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chebyshev pseudospectral method in the reconstruction of orthotropic conductivity

INVERSE PROBLEMS IN SCIENCE AND ENGINEERING 2021年第5期29卷 681-711页

作者： Boos, Everton Luchesi, Vanda M. Bazan, Fermin S. V. Univ Fed Santa Catarina Dept Math Florianopolis SC Brazil Univ Fed Uberlandia Inst Exact & Nat Sci Pontal Ituiutaba Brazil

In this paper, we present a method to reconstruct the spatially varying conductivity tensor in isotropic and orthotropic materials, involved in a two-dimensional transient anisotropic model with Robin boundary conditions. For the reconstruction, the partial differential equation is solved by a semi-discrete method that combines a pseudospectral collocation method for spatial variables and Crank-Nicolson for time. The conductivity tensor is reconstructed through a non-linear least-squares problem solved by Levenberg-Marquardt method (LMM), along with Morozov's discrepancy principle as stopping rule to cope with noise in the data. Unlike classic LMM implementations that mitigate poor conditioning in calculating iterates using nonsingular diagonal scaling matrices, in this paper, singular regularization matrices are used. The impact of such a modification is illustrated with numerical experiments using discrete differential operators as scaling matrices. Numerical results show that accurate conductivity values can be obtained using a fairly small number of discretization points at a very low computational cost.

关键词： Orthotropic conductivity reconstruction chebyshev pseudospectral method Levenberg-Marquardt method Morozov's discrepancy principle inverse problem

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chebyshev pseudospectral method for computing numerical solution of convection-diffusion equation

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APPLIED MATHEMATICS AND COMPUTATION 2008年第2期200卷 537-546页

作者： Bazan, F. S. V. Univ Fed Santa Catarina Dept Math BR-88040900 Florianopolis SC Brazil

A method for computing highly accurate numerical solutions of 1D convection-diffusion equations is proposed. In this method, the equation is first discretized with respect to the spatial variable, transforming the original problem into a set of ordinary differential equations, and then the resulting system is integrated in time by the fourth-order Runge-Kutta method. Spatial discretization is done by using the chebyshev pseudospectral collocation method. Before describing the method, we review a finite difference-based method by Salkuyeh [D. Khojasteh Salkuyeh, On the finite difference approximation to the convection-diffusion equation, Appl. Math. Comput. 179 (2006) 79-86], and, contrary to the proposal of the author, we show that this method is not suitable for problems involving time dependent boundary conditions, which calls for revision. Stability analysis based on pseudoeigenvalues to determine the maximum time step for the proposed method is also carried out. Superiority of the proposed method over a revised version of Salkuyeh's method is verified by numerical examples. (c) 2007 Elsevier Inc. All rights reserved.

关键词： convection diffusion chebyshev pseudospectral method stability region pseudoeigenvalues

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ANALYSIS OF chebyshev pseudospectral method FOR MULTI-DIMENSIONAL GENERALIZED SRLW EQUATIONS

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Applied Mathematics and Mechanics(English Edition) 2003年第10期24卷 1168-1183页

作者：尚亚东郭柏灵 Department of Mathematics Guangzhou University Institute of Applied Physics and Computational Mathematics

The chebyshev pseudospectral approximation of the homogeneous initial boundary value problem for a class of multi-dimensional generalized symmetric regularized long wave (SRLW) equations is considered. The fully discrete chebyshev pseudospectral scheme is constructed. The convergence of the approximation solution and the optimum error of approximation solution are obtained.

关键词： multi-dimensional generalized SRLW equation initial and boundary value problem chebyshev pseudospectral method error estimate

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Costate Computation by a chebyshev pseudospectral method

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JOURNAL OF GUIDANCE CONTROL AND DYNAMICS 2010年第2期33卷 623-628页

作者： Gong, Qi Ross, I. Michael Fahroo, Fariba Univ Calif Santa Cruz Dept Appl Math & Stat Santa Cruz CA 95064 USA USN Postgrad Sch Dept Mech & Astronaut Engn Monterey CA 93943 USA USN Postgrad Sch Dept Appl Math Monterey CA 93943 USA

AMONG the various pseudospectral (PS) methods for optimal control [1], only the Legendre PS method has been mathematically proven to guarantee the feasibility, consistency, and convergence of the approximations [2-5]. As exemplified by its experimental and flight applications in national programs [6-10], it is not surprising that the Legendre PS method has become the method of choice [11-19] in both industry and academia for solving optimal control problems. Efforts to improve the Legendre PS methods by using either other polynomials [20-22] or point distributions [23,24] have not yet resulted in any rigorous framework for convergence of these approximations [24,25].

关键词： chebyshev pseudospectral method Covector Mapping Principle Polynomial Interpolation Boundary Value Problems Control Algorithm

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Fractional chebyshev pseudospectral method for fractional optimal control problems

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OPTIMAL CONTROL APPLICATIONS & methodS 2019年第3期40卷 558-572页

作者： Habibli, M. Skandari, M. H. Noori Shahrood Univ Technol Fac Math Sci Shahrood *** Iran

In this paper, we introduce and apply a fractional pseudospectral method for indirectly solving a generic form of fractional optimal control problems. By employing the fractional Lagrange interpolating functions and discretizing the necessary optimality conditions at chebyshev-Gauss-Lobatto points, the problem is converted into an algebraic system. By solving this system, the optimal solution of the main fractional optimal control problem is approximated. Finally, in some numerical examples, we show the applicability, efficiency, and accuracy of the proposed method comparing with some other methods.

关键词： chebyshev pseudospectral method fractional Lagrange polynomial fractional optimal control optimality conditions

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A numerical solution of open-loop Nash equilibrium in nonlinear differential games based on chebyshev pseudospectral method

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2016年 300卷 369-384页

作者： Nikooeinejad, Z. Delavarkhalafi, A. Heydari, M. Yazd Univ Dept Math POB 89195-741 Yazd Iran

In general, the applications of differential games for solving practical problems have been limited, because all calculations had to be done analytically. In this investigation, a simple and efficient numerical method for solving nonlinear nonzero-sum differential games with finite- and infinite-time horizon is presented. In both cases, derivation of open-loop Nash equilibria solutions usually leads to solving nonlinear boundary value problems for a system of ODEs. The proposed numerical method is based on a combination of minimum principle of Pontryagin and expanding the required approximate solutions as the elements of chebyshev polynomials. Applying chebyshev pseudospectral method, two-point boundary value problems in differential games are reduced to the solution of a system of algebraic equations. Finally, several examples are given to demonstrate the accuracy and efficiency of the proposed method and a comparison is made with the results obtained by fourth order Runge Kutta method. (C) 2016 Elsevier B.V. All rights reserved.

关键词： Nonzero-sum differential games Open-loop Nash equilibria Boundary value problems chebyshev pseudospectral method

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Application of the chebyshev pseudospectral method to van der Waals fluids

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COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION 2012年第9期17卷 3499-3507页

作者： Odeyemi, Tinuade Mohammadian, Abdolmajid Seidou, Ousmane Univ Ottawa Dept Civil Engn Ottawa ON K1N 6N5 Canada

In this paper, we consider a class of van der Waals flows with non-convex flux functions. In these flows, nonclassical under-compressive shock waves can develop. Such waves, which are characterized by kinetic functions, violate classical entropy conditions. We propose to use a chebyshev pseudospectral method for solving the governing equations. A comparison of the results of this method with very high-order (up to 10th-order accurate) finite difference schemes is presented, which shows that the proposed method leads to a lower level of numerical oscillations than other high-order finite difference schemes and also does not exhibit fast-traveling packages of short waves which are usually observed in high-order finite difference methods. The proposed method can thus successfully capture various complex regimes of waves and phase transitions in both elliptic and hyperbolic regimes. (C) 2012 Elsevier B.V. All rights reserved.

关键词： chebyshev pseudospectral method van der Waals flows Non-classical shock waves High-order finite difference methods

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Direct trajectory optimization by a chebyshev pseudospectral method

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JOURNAL OF GUIDANCE CONTROL AND DYNAMICS 2002年第1期25卷 160-166页

作者： Fahroo, F Ross, IM USN Postgrad Sch Dept Math Monterey CA 93943 USA USN Postgrad Sch Dept Aeronaut & Astronaut Monterey CA 93943 USA

We present a chebyshev pseudospectral method for directly solving a generic Bolza optimal control problem with state and control constraints. This method employs Nth-degree Lagrange polynomial approximations for the state and control variables with the values of these variables at the chebyshev-Gauss-Lobatto (CGL) points as the expansion coefficients. This process yields a nonlinear programming problem (NLP) with the state and control values at the CGL points as unknown NLP parameters. Numerical examples demonstrate that this method yields more accurate results than those obtained from the traditional collocation methods.

关键词： chebyshev pseudospectral method Trajectory Optimization Nonlinear Programming Numerical Integration Moon Landing Propellant Guidance Algorithms Soft Landing MATLAB Planets

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Analysis of the dissipation and dispersion properties of the multi-domain chebyshev pseudospectral method

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JOURNAL OF COMPUTATIONAL PHYSICS 2013年 255卷 31-47页

作者： Dragna, D. Bogey, C. Hornikx, M. Blanc-Benon, P. Univ Lyon Ecole Cent Lyon UMR CNRS 5509 Lab Mecan Fluides & Acoust F-69134 Ecully France Eindhoven Univ Technol NL-5600 MB Eindhoven Netherlands

The accuracy of the multi-domain chebyshev pseudospectral method is investigated for wave propagation problems by examining the properties of the method in the wavenumber space theoretically in terms of dispersion and dissipation errors. For a number of (N + 1) points in the subdomains used in the literature, with N typically between 8 to 32, significant errors can be obtained for waves discretized by more than pi points per wavelength. The dispersion and dissipation errors determined from the analysis in the wavenumber space are found to be in good agreement with those obtained in test cases. Accuracy limits based on arbitrary criteria are proposed, yielding minimum resolutions of 7.7, 5.2 and 4.0 points per wavelength for N = 8, 16 and 32 respectively. The numerical efficiency of the method is estimated, showing that it is preferable to choose N between 16 and 32 in practice. The stability of the method is also assessed using the standard fourth-order Runge-Kutta algorithm. Finally, 1-D and 2-D problems involving long-range wave propagation are solved to illustrate the dissipation and dispersion errors for short waves. The error anisotropy is studied in the 2-D case, in particular for a hybrid Fourier-chebyshev configuration. (c) 2013 Elsevier Inc. All rights reserved.

关键词： chebyshev pseudospectral method Wave propagation Multi-domain method

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A FAST POISSON SOLVER BY chebyshev pseudospectral method USING REFLEXIVE DECOMPOSITION

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TAIWANESE JOURNAL OF MATHEMATICS 2013年第4期17卷 1167-1181页

作者： Kuo, Teng-Yao Chen, Hsin-Chu Horng, Tzyy-Leng Feng Chia Univ PhD Program Mech & Aeronaut Engn Taichung 40724 Taiwan Clark Atlanta Univ Dept Comp & Informat Sci Atlanta GA 30314 USA Feng Chia Univ Dept Appl Math Taichung 40724 Taiwan

Poisson equation is frequently encountered in mathematical modeling for scientific and engineering applications. Fast Poisson numerical solvers for 2D and 3D problems are, thus, highly requested. In this paper, we consider solving the Poisson equation del(2)u = f(x, y) in the Cartesian domain Omega = [-1, 1] x [-1, 1], subject to all types of boundary conditions, discretized with the chebyshev pseudospectral method. The main purpose of this paper is to propose a reflexive decomposition scheme for orthogonally decoupling the linear system obtained from the discretization into independent subsystems via the exploration of a special reflexive property inherent in the second-order chebyshev collocation derivative matrix. The decomposition will introduce coarse-grain parallelism suitable for parallel computations. This approach can be applied to more general linear elliptic problems discretized with the chebyshev pseudospectral method, so long as the discretized problems possess reflexive property Numerical examples with error analysis are presented to demonstrate the validity and advantage of the proposed approach.

关键词： Poisson equation chebyshev pseudospectral method chebyshev collocation derivative matrix Coarse-grain parallelism Reflexive property

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