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The Schur complements for SDD1 matrices and their application to linear complementarity problems

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arXiv 2025年

作者： Hu, Yang Liu, Jianzhou Zeng, Wenlong School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan411105 China Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Xiangtan China Department of Mathematics Shanghai University Shanghai200444 China

In this paper we propose a new scaling method to study the Schur complements of SDD1 matrices. Its core is related to the non-negative property of the inverse M-matrix, while numerically improving the Quotient formula. Based on the Schur complement and a novel norm splitting manner, we establish an upper bound for the infinity norm of the inverse of SDD1 matrices, which depends solely on the original matrix entries. We apply the new bound to derive an error bound for linear complementarity problems of B1-matrices. Additionally, new lower and upper bounds for the determinant of SDD1 matrices are presented. Numerical experiments validate the effectiveness and superiority of our results. © 2025, CC BY.

关键词： Matrix algebra

A convergence analysis of a structure-preserving gradient flow method for the all-electron Kohn-Sham model

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arXiv 2023年

作者： Shen, Yedan Wang, Ting Zhou, Jie Hu, Guanghui School of Mathematics and Information Science Guangzhou University Guangzhou China Faculty of Science and Technology University of Macau China Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Xiangtan China School of Mathematical Computational Sciences Xiangtan University Xiangtan China Zhuhai UM Science & Technology Research Institute Zhuhai China Guangdong-Hong Kong-Macao Joint Laboratory for Data-Driven Fluid Mechanics and Engineering Applications University of Macau China

In [Dai et al, Multi. Model. Simul., 2020], a structure-preserving gradient flow method was proposed for the ground state calculation in Kohn–Sham density functional theory, based on which a linearized method was developed in [Hu, et al, EAJAM, accepted] for further improving the numerical efficiency. In this paper, a complete convergence analysis is delivered for such a linearized method for the all-electron Kohn–Sham model. Temporally, the convergence, the asymptotic stability, as well as the structure-preserving property of the linearized numerical scheme in the method is discussed following previous works, while spatially, the convergence of the h-adaptive mesh method is demonstrated following [Chen et al, Multi. Model. Simul., 2014], with a key study on the boundedness of the Kohn–Sham potential for the all-electron Kohn–Sham model. Numerical examples confirm the theoretical results very well. Copyright © 2023, The Authors. All rights reserved.

关键词： Asymptotic stability

Diffeomorphic image registration with an optimal control relaxation and its implementation

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arXiv 2021年

作者： Zhang, Jianping Li, Yanyan School of Mathematics and Computational Science Hunan National Center for Applied Mathematics Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Hunan Xiangtan411105 China School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan411105 China

Image registration has played an important role in image processing problems, especially in medical imaging applications. It is well known that when the deformation is large, many variational models cannot ensure diffeomorphism. In this paper, we propose a new registration model based on an optimal control relaxation constraint for large deformation images, which can theoretically guarantee that the registration mapping is diffeomorphic. We present an analysis of optimal control relaxation for indirectly seeking the diffeomorphic transformation of Jacobian determinant equation and its registration applications, including the construction of diffeomorphic transformation as a special space. We also provide an existence result for the control increment optimization problem in the proposed diffeomorphic image registration model with an optimal control relaxation. Furthermore, a fast iterative scheme based on the augmented Lagrangian multipliers method (ALMM) is analyzed to solve the control increment optimization problem, and a convergence analysis is followed. Finally, a grid unfolding indicator is given, and a robust solving algorithm for using the deformation correction and backtrack strategy is proposed to guarantee that the solution is diffeomorphic. Numerical experiments show that the registration model we proposed can not only get a diffeomorphic mapping when the deformation is large, but also achieves the state-of-the-art performance in quantitative evaluations in comparing with other classical models. Copyright © 2021, The Authors. All rights reserved.

关键词： Dynamical systems

Two-level overlapping Schwarz methods based on local generalized eigenproblems for hermitian variational problems

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arXiv 2021年

作者： Lu, Qing Wang, Junxian Shu, Shi Peng, Jie School of Mathematics and Computational Science Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Hunan Xiangtan411105 China School of Mathematical Sciences South China Normal University Guangzhou510631 China

The research of two-level overlapping Schwarz (TL-OS) method based on constrained energy minimizing coarse space is still in its infancy, and there exist some defects, e.g. mainly for second order elliptic problem and too heavy computational cost of coarse space construction. In this paper, by introducing appropriate assumptions, we propose more concise coarse basis functions for general Hermitian positive and definite discrete systems, and establish the algorithmic and theoretical frameworks of the corresponding TL-OS methods. Furthermore, to enhance the practicability of the algorithm, we design two economical TL-OS preconditioners and prove the condition number estimate. As the first application of the frameworks, we prove that the assumptions hold for the linear finite element discretization of second order elliptic problem with high contrast and oscillatory coefficient and the condition number of the TL-OS preconditioned system is robust with respect to the model and mesh parameters. In particular, we also prove that the condition number of the economically preconditioned system is independent of the jump range under a certain jump distribution. Experimental results show that the first kind of economical preconditioner is more efficient and stable than the existed one. Secondly, we construct TL-OS and the economical TL-OS preconditioners for the plane wave least squares discrete system of Helmholtz equation by using the frameworks. The numerical results for homogeneous and non-homogeneous cases illustrate that the PCG method based on the proposed preconditioners have good stability in terms of the angular frequency, mesh parameters and the number of degrees of freedom in each element. Copyright © 2021, The Authors. All rights reserved.

关键词： Number theory

computationally efficient sandbox algorithm for multifractal analysis of large-scale complex networks with tens of millions of nodes

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Physical Review E 2021年第4期103卷 043303-043303页

作者： Yuemin Ding Jin-Long Liu Xiaohui Li Yu-Chu Tian Zu-Guo Yu Department of Energy and Process Engineering Norwegian University of Science and Technology 7049 Trondheim Norway School of Computer Science Queensland University of Technology Brisbane QLD 4000 Australia Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education and Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Xiangtan Hunan 411105 China School of Information Science and Engineering Wuhan University of Science and Technology Wuhan Hubei 430081 China

Among various algorithms of multifractal analysis (MFA) for complex networks, the sandbox MFA algorithm behaves with the best computational efficiency. However, the existing sandbox algorithm is still computationally expensive for MFA of large-scale networks with tens of millions of nodes. It is also not clear whether MFA results can be improved by a largely increased size of a theoretical network. To tackle these challenges, a computationally efficient sandbox algorithm (CESA) is presented in this paper for MFA of large-scale networks. Distinct from the existing sandbox algorithm that uses the shortest-path distance matrix to obtain the required information for MFA of networks, our CESA employs the compressed sparse row format of the adjacency matrix and the breadth-first search technique to directly search the neighbor nodes of each layer of center nodes, and then to retrieve the required information. A theoretical analysis reveals that the CESA reduces the time complexity of the existing sandbox algorithm from cubic to quadratic, and also improves the space complexity from quadratic to linear. Then the CESA is demonstrated to be effective, efficient, and feasible through the MFA results of (u,v)-flower model networks from the fifth to the 12th generations. It enables us to study the multifractality of networks of the size of about 11 million nodes with a normal desktop computer. Furthermore, we have also found that increasing the size of (u,v)-flower model network does improve the accuracy of MFA results. Finally, our CESA is applied to a few typical real-world networks of large scale.

关键词： Network formation & growth Fractals Fractal dimension characterization

Hm-CONFORMING VIRTUAL ELEMENTS IN ARBITRARY DIMENSION

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arXiv 2021年

作者： Huang, Xuehai Chen, Chunyu Wei, Huayi Hunan Key Laboratory for Computation and Simulation in Science and Engineering School of Mathematics and Computational Science Xiangtan University Xiangtan411105 China School of Mathematics Shanghai University of Finance and Economics Shanghai200433 China

The Hm-conforming virtual elements of any degree k on any shape of polytope in n with m, n ≥ 1 and k ≥ m are recursively constructed by gluing conforming virtual elements on faces in a universal way. For the lowest degree case k = m, the set of degrees of freedom only involves function values and derivatives up to order m − 1 at the vertices of the polytope. The inverse inequality and several norm equivalences for the Hm-conforming virtual elements are rigorously proved. The Hm-conforming virtual elements are then applied to discretize a polyharmonic equation with a lower order term. With the help of the interpolation error estimate and norm equivalences, the optimal error estimates are derived for the Hm-conforming virtual element *** Codes 65N12, 65N15, 65N22, 65N30 Copyright © 2021, The Authors. All rights reserved.

关键词： Degrees of freedom (mechanics)

Invariant-preserving schemes with arbitrarily high-order accuracy for the two-component Camassa–Holm system via Hamiltonian boundary value methods

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Communications in Nonlinear Science and Numerical simulation 2025年 150卷

作者： Qian Luo Aiguo Xiao Min Li Xiaoqiang Yan School of Mathematics and Computational Science & Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Hunan 411105 China Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education Hunan 411105 China School of Mathematics and Physics China University of Geosciences Wuhan 430074 China National Center for Applied Mathematics in Hunan Xiangtan University Hunan 411105 China

In this paper, we present an arbitrarily high-order numerical scheme for the two-component Camassa–Holm system, ensuring the preservation of three invariants: energy and two Casimir functions. The spatial discretization is achieved using Fourier–Galerkin methods, resulting in a semi-discrete system which retains a Hamiltonian structure and approximates the invariants of the original continuous system. Subsequently, an energy-preserving Runge–Kutta method, such as Hamiltonian boundary value methods, is applied to the semi-discrete system, yielding a fully discrete scheme of arbitrarily high order. The proposed scheme is validated through numerical simulations, demonstrating its accuracy and capability in capturing different types of solutions of the two-component Camassa–Holm equation.

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Finite element methods for fractional diffusion equations

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International Journal of Modeling, simulation, and Scientific Computing 2020年第4期11卷 32-52页

作者： Yue Zhao Chen Shen Min Qu Weiping Bu Yifa Tang LSEC ICMSECAcademy of Mathematics and Systems Science Chinese Academy of SciencesBeijing 100190P.R.China School of Mathematical Sciences University of Chinese Academy of Sciences Beijing 100049 P.R.China School of Mathematics and Computational Science&Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Hunan 411105P.R.China

Due to the successful applications in engineering,physics,biology,finance,etc.,there has been substantial interest in fractional diffusion equations over the past few decades,and literatures on developing and analyzing efficient and accurate numerical methods for reliably simulating such equations are vast and fast *** paper gives a concise overview on finite element methods for these equations,which are divided into time fractional,space fractional and time-space fractional diffusion ***,we also involve some relevant topics on the regularity theory,the well-posedness,and the fast algorithm.

关键词： Fractional diffusion equation finite element method regularity theory well-posedness fast algorithm.

NUMERICAL SOLUTIONS OF NONAUTONOMOUS STOCHASTIC DELAY DIFFERENTIAL EQUATIONS BY DISCONTINUOUS GALERKIN METHODS

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Journal of computational Mathematics 2019年第3期37卷 419-436页

作者： Xinjie Dai Aiguo Xiao School of Mathematics and Computational Science Xiangtan UniversityXiangtan 411105China School of Mathematics and Computational Science & Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan UniversityXiangtan 411105China

This paper considers a class of discontinuous Galerkin method,which is constructed by Wong-Zakai approximation with the orthonormal Fourier basis,for numerically solving nonautonomous Stratonovich stochastic delay differential *** prove that the discontinuous Galerkin scheme is strongly convergent:globally stable and analogously asymptotically stable in mean square *** addition,this method can be easily extended to solve nonautonomous Stratonovich stochastic pantograph differential *** tests indicate that the method has first-order and half-order strong mean square convergence,when the diffusion term is without delay and with delay,respectively.

关键词： Discontinuous Galerkin method Wong-Zakai approximation Nonautonomous Stratonovich stochastic delay differential equation