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Efficiency energy-preserving cosine pseudo-spectral algorithms for the sine-Gordon equation with Neumann boundary conditions

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INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS 2022年第12期99卷 2367-2381页

作者： Yue, Xiaopeng Fu, Yayun Xuchang Univ Sch Sci Henan Joint Int Res Lab High Performance Computat Xuchang 461000 Peoples R China

This paper considers the newly introduced generalized scalar auxiliary variable approaches to construct high-efficiency energy-preserving schemes for the sine-Gordon equation with Neumann boundary conditions. The equation is first reformulated into an equivalent system by defining a new auxiliary variable that is not limited to square root. Then, the cosine pseudo-spectral method is applied to the system and derive a semi-discrete conservative scheme. Subsequently, we combine the auxiliary variable with the nonlinear term and use an explicit technique discretization in time to derive a fully-discrete energy-preserving scheme. Furthermore, a fast algorithm based on the discrete cosine transform technique reduces the computational complexity in practical computation. Finally, various numerical experiments are displayed to verify the accuracy, efficiency and conservation of the proposed schemes.

关键词： structure-preserving algorithms generalized scalar auxiliary sine-Gordon equation cosine pseudo-spectral method Neumann boundary

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structure-preserving Galerkin POD reduced-order modeling of Hamiltonian systems

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COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING 2017年 315卷 780-798页

作者： Gong, Yuezheng Wang, Qi Wang, Zhu Beijing Computat Sci Res Ctr Appl & Computat Math Div Beijing 100193 Peoples R China Univ South Carolina Dept Math Columbia SC 29208 USA Nankai Univ Sch Mat Sci & Engn Tianjin 300350 Peoples R China

The proper orthogonal decomposition reduced-order model (POD-ROM) has been widely used as a computationally efficient surrogate model in large-scale numerical simulations of complex systems. However, when it is applied to a Hamiltonian system, a naive application of the POD method can destroy the Hamiltonian structure in the reduced-order modelin this paper, we develop a new reduced-order modeling approach for Hamiltonian systems, which modifies the Galerkin projection-based POD -ROM so that the appropriate Hamiltonian structure is preserved. Since the POD truncation can degrade the approximation of the Hamiltonian function, we propose to use a POD basis from shifted snapshots to improve the approximation to the Hamiltonian function. We further derive a rigorous a priori error estimate for the structure-preserving ROM and demonstrate its effectiveness in several numerical examples. This approach can be readily extended to dissipative Hamiltonian systems, port-Hamiltonian systems, etc. Published by Elsevier B.V.

关键词： Proper orthogonal decomposition Model reduction Hamiltonian systems structure-preserving algorithms

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A Class of Explicit Divergence-Free Methods for Maxwell's Equations With Dirichlet Boundary Conditions

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IEEE ACCESS 2022年 10卷 126188-126198页

作者： Zeng, Xianyang Yang, Hongli Nanjing Inst Technol Ind Ctr Nanjing Peoples R China Nanjing Inst Technol Sch Math & Phys Nanjing Peoples R China

In this paper, we focus on the numerical solutions of Maxwell's equations with Dirichlet boundary conditions in rectangular coordinate. A class of explicit methods is derived by using an effective solver for a system of ordinary differential equations which is obtained by approximating on spatial fields. A significant advantage of this class of methods is their simplicity and their ease of implementation. The error estimates presented in this paper show that the numerical solutions obtained by this class of methods is of high-order. The main advantage of this class of methods is that it is divergence-free.

关键词： Maxwell equations Boundary conditions Time-domain analysis Mathematical models Electromagnetic propagation Ordinary differential equations Numerical models Computation theory Dirichlet boundary value problem divergence-free methods electromagnetic propagation Maxwell's equations time-domain methods structure-preserving algorithms

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An efficient algorithm for the discrete-time algebraic Riccati equation

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL 1999年第6期44卷 1216-1220页

作者： Lu, LZ Lin, WW Pearce, CEM Xiamen Univ Dept Math Xiamen Peoples R China Tsing Hua Univ Dept Math Appl Hsinchu Taiwan Univ Adelaide Dept Appl Math Adelaide SA Australia

In this paper the authors develop a new algorithm to solve the standard discrete-time algebraic Riccati equation by using a skew-Hamiltonian transformation and the square-root method. The algorithm is structure-preserving and efficient because the Hamiltonian structure is fully exploited and only orthogonal transformations are used. The efficiency and stability of the algorithm are analyzed. Numerical examples are included.

关键词： algebraic Riccati equation Hamiltonian structure square-root method structure-preserving algorithms

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An exact in time Fourier pseudospectral method with multiple conservation laws for three-dimensional Maxwell's equations

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ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS 2024年第3期58卷 857-880页

作者： Wang, Bin Jiang, Yaolin Xi An Jiao Tong Univ Sch Math & Stat Xian 710049 Shannxi Peoples R China

Maxwell's equations describe the propagation of electromagnetic waves and are therefore fundamental to understanding many problems encountered in the study of antennas and electromagnetics. The aim of this paper is to propose and analyse an efficient fully discrete scheme for solving three-dimensional Maxwell's equations. This is accomplished by combining Fourier pseudospectral methods in space and exact formulation in time. Fast computation is efficiently implemented in the scheme by using the matrix diagonalisation method and fast Fourier transform algorithm which are well known in scientific computations. An optimal error estimate which is not encumbered by the CFL condition is established and the resulting scheme is proved to be of spectral accuracy in space and exact in time. Furthermore, the scheme is shown to have multiple conservation laws including discrete energy, helicity, momentum, symplecticity, and divergence-free field conservations. All the theoretical results of the accuracy and conservations are numerically illustrated by two numerical tests.

关键词： Maxwell's equations exact in time Fourier pseudospectral method Duhamel's formula structure-preserving algorithms convergence

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Regularizing flows for constrained matrix-valued images

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JOURNAL OF MATHEMATICAL IMAGING AND VISION 2004年第1-2期20卷 147-162页

作者： ChefD'Hotel, C Tschumperlé, D Deriche, R Faugeras, O INRIA Sophia Antipolis Odyssee Lab F-06902 Sophia Antipolis France

Nonlinear diffusion equations are now widely used to restore and enhance images. They allow to eliminate noise and artifacts while preserving large global features, such as object contours. In this context, we propose a differential-geometric framework to define PDEs acting on some manifold constrained datasets. We consider the case of images taking value into matrix manifolds defined by orthogonal and spectral constraints. We directly incorporate the geometry and natural metric of the underlying configuration space ( viewed as a Lie group or a homogeneous space) in the design of the corresponding flows. Our numerical implementation relies on structure-preserving integrators that respect intrinsically the constraints geometry. The efficiency and versatility of this approach are illustrated through the anisotropic smoothing of diffusion tensor volumes in medical imaging.

关键词： image regularization multi-valued data constraints lie groups homogeneous spaces Riemannian metric natural gradient structure-preserving algorithms

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Regularizing flows for constrained matrix-valued images

Regularizing flows for constrained matrix-valued images

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4th Conference on Mathematics and Image Analysis

作者： ChefD'Hotel, C Tschumperlé, D Deriche, R Faugeras, O INRIA Sophia Antipolis Odyssee Lab F-06902 Sophia Antipolis France

关键词： image regularization multi-valued data constraints lie groups homogeneous spaces Riemannian metric natural gradient structure-preserving algorithms

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Iterative Rational Krylov Algorithm for Unstable Dynamical Systems and Genaralized Coprime Factorizations

Iterative Rational Krylov Algorithm for Unstable Dynamical S...

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作者： Sinani, Klajdi Virginia Tech | University

Generally, large-scale dynamical systems pose tremendous computational difficulties when applied in numerical simulations. In order to overcome these challenges we use several model reduction techniques. For stable linear models these techniques work very well and provide good approximations for the full model. However, large-scale unstable systems arise in many applications. Many of the known model reduction methods are not very robust, or in some cases, may not even work if we are dealing with unstable systems. When approximating an unstable sytem by a reduced order model, accuracy is not the only concern. We also need to consider the structure of the reduced order model. Often, it is important that the number of unstable poles in the reduced system is the same as the number of unstable poles in the original system. The Iterative Rational Krylov Algorithm (IRKA) is a robust model reduction technique which is used to locally reduce stable linear dynamical systems optimally in the $mathcal{H}_2$-norm. While we cannot guarantee that IRKA reduces an unstable model optimally, there are no numerical obstacles to the reduction of an unstable model via IRKA. In this thesis, we investigate IRKA’s behavior when it is used to reduce unstable models. We also consider systems for which we cannot obtain a first order realization of the transfer function. We can use Realization-independent IRKA to obtain a reduced order model which does not preserve the structure of the original model. In this paper, we implement a structure preserving algorithm for systems with nonlinear frequency dependency.

关键词： Model Reduction Dynamical Systems IRKA Unstable Systems structure-preserving algorithms Thesis

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Exponential discrete gradient schemes for a class of stochastic differential equations

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2022年 402卷 113797-113797页

作者： Ruan, Jialin Wang, Lijin Wang, Pengjun Univ Chinese Acad Sci Sch Math Sci YuQuan Rd 19 Beijing 100049 Peoples R China

In this paper, we propose a class of stochastic exponential discrete gradient schemes for SDEs with linear and gradient components in the coefficients. The root mean-square errors of the schemes are analyzed, and the structure-preserving properties of the schemes for SDEs with special structures are investigated. Numerical tests are performed to verify the theoretical results and illustrate the numerical behavior of the proposed methods. (C) 2021 Elsevier B.V. All rights reserved.

关键词： Stochastic differential equations Exponential integrators Discrete gradient methods Mean-square convergence structure-preserving algorithms

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非线性Schr(o)dinger方程两个新的多辛格式

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高等学校计算数学学报 2015年第3期37卷 203-227页

作者：鞠巍王雨顺张雨泽江苏省大规模复杂系统数值模拟重点实验室南京师范大学数学科学学院南京210023 江苏省大规模复杂系统数值模拟重点实验室南京师范大学数学科学学院南京210023 江苏省大规模复杂系统数值模拟重点实验室南京师范大学数学科学学院南京210023

偏微分方程的局部保结构算法概念是哈密尔顿辛几何算法在偏微分方程上的自然推广,它给数值研究保守的偏微分方程提供一个新的工具,也给分析保结构算法提供了一个新的视角.本文讨论了非线性Schrodinger方程的局部保结构算法,构造了两种... 详细信息

偏微分方程的局部保结构算法概念是哈密尔顿辛几何算法在偏微分方程上的自然推广,它给数值研究保守的偏微分方程提供一个新的工具,也给分析保结构算法提供了一个新的视角.本文讨论了非线性Schrodinger方程的局部保结构算法,构造了两种多辛格式.理论分析和数值实验表明了局部保结构算法应用于非线性Schrodinger方程是行之有效的,所构造的两种多辛格式是可取的.

关键词： nonlinear Schr(o)dinger equation structure-preserving algorithms concatenating method multi-symplectic scheme numerical experiments

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