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FOURIER NEURAL SOLVER FOR LARGE SPARSE LINEAR ALGEBRAIC SYSTEMS

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arXiv 2022年

作者： Cui, Chen Jiang, Kai Liu, Yun Shu, Shi Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan411105 China

Large sparse linear algebraic systems can be found in a variety of scientific and engineering fields, and many scientists strive to solve them in an efficient and robust manner. In this paper, we propose an interpretable neural solver, the Fourier Neural Solver (FNS), to address them. FNS is based on deep learning and Fast Fourier transform. Because the error between the iterative solution and the ground truth involves a wide range of frequency modes, FNS combines a stationary iterative method and frequency space correction to eliminate different components of the error. Local Fourier analysis reveals that the FNS can pick up on the error components in frequency space that are challenging to eliminate with stationary methods. Numerical experiments on the anisotropy diffusion equation, convection-diffusion equation, and Helmholtz equation show that FNS is more efficient and more robust than the state-of-the-art neural *** Codes 65F1. 65N22 68T07 35Q68 Copyright © 2022, The Authors. All rights reserved.

关键词： Fourier analysis

DLTree: efficient and accurate phylogeny reconstruction using the dynamical language method

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Bioinformatics (Oxford, England) 2017年第14期33卷 2214-2215页

作者： Qi Wu Zu-Guo Yu Jianyi Yang Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Hunan China. School of Mathematical Sciences Nankai University Tianjin China.

SUMMARY:A number of alignment-free methods have been proposed for phylogeny reconstruction over the past two decades. But there are some long-standing challenges in these methods, including requirement of huge computer memory and CPU time, and existence of duplicate computations. In this article, we address these challenges with the idea of compressed vector, fingerprint and scalable memory management. With these ideas we developed the DLTree algorithm for efficient implementation of the dynamical language model and whole genome-based phylogenetic analysis. The DLTree algorithm was compared with other alignment-free tools, demonstrating that it is more efficient and accurate for phylogeny reconstruction. AVAILABILITY AND IMPLEMENTATION:The DLTree algorithm is freely available at http://***. CONTACT:yuzuguo@*** or yangjy@***. SUPPLEMENTARY INFORMATION:Supplementary data are available at Bioinformatics online.

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A virtual element method with IMEX-SAV scheme for the incompressible magnetohydrodynamics equations

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arXiv 2024年

作者： Dong, Xiaojing Huang, Yunqing Wang, Tianwen Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing & Information Processing of Ministry of Education School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan411105 China

This paper proposes a virtual element method (VEM) combined with a second-order implicit-explicit scheme based on the scalar auxiliary variable (SAV) method for the incompressible magnetohydrodynamics (MHD) equations. We employ the BDF2 scheme for time discretization and a conservative VEM for spatial discretization, in which the mass conservation in the velocity field is kept by taking advantage of the virtual element method’s adaptability and its divergence-free characteristics. In our scheme, the nonlinear terms are handled explicitly using the SAV method, and the magnetic field is decoupled from the velocity and pressure. This decoupling only requires solving a sequence of linear systems with constant coefficient at each time step. The stability estimate of the fully discrete scheme is developed, demonstrating the scheme is unconditionally stable. Moreover, rigorous error estimates for the velocity and magnetic field are provided. Finally, numerical experiments are presented to verify the valid of theoretical analysis. © 2024, CC BY-NC-ND.

关键词： Discrete element methods

A preconditioned iteration method for solving saddle point problems

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arXiv 2024年

作者： Zhang, Juan Luo, Yiyi Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education School of Mathematics and Computational Science Xiangtan University China

This paper introduces a preconditioned method designed to comprehensively address the saddle point system with the aim of improving convergence efficiency. In the preprocessor construction phase, a technical approach for solving the approximate inverse matrix of sparse matrices is presented. The effectiveness of the proposed method is demonstrated through numerical examples, emphasizing its efficacy in approximating the inverse matrix. Furthermore, the preprocessing technology includes a low-rank processing step, effectively reducing algorithmic complexity. Numerical experiments validate the effectiveness and feasibility of PSLR-GMRES in solving the saddle point system. © 2024, CC BY.

关键词： Iterative methods

Maximum-norm error analysis of a difference scheme for the space fractional CNLS

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Applied Mathematics and computation 2015年 257卷 241-251页

作者： Dongling Wang Aiguo Xiao Wei Yang Department of Mathematics and Center for Nonlinear Studies Northwest University Xi’an Shaanxi 710127 PR China Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Xiangtan Hunan 411105 PR China

The difference method for the space fractional coupled nonlinear Schrödinger equations (CNLS) is studied. The fractional centered difference is used to approximate the space fractional Laplacian. This scheme conserves the discrete mass and energy. Due to the nonlocal nature of fractional Laplacian, in the classic Sobolev space, it is hard to obtain the error estimation in l ∞ . To overcome this difficulty, the fractional Sobolev space H α / 2 and a fractional norm equivalence in H α / 2 are introduced. Then the convergence of order O ( h 2 + τ 2 ) in l ∞ is proved by fractional Sobolev inequality, where h is the mesh size and τ is the time step. Numerical examples are given to illustrate the theoretical results at last.

关键词： Fractional Schrödinger equations Fractional centered difference Convergence analysis

Preprocessed GMRES for fast solution of linear equations

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arXiv 2024年

The article mainly introduces preprocessing algorithms for solving linear equation systems. This algorithm uses three algorithms as inner iterations, namely RPCG algorithm, ADI algorithm, and Kaczmarz algorithm. Then, it uses BA-GMRES as an outer iteration to solve the linear equation system. These three algorithms can indirectly generate preprocessing matrices, which are used for solving equation systems. In addition, we provide corresponding convergence analysis and numerical examples. Through numerical examples, we demonstrate the effectiveness and feasibility of these preprocessing methods. Furthermore, in the Kaczmarz algorithm, we introduce both constant step size and adaptive step size, and extend the parameter range of the Kaczmarz algorithm to α ∈ (0, ∞). We also study the solution rate of linear equation systems using different step sizes. Numerical examples show that both constant step size and adaptive step size have higher solution efficiency than the solving algorithm without preprocessing. © 2024, CC BY.

关键词： Linear equations

Elliptical characteristics and numerical simulation of the PSEs of ferrofluid

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Elliptical characteristics and numerical simulation of the P...

中国力学大会（CCTAM201.）

作者： Li Mingjun Tang Shujiang Hiroshi Yamaguchi Jie Wu School of Mathematics and Computational Science Xiangtan University Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Department of Mechanical Engineering Doshisha University

Ferrofluid is a colloidal suspension system with magnetic nanoparticles, the overall field of study in ferrofluid has a highly interdisciplinary character in practical applications. In this paper a set of parabolized stability equations(PSEs)of ferrofluid is represented by Rosensweig equations. The characteristic analysis show that in subsonic region the nonlinear PSEs of ferrofluid are parabolical, and in supersonic region are elliptical. The PSEs of ferrofluid is solved numerically for heat conduction process and RT instability problem, respectively. The results of characteristic analysis and numerical simulation all verify that the basic characteristic of ferrofluid are not influenced by the external magnetic field, and the viscosity is the only reason for the variation of ellipticity of PSEs of ferrofluid. That is to say, the external magnetic field and the viscosity are essentially different.

关键词： Ferrofluid Magnetic nanoparticles Parabolized stability equations Ellipticity

Tilt grain boundaries of hexagonal structures: a spectral viewpoint

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arXiv 2021年

作者： Jiang, Kai Si, Wei Xu, Jie School of Mathematics and Computational Science Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Xiangtan Hunan411105 China Chinese Academy of Sciences Beijing China

We propose a spectral viewpoint for grain boundaries that are generally quasiperiodic. To accurately capture the spectra computationally, it is crucial to adopt the projection method for quasiperiodic functions. Armed with the Lifshitz–Petrich free energy, we take the spectral viewpoint to examine tilt grain boundaries of the hexagonal phase. Several ingredients of grain boundaries are extracted, which are not easy to obtain from real-space profiles. We find that only a few spectra substantially contribute to the formation of grain boundaries. Their linear relation to the intrinsic spectra of the bulk hexagonal phase is independent of the tilt angle. By examining the feature of the spectral intensities, we propose a definition of the interface width. The widths calculated from this definition are consistent with visual *** Codes 65N35, 70H1., 82B24 © 2021. CC BY.

关键词： Grain boundaries

Svgd: A Virtual Gradients Descent Method for Stochastic Optimization

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arXiv 2019年

作者： Li, Zheng Shu, Shi School of Mathematics and Computational Science Xiangtan University Xiangtan Hunan411105 China Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Xiangtan411105 China

Inspired by dynamic programming, we propose Stochastic Virtual Gradient Descent (SVGD) algorithm where the Virtual Gradient is defined by computational graph and automatic differentiation. The method is computationally efficient and has little memory requirements. We also analyze the theoretical convergence properties and implementation of the algorithm. Experimental results on multiple datasets and network models show that SVGD has advantages over other stochastic optimization methods. Copyright © 201., The Authors. All rights reserved.

关键词： Machine learning