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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

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

arXiv

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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

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A preconditioned iteration method for solving saddle point problems

arXiv

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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

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Preprocessed GMRES for fast solution of linear equations

arXiv

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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

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

arXiv

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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

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Adaptive Edge Finite Element Method and Numerical Design for Metasurface Cloak

SSRN

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SSRN 2023年

作者： Yang, Wei Wang, Tiancheng Mao, Jiangqiong Hunan Key Laboratory for Computation and Simulation in Science and Engineering School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan411105 China

Metasurfaces are a type of metamaterial that have two-dimensional structures. In comparison to traditional metamaterials, metasurfaces have several advantages such as being lightweight, easily controllable, and simple to design. This paper outlines the phase matching principle and constitutive parameter theory of electromagnetic (EM) metasurfaces, as well as the constitutive parameter analysis and model equations for metasurface cloaks. Due to the singularity of metasurface cloaks, an adaptive edge finite element method (AEFEM) is developed in this paper. Additionally, the convergence theory of time-harmonic Maxwell’s equations for metasurface cloak is established under the T-coercivity assumption. The numerical simulations of arc-shaped metasurface cloaks have confirmed the accuracy of our model and numerical theory. Furthermore, we have utilized our numerical method to design a triangular-shaped metasurface cloak with fewer singularities. © 2023, The Authors. All rights reserved.

关键词： Maxwell equations

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New Upper Solution Bounds of a Class of the Continuous Algebraic Riccati Equations and Their Application in Redundant Control Inputs Systems

SSRN

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SSRN 2023年

作者： Liu, Jianzhou Tang, Fuying School of Mathematics and Computational Science Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Hunan Xiangtan411105 China

In the redundant control inputs problems, the norm of the feedback matrix is smaller, the controlled system is better for control. Therefore, it is the key to solve these problems for studying upper and lower bounds of the symmetric positive definite solutions of the continuous algebra Riccati equations(CAREs) corresponding to the controlled systems. In this paper, we obtain a convergent matrix series as upper bound of the symmetric positive definite solution for a special class of the CAREs. It is the limit of monotonically decreasing upper bound sequence of the symmetric positive definite solution. We use numerous numerical experiments to illustrate that upper bounds obtained are better than the relevant results in the existing literature when the terms are sufficiently large. Moreover, we apply the main theoretical results to the related redundant control inputs systems. Under a wider range of conditions, a class of redundant control inputs problems is solved, and the recent relevant results are improved. © 2023, The Authors. All rights reserved.

关键词： Riccati equations

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Multiscale Adaptive Multifractal Cross-Correlation Analysis of Multivariate Time Series

SSRN

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SSRN 2023年

作者： Jiang, Huanwen Wang, Xinyao Han, Guosheng Key Laboratory of Intelligent Computing and Information Processing Ministry of Education Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Hunan Xiangtan411105 China

In the real world, most of the time series generated from complex systems are nonlinear. To effectively study its fractal properties, in this work, we first generalize the adaptive fractal analysis (AFA) to the adaptive multifractal cross-correlation analysis (AMFCCA), which can be used to study the multifractal cross-correlation between two time series. Considering the complexity of time series from complex systems, we extend AMFCCA to the case of multivariate time series, namely multivariate adaptive multifractal cross-correlation analysis (MV-AMFCCA). In order to detect the multifractality on different time scales, we propose multiscale adaptive multifractal cross-correlation analysis of multivariate time series (MMV-AMFCCA). By the numerical simulation on synthetic multivariate processes, our method shows the theoretical validity. Furthermore, we applied these methods to the multifractal analysis of three pollutants in urban and suburban areas in Beijing. After removing the seasonal trend, we find that the urban and suburban systems both have multifractality, especially the multifractality of the urban system are more evident than the suburban systems, the degree of multifractality in spring and winter stronger than that in summer and autumn. Therefore, our methods are suitable for systems with multiple outputs and provide more comprehensive methods for characterizing autocorrelation and cross-correlation in multivariate time series. © 2023, The Authors. All rights reserved.

关键词： Time series

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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.

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High-Order Interior Penalty Finite Element Methods for Fourth-Order Phase-Field Models in Fracture Analysis

arXiv

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arXiv 2025年

作者： Tian, Tian Chunyu, Chen Wei, Huayi School of Mathematics and Statistics Xiangtan University Hunan Xiangtan411105 China National Center of Applied Mathematics in Hunan Hunan Xiangtan411105 China Hunan Key Laboratory for Computation and Simulation in Science and Engineering Hunan Xiangtan411105 China

This paper presents a novel approach for solving fourth-order phase-field models in brittle fracture mechanics using the Interior Penalty Finite Element Method (IP-FEM). The fourth-order model improves numerical stability and accuracy compared to traditional second-order phase-field models, particularly when simulating complex crack paths. The IP-FEM provides an efficient framework for discretizing these models, effectively handling nonconforming trial functions and complex boundary conditions. In this study, we leverage the FEALPy framework to implement a flexible computational tool that supports high-order IP-FEM discretizations. Our results show that as the polynomial order increases, the mesh dependence of the phase-field model decreases, offering improved accuracy and faster convergence. Additionally, we explore the trade-offs between computational cost and accuracy with varying polynomial orders and mesh sizes. The findings offer valuable insights for optimizing numerical simulations of brittle fracture in practical engineering applications. Copyright © 2025, The Authors. All rights reserved.

关键词： Brittle fracture

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