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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.

THREE-PRECISION ITERATIVE REFINEMENT WITH PARAMETER REGULARIZATION AND PREDICTION FOR SOLVING LARGE SPARSE LINEAR SYSTEMS

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arXiv 2025年

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

This study presents a novel mixed-precision iterative refinement algorithm, GADI-IR, within the general alternating-direction implicit (GADI) framework, designed for efficiently solving large-scale sparse linear systems. By employing low-precision arithmetic, particularly half-precision (FP1.), for computationally intensive inner iterations, the method achieves substantial acceleration while maintaining high numerical accuracy. {1. challenges such as overflow in half-precision and convergence issues for low precision are addressed through careful backward error analysis and the application of a regularization parameter α. Furthermore, the integration of low-precision arithmetic into the parameter prediction process, using Gaussian Process Regression (GPR), significantly reduces computational time without degrading performance. The method is particularly effective for large-scale linear systems arising from discretized partial differential equations and other high-dimensional problems, where both accuracy and efficiency are critical. Numerical experiments demonstrate that the use of mixed-precision strategies not only accelerates computation but also ensures robust convergence, making the approach advantageous for various applications. The results highlight the potential of leveraging lower-precision arithmetic to achieve superior computational efficiency in high-performance computing. © 2025, CC BY.

关键词： Convergence of numerical methods

Analysis of a two-grid method for semiconductor device problem

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Applied Mathematics and Mechanics(English Edition) 2021年第1期42卷 143-158页

作者： Ying LIU Yanping CHEN Yunqing HUANG Qingfeng LI School of Mathematics and Computational Science Hunan Key Laboratory for Computation and Simulation in Science and EngineeringXiangtan UniversityXiangtan 411105Hunan ProvinceChina College of Information and Intelligence Hunan Agricultural UniversityChangsha 410128China School of Mathematical Sciences South China Normal UniversityGuangzhou 510631China

The mathematical model of a semiconductor device is governed by a system of quasi-linear partial differential *** electric potential equation is approximated by a mixed finite element method,and the concentration equations are approximated by a standard Galerkin *** estimate the error of the numerical solutions in the sense of the *** linearize the full discrete scheme of the problem,we present an efficient two-grid method based on the idea of Newton *** main procedures are to solve the small scaled nonlinear equations on the coarse grid and then deal with the linear equations on the fine *** estimation for the two-grid solutions is analyzed in *** is shown that this method still achieves asymptotically optimal approximations as long as a mesh size satisfies H=O(h^1.2).Numerical experiments are given to illustrate the efficiency of the two-grid method.

关键词： two-grid method semiconductor device mixed finite element method Galerkin method L^q error estimate

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(201.)653–680]and[***,***,***,***.442(2021.1.051.],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

Investigation on a quantum algorithm for linear differential equations

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arXiv 2024年

作者： Dong, Xiaojing Peng, Yizhe Tang, Qili Yang, Yin Yu, Yue Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education National Center for Applied Mathematics in Hunan School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan411105 China

Ref. [BCOW1.] introduced a pioneering quantum approach (coined BCOW algorithm) for solving linear differential equations with optimal error tolerance. Originally designed for a specific class of diagonalizable linear differential equations, the algorithm was extended by Krovi in [Kro23] to encompass broader classes, including non-diagonalizable and even singular matrices. Despite the common misconception, the original algorithm is indeed applicable to non-diagonalizable matrices, with diagonalisation primarily serving for theoretical analyses to establish bounds on condition number and solution error. By leveraging basic estimates from [Kro23], we derive bounds comparable to those outlined in the Krovi algorithm, thereby reinstating the advantages of the BCOW approach. Furthermore, we extend the BCOW algorithm to address time-dependent linear differential equations by transforming non-autonomous systems into higher-dimensional autonomous ones, a technique also applicable for the Krovi algorithm. © 2024, CC BY-NC-ND.

关键词： Consensus algorithm

The grad-div conforming virtual element method for the quad-div problem in three dimensions

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arXiv 2024年

作者： Dong, Xiaojing Han, Yibing Huang, Yunqing 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 Er huan Road Hunan Xiangtan411105 China

We propose a new stable variational formulation for the quad-div problem in three dimensions and prove its well-posedness. Using this weak form, we develop and analyze the H(grad-div)-conforming virtual element method of arbitrary approximation orders on polyhedral meshes. Three families of H(grad-div)conforming virtual elements are constructed based on the structure of a de Rham sub-complex with enhanced smoothness, resulting in an exact discrete virtual element complex. In the lowest-order case, the simplest element has only one degree of freedom at each vertex and face, respectively. We rigorously prove the interpolation error estimates, the stability of discrete bilinear forms, the well-posedness of discrete formulation and the optimal error estimates. Some numerical examples are shown to verify the theoretical results. © 2024, CC BY-NC-ND.

关键词： Mesh generation

Low-rank generalized alternating direction implicit iteration method for solving matrix equations

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arXiv 2024年

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

This paper presents an effective low-rank generalized alternating direction implicit iteration (R-GADI) method for solving large-scale sparse and stable Lyapunov matrix equations and continuous-time algebraic Riccati matrix equations. The method is based on generalized alternating direction implicit iteration (GADI), which exploits the low-rank property of matrices and utilizes the Cholesky factorization approach for solving. The advantage of the new algorithm lies in its direct and efficient low-rank formulation, which is a variant of the Cholesky decomposition in the Lyapunov GADI method, saving storage space and making it computationally effective. When solving the continuous-time algebraic Riccati matrix equation, the Riccati equation is first simplified to a Lyapunov equation using the Newton method, and then the R-GADI method is employed for computation. Additionally, we analyze the convergence of the R-GADI method and prove its consistency with the convergence of the GADI method. Finally, the effectiveness of the new algorithm is demonstrated through corresponding numerical experiments. Copyright © 2024, The Authors. All rights reserved.

关键词： Riccati equations

Multiscale Adaptive Multifractal Cross-Correlation Analysis of Multivariate Time Series

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SSRN

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

A general alternating direction implicit iteration method for solving complex symmetric linear systems

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arXiv 2024年

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

We have introduced the generalized alternating direction implicit iteration (GADI) method for solving large sparse complex symmetric linear systems and proved its convergence properties. Additionally, some numerical results have demonstrated the effectiveness of this algorithm. Furthermore, as an application of the GADI method in solving complex symmetric linear systems, we utilized the flattening operator and Kronecker product properties to solve Lyapunov and Riccati equations with complex coefficients using the GADI method. In solving the Riccati equation, we combined inner and outer iterations, first simplifying the Riccati equation into a Lyapunov equation using the Newton method, and then applying the GADI method for solution. Finally, we provided convergence analysis of the method and corresponding numerical results. Copyright © 2024, The Authors. All rights reserved.

关键词： Riccati equations

Low-rank alternating direction doubling algorithm for solving large-scale continuous time algebraic Riccati equations

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arXiv 2024年

This paper proposes an effective low-rank alternating direction doubling algorithm (R-ADDA) for computing numerical low-rank solutions to large-scale sparse continuous-time algebraic Riccati matrix equations. The method is based on the alternating direction doubling algorithm (ADDA), utilizing the low-rank property of matrices and employing Cholesky factorization for solving. The advantage of the new algorithm lies in computing only the 2k-th approximation during the iterative process, instead of every approximation. Its efficient low-rank formula saves storage space and is highly effective from a computational perspective. Finally, the effectiveness of the new algorithm is demonstrated through theoretical analysis and numerical experiments. Copyright © 2024, The Authors. All rights reserved.

关键词： Matrix algebra