probability distributions (or equivalently densities of states) of macroscopic or mesoscopic variables defined on the space of microstates of a statistical mechanical system. This establishes this method as a systematic way for coarse graining a model system, or, in other words, for performing a renormalization group step on a model. We discuss the formulation of the Kadanoff block spin transformation and the coarse-graining procedure for polymer models in this language. We also apply it to a standard case in the literature of two-dimensional densities of states, where two competing energetic effects are present g(E_{1},E_{2}). We show when and why care has to be exercised when obtaining the microcanonical density of states g(E_{1}+E_{2}) from g(E_{1},E_{2}). PMID:27415383
Interplay of approximate planning strategies
Huys, Quentin J. M.; Lally, Níall; Faulkner, Paul; Eshel, Neir; Seifritz, Erich; Gershman, Samuel J.; Dayan, Peter; Roiser, Jonathan P.
2015-01-01
Humans routinely formulate plans in domains so complex that even the most powerful computers are taxed. To do so, they seem to avail themselves of many strategies and heuristics that efficiently simplify, approximate, and hierarchically decompose hard tasks into simpler subtasks. Theoretical and cognitive research has revealed several such strategies; however, little is known about their establishment, interaction, and efficiency. Here, we use model-based behavioral analysis to provide a detailed examination of the performance of human subjects in a moderately deep planning task. We find that subjects exploit the structure of the domain to establish subgoals in a way that achieves a nearly maximal reduction in the cost of computing values of choices, but then combine partial searches with greedy local steps to solve subtasks, and maladaptively prune the decision trees of subtasks in a reflexive manner upon encountering salient losses. Subjects come idiosyncratically to favor particular sequences of actions to achieve subgoals, creating novel complex actions or “options.” PMID:25675480
Multidimensional stochastic approximation Monte Carlo
NASA Astrophysics Data System (ADS)
Zablotskiy, Sergey V.; Ivanov, Victor A.; Paul, Wolfgang
2016-06-01
Stochastic Approximation Monte Carlo (SAMC) has been established as a mathematically founded powerful flat-histogram Monte Carlo method, used to determine the density of states, g (E ) , of a model system. We show here how it can be generalized for the determination of multidimensional probability distributions (or equivalently densities of states) of macroscopic or mesoscopic variables defined on the space of microstates of a statistical mechanical system. This establishes this method as a systematic way for coarse graining a model system, or, in other words, for performing a renormalization group step on a model. We discuss the formulation of the Kadanoff block spin transformation and the coarse-graining procedure for polymer models in this language. We also apply it to a standard case in the literature of two-dimensional densities of states, where two competing energetic effects are present g (E1,E2) . We show when and why care has to be exercised when obtaining the microcanonical density of states g (E1+E2) from g (E1,E2) .
Semiclassics beyond the diagonal approximation
NASA Astrophysics Data System (ADS)
Turek, Marko
2004-05-01
The statistical properties of the energy spectrum of classically chaotic closed quantum systems are the central subject of this thesis. It has been conjectured by O.Bohigas, M.-J.Giannoni and C.Schmit that the spectral statistics of chaotic systems is universal and can be described by random-matrix theory. This conjecture has been confirmed in many experiments and numerical studies but a formal proof is still lacking. In this thesis we present a semiclassical evaluation of the spectral form factor which goes beyond M.V.Berry's diagonal approximation. To this end we extend a method developed by M.Sieber and K.Richter for a specific system: the motion of a particle on a two-dimensional surface of constant negative curvature. In particular we prove that these semiclassical methods reproduce the random-matrix theory predictions for the next to leading order correction also for a much wider class of systems, namely non-uniformly hyperbolic systems with f>2 degrees of freedom. We achieve this result by extending the configuration-space approach of M.Sieber and K.Richter to a canonically invariant phase-space approach.
Randomized approximate nearest neighbors algorithm.
Jones, Peter Wilcox; Osipov, Andrei; Rokhlin, Vladimir
2011-09-20
We present a randomized algorithm for the approximate nearest neighbor problem in d-dimensional Euclidean space. Given N points {x(j)} in R(d), the algorithm attempts to find k nearest neighbors for each of x(j), where k is a user-specified integer parameter. The algorithm is iterative, and its running time requirements are proportional to T·N·(d·(log d) + k·(d + log k)·(log N)) + N·k(2)·(d + log k), with T the number of iterations performed. The memory requirements of the procedure are of the order N·(d + k). A by-product of the scheme is a data structure, permitting a rapid search for the k nearest neighbors among {x(j)} for an arbitrary point x ∈ R(d). The cost of each such query is proportional to T·(d·(log d) + log(N/k)·k·(d + log k)), and the memory requirements for the requisite data structure are of the order N·(d + k) + T·(d + N). The algorithm utilizes random rotations and a basic divide-and-conquer scheme, followed by a local graph search. We analyze the scheme's behavior for certain types of distributions of {x(j)} and illustrate its performance via several numerical examples.
A quantum relaxation-time approximation for finite fermion systems
Reinhard, P.-G.; Suraud, E.
2015-03-15
We propose a relaxation time approximation for the description of the dynamics of strongly excited fermion systems. Our approach is based on time-dependent density functional theory at the level of the local density approximation. This mean-field picture is augmented by collisional correlations handled in relaxation time approximation which is inspired from the corresponding semi-classical picture. The method involves the estimate of microscopic relaxation rates/times which is presently taken from the well established semi-classical experience. The relaxation time approximation implies evaluation of the instantaneous equilibrium state towards which the dynamical state is progressively driven at the pace of the microscopic relaxation time. As test case, we consider Na clusters of various sizes excited either by a swift ion projectile or by a short and intense laser pulse, driven in various dynamical regimes ranging from linear to strongly non-linear reactions. We observe a strong effect of dissipation on sensitive observables such as net ionization and angular distributions of emitted electrons. The effect is especially large for moderate excitations where typical relaxation/dissipation time scales efficiently compete with ionization for dissipating the available excitation energy. Technical details on the actual procedure to implement a working recipe of such a quantum relaxation approximation are given in appendices for completeness.
Decoupling approximation design using the peak to peak gain
NASA Astrophysics Data System (ADS)
Sultan, Cornel
2013-04-01
Linear system design for accurate decoupling approximation is examined using the peak to peak gain of the error system. The design problem consists in finding values of system parameters to ensure that this gain is small. For this purpose a computationally inexpensive upper bound on the peak to peak gain, namely the star norm, is minimized using a stochastic method. Examples of the methodology's application to tensegrity structures design are presented. Connections between the accuracy of the approximation, the damping matrix, and the natural frequencies of the system are examined, as well as decoupling in the context of open and closed loop control.
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.
Comparison of dynamical approximation schemes for nonlinear gravitaional clustering
NASA Technical Reports Server (NTRS)
Melott, Adrian L.
1994-01-01
We have recently conducted a controlled comparison of a number of approximations for gravitational clustering against the same n-body simulations. These include ordinary linear perturbation theory (Eulerian), the lognormal approximation, the adhesion approximation, the frozen-flow approximation, the Zel'dovich approximation (describable as first-order Lagrangian perturbation theory), and its second-order generalization. In the last two cases we also created new versions of the approximation by truncation, i.e., by smoothing the initial conditions with various smoothing window shapes and varying their sizes. The primary tool for comparing simulations to approximation schemes was cross-correlation of the evolved mass density fields, testing the extent to which mass was moved to the right place. The Zel'dovich approximation, with initial convolution with a Gaussian e(exp -k(exp 2)/k(sub G(exp 2)), 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. We recommend either n-body simulations or our modified versions of the Zel'dovich approximation, depending upon the purpose. The theoretical implication is that pancaking is implicit in all cosmological gravitational clustering, at least from Gaussian initial conditions, even when subcondensations are present. This in turn provides a natural explanation for the presence of sheets and filaments in the observed galaxy distribution. Use of the approximation scheme can permit extremely rapid generation of large numbers of realizations of model universes with good accuracy down to galaxy group mass scales.
Lanzarotta, Adam
2015-01-01
Theoretical and empirical detection limits have been estimated for aripiprazole (analyte) in alpha lactose monohydrate (matrix model pharmaceutical formulation) using a micro-attenuated total reflection Fourier transform infrared (ATR FT-IR) spectroscopic imaging instrument equipped with a linear array detector and a 1.5 mm germanium hemisphere internal reflection element (IRE). The instrument yielded a theoretical detection limit of 0.0035% (35 parts per million (ppm)) when operating under diffraction-limited conditions, which was 49 times lower than what was achieved with a traditional macro-ATR instrument operating under practical conditions (0.17%, 1700 ppm). However, these results may not be achievable for most analyses because the detection limits will be particle size limited, rather than diffraction limited, for mixtures with average particle diameters greater than 8.3 μm (most pharmaceutical samples). For example, a theoretical detection limit of 0.028% (280 ppm) was calculated for an experiment operating under particle size-limited conditions where the average particle size was 23.4 μm. These conditions yielded a detection limit of 0.022% (220 ppm) when measured empirically, which was close to the theoretical value and only eight times lower than that of a faster, more simplistic macro-ATR instrument. Considering the longer data acquisition and processing times characteristic of the micro-ATR imaging approach (minutes or even hours versus seconds), the cost-benefit ratio may not often be favorable for the analysis of analytes in matrices that exhibit only a few overlapping absorptions (low-interfering matrices such as alpha lactose monohydrate) using this technique compared to what can be achieved using macro-ATR. However, the advantage was significant for detecting analytes in more complex matrices (those that exhibited several overlapping absorptions with the analyte) because the detection limit of the macro-ATR approach was highly formulation
Lanzarotta, Adam
2015-01-01
Theoretical and empirical detection limits have been estimated for aripiprazole (analyte) in alpha lactose monohydrate (matrix model pharmaceutical formulation) using a micro-attenuated total reflection Fourier transform infrared (ATR FT-IR) spectroscopic imaging instrument equipped with a linear array detector and a 1.5 mm germanium hemisphere internal reflection element (IRE). The instrument yielded a theoretical detection limit of 0.0035% (35 parts per million (ppm)) when operating under diffraction-limited conditions, which was 49 times lower than what was achieved with a traditional macro-ATR instrument operating under practical conditions (0.17%, 1700 ppm). However, these results may not be achievable for most analyses because the detection limits will be particle size limited, rather than diffraction limited, for mixtures with average particle diameters greater than 8.3 μm (most pharmaceutical samples). For example, a theoretical detection limit of 0.028% (280 ppm) was calculated for an experiment operating under particle size-limited conditions where the average particle size was 23.4 μm. These conditions yielded a detection limit of 0.022% (220 ppm) when measured empirically, which was close to the theoretical value and only eight times lower than that of a faster, more simplistic macro-ATR instrument. Considering the longer data acquisition and processing times characteristic of the micro-ATR imaging approach (minutes or even hours versus seconds), the cost-benefit ratio may not often be favorable for the analysis of analytes in matrices that exhibit only a few overlapping absorptions (low-interfering matrices such as alpha lactose monohydrate) using this technique compared to what can be achieved using macro-ATR. However, the advantage was significant for detecting analytes in more complex matrices (those that exhibited several overlapping absorptions with the analyte) because the detection limit of the macro-ATR approach was highly formulation
Approximate algorithms for partitioning and assignment problems
NASA Technical Reports Server (NTRS)
Iqbal, M. A.
1986-01-01
The problem of optimally assigning the modules of a parallel/pipelined program over the processors of a multiple computer system under certain restrictions on the interconnection structure of the program as well as the multiple computer system was considered. For a variety of such programs it is possible to find linear time if a partition of the program exists in which the load on any processor is within a certain bound. This method, when combined with a binary search over a finite range, provides an approximate solution to the partitioning problem. The specific problems considered were: a chain structured parallel program over a chain-like computer system, multiple chain-like programs over a host-satellite system, and a tree structured parallel program over a host-satellite system. For a problem with m modules and n processors, the complexity of the algorithm is no worse than O(mnlog(W sub T/epsilon)), where W sub T is the cost of assigning all modules to one processor and epsilon the desired accuracy.
Numerical linear algebra in data mining
NASA Astrophysics Data System (ADS)
Eldén, Lars
Ideas and algorithms from numerical linear algebra are important in several areas of data mining. We give an overview of linear algebra methods in text mining (information retrieval), pattern recognition (classification of handwritten digits), and PageRank computations for web search engines. The emphasis is on rank reduction as a method of extracting information from a data matrix, low-rank approximation of matrices using the singular value decomposition and clustering, and on eigenvalue methods for network analysis.
NASA Astrophysics Data System (ADS)
2001-05-01
the reason for this strange behaviour: the comet's "dirty snowball" nucleus had split into two pieces (IAUC 7616 [1]). During the splitting of the nucleus, fresh material from the interior of this frozen body is suddenly exposed to the sunlight, causing a rapid increase in the evaporation process. More cometary material is released and the overall brightness increases, as more sunlight is reflected off the dust around the nucleus. The VLT observes three fragments But Comet LINEAR has just shown that it is good for another surprise. When astronomers at ESO's Paranal Observatory [2] turned the 8.2-m VLT MELIPAL telescope (UT3) towards that object in the evening of May 14, they noted that one of the two pieces of the nucleus appeared somewhat elongated. The comet is rapidly approaching the Sun - it will pass through its perihelion (the point closest to the Sun) on May 25, and it was quite low in the sky (about 20° above the western horizon). Accordingly, the image quality was not perfect, but there was no doubt that something was going on with the fragment that was closest to the Sun (denoted "B"). And indeed, when the 8.2-m VLT YEPUN telescope (UT4) obtained another image of the comet in the evening of May 16, it was obvious that fragment "B" had split into two, see PR Photos 18a-b/01 . In fact, the astronomers suspect that there may be other, smaller pieces. The distance between the two pieces of nucleus "B" of Comet LINEAR (now denoted "B1" and "B2") was only about 1 arcsec, or approximately 500 km (projected) at the present distance of the comet from the Earth (about 100 million km). The distance between these and the other nucleus ("A") increased from about 6000 km (May 14) to 7000 km (May 16). The ESO astronomers have reported their detailed findings in IAU Circular 7627 [1]. They also note that the shape of the bright cloud (the "coma") around components "B1" and "B2" is quite unsual - this is well visible on the false-colour PR Photo 18b/01 . They interpret this
Parameterized Linear Longitudinal Airship Model
NASA Technical Reports Server (NTRS)
Kulczycki, Eric; Elfes, Alberto; Bayard, David; Quadrelli, Marco; Johnson, Joseph
2010-01-01
A parameterized linear mathematical model of the longitudinal dynamics of an airship is undergoing development. This model is intended to be used in designing control systems for future airships that would operate in the atmospheres of Earth and remote planets. Heretofore, the development of linearized models of the longitudinal dynamics of airships has been costly in that it has been necessary to perform extensive flight testing and to use system-identification techniques to construct models that fit the flight-test data. The present model is a generic one that can be relatively easily specialized to approximate the dynamics of specific airships at specific operating points, without need for further system identification, and with significantly less flight testing. The approach taken in the present development is to merge the linearized dynamical equations of an airship with techniques for estimation of aircraft stability derivatives, and to thereby make it possible to construct a linearized dynamical model of the longitudinal dynamics of a specific airship from geometric and aerodynamic data pertaining to that airship. (It is also planned to develop a model of the lateral dynamics by use of the same methods.) All of the aerodynamic data needed to construct the model of a specific airship can be obtained from wind-tunnel testing and computational fluid dynamics
Approximate Green's function methods for HZE transport in multilayered materials
NASA Technical Reports Server (NTRS)
Wilson, John W.; Badavi, Francis F.; Shinn, Judy L.; Costen, Robert C.
1993-01-01
A nonperturbative analytic solution of the high charge and energy (HZE) Green's function is used to implement a computer code for laboratory ion beam transport in multilayered materials. The code is established to operate on the Langley nuclear fragmentation model used in engineering applications. Computational procedures are established to generate linear energy transfer (LET) distributions for a specified ion beam and target for comparison with experimental measurements. The code was found to be highly efficient and compared well with the perturbation approximation.
The Oberbeck-Boussinesq approximation as a constitutive limit
NASA Astrophysics Data System (ADS)
Kagei, Yoshiyuki; Růžička, Michael
2015-12-01
We derive the usual Oberbeck-Boussinesq approximation as a constitutive limit of the full system describing the motion of an compressible linearly viscous fluid. To this end, the starting system is written, using the Gibbs free energy, in the variables v, θ and p. The Oberbeck-Boussinesq system is then obtained as the thermal expansion coefficient α and the isothermal compressibility coefficient β tend to zero.
The Oberbeck-Boussinesq approximation as a constitutive limit
NASA Astrophysics Data System (ADS)
Kagei, Yoshiyuki; Růžička, Michael
2016-09-01
We derive the usual Oberbeck-Boussinesq approximation as a constitutive limit of the full system describing the motion of an compressible linearly viscous fluid. To this end, the starting system is written, using the Gibbs free energy, in the variables v, θ and p. The Oberbeck-Boussinesq system is then obtained as the thermal expansion coefficient α and the isothermal compressibility coefficient β tend to zero.
Electrothermal linear actuator
NASA Technical Reports Server (NTRS)
Derr, L. J.; Tobias, R. A.
1969-01-01
Converting electric power into powerful linear thrust without generation of magnetic fields is accomplished with an electrothermal linear actuator. When treated by an energized filament, a stack of bimetallic washers expands and drives the end of the shaft upward.
NASA Technical Reports Server (NTRS)
Tuey, R. C.
1972-01-01
Computer solutions of linear programming problems are outlined. Information covers vector spaces, convex sets, and matrix algebra elements for solving simultaneous linear equations. Dual problems, reduced cost analysis, ranges, and error analysis are illustrated.
Periodic standing-wave approximation: Post-Minkowski computations
Beetle, Christopher; Bromley, Benjamin; Hernandez, Napoleon; Price, Richard H.
2007-10-15
The periodic standing-wave method studies circular orbits of compact objects coupled to helically symmetric standing-wave gravitational fields. From this solution an approximation is extracted for the strong field, slowly inspiralling motion of black holes and binary stars. Previous work on this model has dealt with nonlinear scalar models, and with linearized general relativity. Here we present the results of the method for the post-Minkowski (PM) approximation to general relativity, the first step beyond linearized gravity. We compute the PM approximation in two ways: first, via the standard approach of computing linearized gravitational fields and constructing from them quadratic driving sources for second-order fields, and second, by solving the second-order equations as an 'exact' nonlinear system. The results of these computations have two distinct applications: (i) The computational infrastructure for the exact PM solution will be directly applicable to full general relativity. (ii) The results will allow us to begin supplying initial data to collaborators running general relativistic evolution codes.
NASA Technical Reports Server (NTRS)
Lawson, C. L.; Krogh, F. T.; Gold, S. S.; Kincaid, D. R.; Sullivan, J.; Williams, E.; Hanson, R. J.; Haskell, K.; Dongarra, J.; Moler, C. B.
1982-01-01
The Basic Linear Algebra Subprograms (BLAS) library is a collection of 38 FORTRAN-callable routines for performing basic operations of numerical linear algebra. BLAS library is portable and efficient source of basic operations for designers of programs involving linear algebriac computations. BLAS library is supplied in portable FORTRAN and Assembler code versions for IBM 370, UNIVAC 1100 and CDC 6000 series computers.
Adapting the range of validity for the Carleman linearization
NASA Astrophysics Data System (ADS)
Weber, Harry; Mathis, Wolfgang
2016-09-01
In this contribution, the limitations of the Carleman linearization approach are presented and discussed. The Carleman linearization transforms an ordinary nonlinear differential equation into an infinite system of linear differential equations. In order to transform the nonlinear differential equation, orthogonal polynomials which represent solutions of a Sturm-Liouville problem are used as basis. The determination of the time derivate of this basis yields an infinite dimensional linear system that depends on the considered nonlinear differential equation. The infinite linear system has the same properties as the nonlinear differential equation such as limit cycles or chaotic behavior. In general, the infinite dimensional linear system cannot be solved. Therefore, the infinite dimensional linear system has to be approximated by a finite dimensional linear system. Due to limitation of dimension the solution of the finite dimensional linear system does not represent the global behavior of the nonlinear differential equation. In fact, the accuracy of the approximation depends on the considered nonlinear system and the initial value. The idea of this contribution is to adapt the range of validity for the Carleman linearization in order to increase the accuracy of the approximation for different ranges of initial values. Instead of truncating the infinite dimensional system after a certain order a Taylor series approach is used to approximate the behavior of the nonlinear differential equation about different equilibrium points. Thus, the adapted finite linear system describes the local behavior of the solution of the nonlinear differential equation.
The models of cosmological inflation in the context of kinetic approximation
NASA Astrophysics Data System (ADS)
Fomin, I.
2016-07-01
In this work the building of models of cosmological inflation with approximate linear dependence of the scalar field kinetic energy on the state parameter is considered. The key parameters of cosmological perturbations are also calculated.