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On greedy randomized coordinate updating iteration methods for solving symmetric eigenvalue problems

Applied Numerical mathematics 2025年 216卷 76-97页

作者： Bai, Zhong-Zhi State Key Laboratory of Mathematical Sciences Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China State Key Laboratory of Scientific/Engineering Computing Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences P.O. Box 2719 Beijing100190 China School of Mathematical Sciences University of Chinese Academy of Sciences Beijing100049 China

In order to compute the smallest eigenvalue and its corresponding eigenvector of a large-scale, real, and symmetric matrix, we propose a class of greedy randomized coordinate updating iteration methods based on the principle that the indices of larger entries in absolute value of the current residual are selected with a higher probability and, with respect to this index set, the next iterate is updated from the current iterate along with the selected coordinate such that the corresponding entry of the residual is annihilated, resulting in fast convergence rates of the proposed iteration methods. Under appropriate conditions, we prove the convergence of both sequences of the Rayleigh-quotients and the acute angles between the iterates and the eigenvector in terms of the expectation. By numerical experiments, we show that this class of greedy randomized coordinate updating iteration methods are advantageous over the parameterized power method and the coordinate descent method in both iteration counts and computing times. Moreover, with theoretical analysis and computational performance, we confirm that the convergence property of this class of iteration methods can be improved significantly by suitably choosing the arbitrary nonnegative parameter involved in the greedy probability criterion. © 2025 IMACS

关键词： Eigenvalues and eigenfunctions

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A Stochastic Gradient Descent Method for computational Design of Random Rough Surfaces in Solar Cells

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Communications in computational Physics 2023年第10期34卷 1361-1390页

作者： Qiang Li Gang Bao Yanzhao Cao Junshan Lin Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing 100190China Department of Mathematics Zhejiang UniversityChina Department of Mathematics and Statistics Auburn UniversityAuburnAL 36849USA

In this work,we develop a stochastic gradient descent method for the computational optimal design of random rough surfaces in thin-film solar *** formulate the design problems as random PDE-constrained optimization problems and seek the optimal statistical parameters for the random *** optimizations at fixed frequency as well as at multiple frequencies and multiple incident angles are *** evaluate the gradient of the objective function,we derive the shape derivatives for the interfaces and apply the adjoint state method to perform the *** stochastic gradient descent method evaluates the gradient of the objective function only at a few samples for each iteration,which reduces the computational cost *** numerical experiments are conducted to illustrate the efficiency of the method and significant increases of the absorptance for the optimal random *** also examine the convergence of the stochastic gradient descent algorithm theoretically and prove that the numerical method is convergent under certain assumptions for the random interfaces.

关键词： Optimal design random rough surface solar cell Helmholtz equation stochastic gradient descent method

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A VARIATIONAL ANALYSIS FOR THE MOVING FINITE ELEMENT METHOD FOR GRADIENTFLOWS

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Journal of computational mathematics 2023年第2期41卷 191-210页

作者： Xianmin Xu LSEC Institute of Computational Mathematics and Scientific/Engineering ComputingNCMISAMSSChinese Academy of SciencesBeijing 100190China School of Mathematical Sciences University of Chinese Academy of SciencesBeijing 100049China

By using the Onsager principle as an approximation tool,we give a novel derivation for the moving finite element method for gradient flow *** show that the discretized problem has the same energy dissipation structure as the continuous *** enables us to do numerical analysis for the stationary solution of a nonlinear reaction diffusion equation using the approximation theory of free-knot piecewise *** show that under certain conditions the solution obtained by the moving finite element method converges to a local minimizer of the total energy when time goes to *** global minimizer,once it is detected by the discrete scheme,approximates the continuous stationary solution in optimal *** examples for a linear diffusion equation and a nonlinear Allen-Cahn equation are given to verify the analytical results.

关键词： Moving finite element method Convergence analysis Onsager principle

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A New Sixth-Order WENO Scheme for Solving Hyperbolic Conservation Laws

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Communications on Applied mathematics and Computation 2023年第1期5卷 3-30页

作者： Kunlei Zhao Yulong Du Li Yuan State Key Laboratory of Scientific and Engineering Computing(LSEC)and Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing 100190China School of Mathematical Sciences University of Chinese Academy of SciencesBeijing 100190China School of Mathematical Sciences Beihang UniversityBeijing 100191China

In this paper,we develop a new sixth-order WENO scheme by adopting a convex combina-tion of a sixth-order global reconstruction and four low-order local *** the classical WENO schemes,the associated linear weights of the new scheme can be any positive numbers with the only requirement that their sum equals ***,a very simple smoothness indicator for the global stencil is *** new scheme can achieve sixth-order accuracy in smooth *** tests in some one-and two-dimensional bench-mark problems show that the new scheme has a little bit higher resolution compared with the recently developed sixth-order WENO-Z6 scheme,and it is more efficient than the classical fifth-order WENO-JS5 scheme and the recently developed sixth-order WENO6-S scheme.

关键词： Global smoothness indicator Linear weights Sixth-order accuracy WENO

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Positive definiteness of real quadratic forms resulting from the variable-step L1-type approximations of convolution operators

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Science China mathematics 2024年第2期67卷 237-252页

作者： Hong-Lin Liao Tao Tang Tao Zhou School of Mathematics Nanjing University of Aeronautics and AstronauticsNanjing 211106China Division of Science and Technology BNU-HKBU United International CollegeZhuhai 519087China SUSTech International Center for Mathematics Shenzhen 518055China NCMIS&LSEC Institute of Computational Mathematics and Scientific/Engineering ComputingAcademy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing 100190China

The positive definiteness of real quadratic forms with convolution structures plays an important rolein stability analysis for time-stepping schemes for nonlocal operators. In this work, we present a novel analysistool to handle discrete convolution kernels resulting from variable-step approximations for convolution *** precisely, for a class of discrete convolution kernels relevant to variable-step L1-type time discretizations, weshow that the associated quadratic form is positive definite under some easy-to-check algebraic conditions. Ourproof is based on an elementary constructing strategy by using the properties of discrete orthogonal convolutionkernels and discrete complementary convolution kernels. To our knowledge, this is the first general result onsimple algebraic conditions for the positive definiteness of variable-step discrete convolution kernels. Using theunified theory, we obtain the stability for some simple nonuniform time-stepping schemes straightforwardly.

关键词： discrete convolution kernels positive definiteness variable time-stepping orthogonal convolution kernels complementary convolution kernels

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METRICALLY REGULAR MAPPING AND ITS UTILIZATION TO CONVERGENCE ANALYSIS OF A RESTRICTED INEXACT NEWTON-TYPE METHOD

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Journal of computational mathematics 2022年第1期40卷 44-69页

作者： Mohammed Harunor Rashid Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing 100190China Department of Mathematics Faculty of ScienceUniversity of RajshahiRajshahi-6205Bangladesh

In the present paper,we study the restricted inexact Newton-type method for solving the generalized equation 0∈f(x)+F(x),where X and Y are Banach spaces,f:X→Y is a Frechet differentiable function and F:X■Y is a set-valued mapping with closed *** establish the convergence criteria of the restricted inexact Newton-type method,which guarantees the existence of any sequence generated by this method and show this generated sequence is convergent linearly and quadratically according to the particular assumptions on the Frechet derivative of ***,we obtain semilocal and local convergence results of restricted inexact Newton-type method for solving the above generalized equation when the Frechet derivative of f is continuous and Lipschitz continuous as well as f+F is metrically *** application of this method to variational inequality is *** addition,a numerical experiment is given which illustrates the theoretical result.

关键词： Generalized equation Restricted inexact Newton-type method Metrically regular mapping Partial Lipschitz-like mapping Semilocal convergence.

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A Derivative-free Trust-region Method for Optimization on the Ellipsoid

A Derivative-free Trust-region Method for Optimization on th...

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2023 International Conference on Advances in Computer Science and Engineering Technology, ACSE 2023

作者： Xie, Pengcheng State Key Laboratory of Scientific and Engineering Computing Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences University of Chinese Academy of Sciences ZhongGuanCun East Road No. 55 Beijing China

Optimization methods play a crucial role in various fields and applications. In some optimization problems, the derivative information of the objective function is unavailable. Such black-box optimization problems need to be solved by derivative-free optimization methods. At the same time, optimization problems with ellipsoidal constraints are important and have widespread applications in various fields as well. Following the development of the late professor M. J. D. Powell's efficient derivative-free trust-region optimization methods, this paper considers solving derivative-free optimization problems on the ellipsoid. Our new optimization solver EC-NEWUOA for problems on the ellipsoid in ., n is designed based on Powell's derivative-free software NEWUOA for unconstrained optimization problems. The proposed techniques for our new method mainly include using the Courant penalty function, the augmented Lagrangian method, and the projection technique. Details about the method and theoretical analysis are included in this paper. We also compare our new method with other algorithms by solving test problems and then show the numerical advantages of our new method. © Published under licence by IOP Publishing Ltd.

关键词： Lagrange multipliers

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NVERSE CONDUCTIVITY PROBLEM WITH INTERNAL DATA

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Journal of computational mathematics 2023年第3期41卷 482-500页

作者： Faouzi Triki Tao Yin Laboratoire Jean Kuntzmann UMR CNRS 5224Université Grenoble-Alpes700 Avenue Centrale38401 Saint-Martin-d’HèresFrance Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems ScienceChinese Academy of ScienceBeijing 100190China

This paper concerns the reconstruction of a scalar coefficient of a second-order elliptic equation in divergence form posed on a bounded domain from internal *** problem finds applications in multi-wave imaging,greedy methods to approximate parameter-dependent elliptic problems,and image treatment with partial differential *** first show that the inverse problem for smooth coefficients can be rewritten as a linear transport *** that the coefficient is known near the boundary,we study the well-posedness of associated transport equation as well as its numerical resolution using discontinuous Galerkin *** propose a regularized transport equation that allow us to derive rigorous convergence rates of the numerical method in terms of the order of the polynomial approximation as well as the regularization *** finally provide numerical examples for the inversion assuming a lower regularity of the coefficient,and using synthetic data.

关键词： Inverse problems Multi-wave imaging Static transport equation Internal data Diffusion coeffcient Stability estimates Regularization

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DG-GMsFEM for Dual Continuum Transport Problem in Perforated Domains

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Lobachevskii Journal of mathematics 2024年第11期45卷 5329-5342页

作者： Alekseev, V.N. Xie, W. Laboratory ‘‘Computing Technologies and Artificial Intelligence’’ North-Eastern Federal University Yakutsk 677980 Russian Federation School of Mathematics and Computational Science Xiangtan University National Center for Applied Mathematics in Hunan Hunan International Scientific and Technological Innovation Cooperation Base of Computational Science Xiangtan Hunan 677980 China

In this paper, we develop a Discontinuous Galerkin Generalized Multiscale Finite Element Method (DG-GMsFEM) for solving dual continuum transport problems in perforated domains. The mathematical model includes the convection-diffusion equations for concentrations, with flow described by the Stokes equations. The model equations are derived by homogenizing small perforations using multicontinuum homogenization techniques. After performing this homogenization, we obtain appropriate parameters for the dual continuum transport model. The resulting system includes larger perforations, which are resolved via multiscale basis functions. For the coarse grid approximation, we introduce a multiscale model reduction technique that constructs local multiscale basis functions to generate a lower-dimensional model. The proposed reduction technique is based on DG-GMsFEM and involves two steps: (1) constructing a snapshot space and (2) solving local spectral problems for dimension reduction in this space. We present numerical results for a two-dimensional problem to demonstrate the method’s performance. Our numerical results show that the proposed method provides good results with a significant reduction of the system size. © Pleiades Publishing, Ltd. 2024.

关键词： convection-diffusion equation DG-GMsFEM flow problem IPDG multiscale method perforated domains transport problems

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Rigorous biaxial limit of a molecular-theory-based two-tensor hydrodynamics

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Journal of Differential Equations 2023年第1期366卷 862-911页

作者： Li, Sirui Xu, Jie School of Mathematics and Statistics Guizhou University Guiyang 550025 China LSEC NCMIS Institute of Computational Mathematics and Scientific/Engineering Computing (ICMSEC) Academy of Mathematics and Systems Science (AMSS) Chinese Academy of Sciences Beijing China

We consider a two-tensor hydrodynamics derived from the molecular model, where high-order tensors are determined by closure approximation through the maximum entropy state or the quasi-entropy. We prove the existence and uniqueness of local in time smooth solutions to the two-tensor system. Then, we rigorously justify the connection between the molecular-theory-based two-tensor hydrodynamics and the biaxial frame hydrodynamics. More specifically, in the framework of Hilbert expansion, we show the convergence of the solution to the two-tensor hydrodynamics to the solution to the frame hydrodynamics. © 2023 Elsevier Inc.

关键词： Biaxial limit Frame hydrodynamics Liquid crystals Two-tensor hydrydynamics

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