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structure-preserving algorithms for Birkhoffian systems

JOURNAL OF GEOMETRY AND PHYSICS 2012年第5期62卷 1157-1166页

作者： Kong, Xinlei Wu, Huibin Mei, Fengxiang Beijing Inst Technol Sch Math Beijing 100081 Peoples R China Beijing Inst Technol Sch Aerosp Engn Beijing 100081 Peoples R China

We propose a new approach to construct structure-preserving algorithms for Birkhoffian systems. First, the Pfaff-Birkhoff variational principle is discretized, and based on the discrete variational principle the discrete Birkhoffian equations are obtained. Then, taking the discrete equations as an algorithm, the corresponding discrete flow is proved to be symplectic. That means the algorithm preserves the symplectic structure of Birkhoffian systems. Simulation results of the given example indicate that structure-preserving algorithms obtained by this method have great advantage in conserving conserved quantities. (C) 2011 Elsevier B.V. All rights reserved.

关键词： Birkhoffian system Discrete Birkhoffian equations structure-preserving algorithms Variational integrator

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The accurate particle tracer code

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COMPUTER PHYSICS COMMUNICATIONS 2017年 220卷 212-229页

作者： Wang, Yulei Liu, Jian Qin, Hong Yu, Zhi Yao, Yicun Univ Sci & Technol China Sch Nucl Sci & Technol Hefei 230026 Anhui Peoples R China Univ Sci & Technol China Dept Modern Phys Hefei 230026 Anhui Peoples R China Princeton Univ Plasma Phys Lab POB 451 Princeton NJ 08543 USA Chinese Acad Sci Inst Plasma Phys Theory & Simulat Div Hefei 230031 Anhui Peoples R China

The Accurate Particle Tracer (APT) code is designed for systematic large-scale applications of geometric algorithms for particle dynamical simulations. Based on a large variety of advanced geometric algorithms, APT possesses long-term numerical accuracy and stability, which are critical for solving multi-scale and nonlinear problems. To provide a flexible and convenient I/O interface, the libraries of Lua and Hdf5 are used. Following a three-step procedure, users can efficiently extend the libraries of electromagnetic configurations, external non-electromagnetic forces, particle pushers, and initialization approaches by use of the extendible module. APT has been used in simulations of key physical problems, such as runaway electrons in tokamaks and energetic particles in Van Allen belt. As an important realization, the APT SW version has been successfully distributed on the world's fastest computer, the Sunway TaihuLight supercomputer, by supporting master slave architecture of Sunway many-core processors. Based on large-scale simulations of a runaway beam under parameters of the ITER tokamak, it is revealed that the magnetic ripple field can disperse the pitch-angle distribution significantly and improve the confinement of energetic runaway beam on the same time. (C) 2017 Published by Elsevier B.V.

关键词： structure-preserving algorithms Plasma simulation Multi-timescale dynamics Large-scale simulation

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Sixth-order symplectic and symmetric explicit ERKN schemes for solving multi-frequency oscillatory nonlinear Hamiltonian equations

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CALCOLO 2017年第1期54卷 117-140页

作者： Wang, Bin Yang, Hongli Meng, Fanwei Qufu Normal Univ Sch Math Sci Qufu 273165 Shandong Peoples R China Nanjing Inst Technol Dept Math & Phys Nanjing 211167 Jiangsu Peoples R China

This paper is devoted to the analysis of the sixth-order symplectic and symmetric explicit extended Runge-Kutta-Nystrom (ERKN) schemes for solving multi-frequency oscillatory nonlinear Hamiltonian equations. Fourteen practical sixth-order symplectic and symmetric explicit ERKN schemes are constructed, and their phase properties are investigated. The paper is accompanied by five numerical experiments, including a nonlinear two-dimensional wave equation. The numerical results in comparison with the sixth-order symplectic and symmetric Runge-Kutta-Nystrom methods and a Gautschi-type method demonstrate the efficiency and robustness of the new explicit schemes for solving multi-frequency oscillatory nonlinear Hamiltonian equations.

关键词： Symplectic and symmetric explicit schemes Sixth-order extended Runge-Kutta-Nystrom schemes Multi-frequency oscillatory nonlinear Hamiltonian equations structure-preserving algorithms

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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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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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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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非线性Schr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 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

1引言在数值算法应尽可能多地保持原问题的本质特征的指导原则下,冯康先生;首先提出保结构算法的思想和概念,他和他的研究小组在Hamilton系统辛算法的构造算法和理论分析方面都取得了一系列成果;.计算实验显示,辛算法优异的稳定性和长... 详细信息

1引言在数值算法应尽可能多地保持原问题的本质特征的指导原则下,冯康先生;首先提出保结构算法的思想和概念,他和他的研究小组在Hamilton系统辛算法的构造算法和理论分析方面都取得了一系列成果;.计算实验显示,辛算法优异的稳定性和长时间跟踪能力有着重要的应用前景;.为了用保结构算法求解偏微分方程,王雨顺;]等人提出了偏微分方程局部保结构算法的概念.局部保结构算法仍属于保结构算法的范畴,其基

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

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Novel improved multidimensional Stormer-Verlet formulas with applications to four aspects in scientific computation

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MATHEMATICAL AND COMPUTER MODELLING 2013年第3-4期57卷 857-872页

作者： Wang, Bin Wu, Xinyuan Zhao, Hua Nanjing Univ Dept Math State Key Lab Novel Software Technol Nanjing 210093 Jiangsu Peoples R China Univ Cambridge Dept Appl Math & Theoret Phys Cambridge CB3 0WA England Beijing Inst Tracking & Tele Commun Technol Beijing 100094 Peoples R China

This paper presents two novel improved multidimensional Stormer-Verlet formulas with four applications to time-independent Schrodinger equations, wave equations, orbital problems and the problem of Fermi, Pasta & Ulam. For solving the system of second-order ordinary differential equations y '' + My = f (t, y) with M is an element of R-mxm, the multidimensional ARKN methods (adapted Runge-Kutta-Nystrom methods) were formulated by Wu et al. (2009) [1]. Very recently, the multidimensional ERKN methods (extended Runge-Kutta-Nystrom methods) were proposed by Wu et al. (2010) [26]. Both the ARKN methods and the ERKN methods perform numerically much better than the classical Runge-Kutta-Nystrom methods due to the use of the special structure of the equation brought by the linear term My. Based on the two kinds of multidimensional schemes, we derive two novel improved multidimensional Stormer-Verlet formulas, which are shown to be symplectic and of order two. Each new formula is a blend of existing trigonometric integrators and symplectic integrators. Meantime, the symplecticity conditions for the one-stage explicit multidimensional ARKN methods are presented. Stability and phase properties of the two improved formulas are analyzed. Numerical experiments demonstrate that the two improved multidimensional Stormer-Verlet formulas are more efficient than the classical Stormer-Verlet formula and the two other improved Stormer-Verlet methods appeared in the literature. In particular, when applied to a Hamiltonian system, the two symplectic improved multidimensional Stormer-Verlet formulas preserve well the Hamiltonian in the sense of numerical approximation, and have better accuracy than the classical Stormer-Verlet formula and the two other improved Stormer-Verlet methods with the same computational cost. (C) 2012 Elsevier Ltd. All rights reserved.

关键词： Schrodinger equations Wave equations Orbital problems Fermi-Pasta-Ulam problem structure-preserving algorithms Improved multidimensional Stormer-Verlet formulas

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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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