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A Hybrid Iterative Neural Solver Based on Spectral Analysis for Parametric PDEs

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arXiv 2024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

Recently, deep learning-based hybrid iterative methods (DL-HIM) have emerged as a promising approach for designing fast neural solvers to tackle large-scale sparse linear systems. DL-HIM combine the smoothing effect of simple iterative methods with the spectral bias of neural networks, which allows them to effectively eliminate both high-frequency and low-frequency error components. However, their efficiency may decrease if simple iterative methods can not provide effective smoothing, making it difficult for the neural network to learn mid-frequency and high-frequency components. This paper first conducts a convergence analysis for general DL-HIM from a spectral viewpoint, concluding that under reasonable assumptions, DL-HIM exhibit a convergence rate independent of grid size h and physical parameters µ. To meet these assumptions, we design a neural network from an eigen perspective, focusing on learning the eigenvalues and eigenvectors corresponding to error components that simple iterative methods struggle to eliminate. Specifically, the eigenvalues are learned by a meta subnet, while the eigenvectors are approximated using Fourier modes with a transition matrix provided by another meta subnet. The resulting DL-HIM, termed the Fourier Neural Solver (FNS), can be trained to achieve a convergence rate independent of grid size and PDE parameters by designing a loss function that ensures the neural network complements the smoothing effect of Jacobi iterative methods. We verify the performance of FNS on six types of linear parametric *** Codes 65F10, 65F08, 68T07, 65N22, 90C06 Copyright © 2024, The Authors. All rights reserved.

关键词： Eigenvalues and eigenfunctions

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 discrete Perfectly Matched Layer for peridynamic scalar waves in two-dimensional viscous media

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arXiv 2025年

作者： Du, Yu Li, Yonglin Zhang, Jiwei School of Mathematics and Computational Science Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Hunan411105 China School of Mathematics and Statistics Hubei Key Laboratory of Computational Science Wuhan University Wuhan430072 China

In this paper, we propose a discrete perfectly matched layer (PML) for the peridynamic scalar wave-type problems in viscous media. Constructing PMLs for nonlocal models is often challenging, mainly due to the fact that nonlocal operators are usually associated with various kernels. We first convert the continua model to a spatial semi-discretized version by adopting quadrature-based finite difference scheme, and then derive the PML equations from the semi-discretized equations using discrete analytic continuation. The harmonic exponential fundamental solutions (plane wave modes) of the semi-discretized equations are absorbed by the PML layer without reflection and are exponentially damped. The excellent efficiency and stability of discrete PML are demonstrated in numerical tests by comparison with exact absorbing boundary *** Codes 65N30, 65R20, 46N20, 45A05, 78A40 Copyright © 2025, The Authors. All rights reserved.

关键词： Boundary conditions

A Robust Modified Weak Galerkin Finite Element Method for Reaction-Diffusion Equations

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Numerical Mathematics(Theory,Methods and Applications) 2022年第1期15卷 68-90页

作者： Guanrong Li Yanping Chen Yunqing Huang School of Mathematics and Statistics Lingnan Normal UniversityZhanjiang 524048GuangdongP.R.China School of Mathematical Science South China Normal UniversityGuangzhou 510631GuangdongP.R.China Hunan Key Laboratory for Computation and Simulation in Science and Engineering School of Mathematics and Computational ScienceXiangtan UniversityXiangtan 411105HunanP.R.China

In this paper,a robust modified weak Galerkin(MWG)finite element method for reaction-diffusion equations is proposed and *** advantage of this method is that it can deal with the singularly perturbed reaction-diffusion *** advantage of this method is that it produces fewer degrees of freedom than the traditional WG method by eliminating the element boundaries *** is worth pointing out that,in our method,the test functions space is the same as the finite element space,which is helpful for the error *** error estimates are established for the corresponding numerical approximation in various *** numerical results are reported to confirm the theory.

关键词： Reaction-diffusion equations singular perturbation modified weak Galerkin finite element methods discrete gradient

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

Two Physics-Based Schwarz Preconditioners for Three-Temperature Radiation Diffusion Equations in High Dimensions

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Communications in Computational Physics 2022年第8期32卷 829-849页

作者： Xiaoqiang Yue Jianmeng He Xiaowen Xu Shi Shu Libo Wang National Center for Applied Mathematics in Hunan Key Laboratory of Intelligent Computing&Information Processing of Ministry of EducationHunan Key Laboratory for Computation and Simulation in Science and EngineeringXiangtan UniversityXiangtan 411105P.R.China Laboratory of Computational Physics Institute of Applied Physics and ComputationalMathematicsBeijing 100094P.R.China College of Information Science and Technology JinanUniversityGuangzhou 510632P.R.China

We concentrate on the parallel,fully coupled and fully implicit solution of the sequence of 3-by-3 block-structured linear systems arising from the symmetrypreserving finite volume element discretization of the unsteady three-temperature radiation diffusion equations in high *** this article,motivated by[***,***,***,SIAM *** ***.33(2012)653–680]and[***,***,***,***.442(2021)110513],we aim to develop the additive and multiplicative Schwarz preconditioners subdividing the physical quantities rather than the underlying domain,and consider their sequential and parallel implementations using a simplified explicit decoupling factor approximation and algebraic multigrid subsolves to address such linear ***,computational efficiencies and parallel scalabilities of the proposed approaches are numerically tested in a number of representative real-world capsule implosion benchmarks.

关键词： Radiation diffusion equations Schwarz methods algebraic multigrid parallel and distributed computing

Comments and extensions on "State-equivalent form and minimum-order compensator design for rectangular descriptor systems"

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arXiv 2025年

作者： Shi, Shuo Zhang, Juan Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing and Information Processing Ministry of Education School of Mathematics and Computational Science Xiangtan University Xiangtan411105 China

This technical note presents a counterexample showing that the equivalence conditions proposed by Geng et al. (IEEE Trans. Automat. Control, 2024), which use a minimum-order compensator (MOC) to achieve desired designs, including generalized regularity or both generalized regularity and free of impulse, are sufficient but not necessary. Furthermore, revised equivalence conditions are introduced, along with an equivalence condition ensuring the closed-loop system remains generalized regular, impulse-free, and stable using MOC. Additionally, it is shown that output feedback can replace the MOC, achieving the same design without increasing dimensionality. These findings are validated through a circuit *** Codes 93B05, 93B52, 15A03 Copyright © 2025, The Authors. All rights reserved.

关键词： Feedback

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

Legendre Neural Network for Solving Linear Variable Coefficients Delay Differential-Algebraic Equations with Weak Discontinuities

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Advances in Applied Mathematics and Mechanics 2021年第1期13卷 101-118页

作者： Hongliang Liu Jingwen Song Huini Liu Jie Xu Lijuan Li School of Mathematics and Computational Science&Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan UniversityXiangtanHunan 411105China School of Automation and Electronic Information Xiangtan UniversityXiangtanHunan 411105China

In this paper,we propose a novel Legendre neural network combined with the extreme learning machine algorithm to solve variable coefficients linear delay differential-algebraic equations with weak ***,the solution interval is divided into multiple subintervals by weak discontinuity ***,Legendre neural network is used to eliminate the hidden layer by expanding the input pattern using Legendre polynomials on each ***,the parameters of the neural network are obtained by training with the extreme learning *** numerical examples show that the proposed method can effectively deal with the difficulty of numerical simulation caused by the discontinuities.

关键词： Convergence delay differential-algebraic equations Legendre activation function,neural network.