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SEMI-PROXIMAL POINT METHOD FOR NONSMOOTH CONVEX-CONCAVE MINIMAX OPTIMIZATION

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Journal of Computational Mathematics 2024年第3期42卷 617-637页

作者： Yuhong Dai Jiani Wang Liwei zhang lsec ICMSECAMSSChinese Academy of SciencesBeijing 100190China School of Mathematical SciencesUniversity of Chinese Academy of SciencesBeijing 100049China Academy of Mathematics and Systems Science Chinese Academy of SciencesBeijing 100190China School of Mathematical Sciences Dalian University of TechnologyDalian 116024China

Minimax optimization problems are an important class of optimization problems arising from modern machine learning and traditional research areas.While there have been many numerical algorithms for solving smooth convex-concave minimax problems,numerical algorithms for nonsmooth convex-concave minimax problems are rare.This paper aims to develop an efficient numerical algorithm for a structured nonsmooth convex-concave minimax problem.A semi-proximal point method(SPP)is proposed,in which a quadratic convex-concave function is adopted for approximating the smooth part of the objective function and semi-proximal terms are added in each subproblem.This construction enables the subproblems at each iteration are solvable and even easily solved when the semiproximal terms are cleverly chosen.We prove the global convergence of our algorithm under mild assumptions,without requiring strong convexity-concavity condition.Under the locally metrical subregularity of the solution mapping,we prove that our algorithm has the linear rate of convergence.Preliminary numerical results are reported to verify the efficiency of our algorithm.

关键词： Minimax optimization Convexity-concavity Global convergence Rate of con-vergence Locally metrical subregularity

ORDER REDUCED METHODS FOR QUAD-CURL EQUATIONS WITH NAVIER TYPE BOUNDARY CONDITIONS

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Journal of Computational Mathematics 2020年第4期38卷 565-579页

作者： Weifeng zhang shuo zhang Department of Engineering Mechanics and Department of Applied Mathematics Kunming University of Science and TechnologyKunming 650500China shuo zhang lsec Institute of Computational Mathematics and Scientific/Engineering ComputingAcademy of Mathematics and System SciencesChinese Academy of SciencesBeijing 100190China

Quad-curl equations with Navier type boundary conditions are studied in this paper.Stable order reduced formulations equivalent to the model problems are presented,and finite element discretizations are designed.Optim... 详细信息

关键词： Quad-curl equation Order reduced scheme Regularity analysis Finite element method

Convergent and Orthogonality Preserving Schemes for Approximating the Kohn-Sham Orbitals

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Numerical Mathematics(Theory,Methods and Applications) 2023年第1期16卷 1-25页

作者： Xiaoying Dai Liwei zhang Aihui Zhou lsec Institute of Computational Mathematics and Scientific/Engineering ComputingAcademy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing 100190China School of Mathematical Sciences University of Chinese Academy of SciencesBeijing 100049China

To obtain convergent numerical approximations without using any orthogonalization operations is of great importance in electronic structure calculations.In this paper,we propose and analyze a class of iteration schemes for the discretized Kohn-Sham Density Functional Theory model,with which the iterative approximations are guaranteed to converge to the Kohn-Sham orbitals without any orthogonalization as long as the initial orbitals are orthogonal and the time step sizes are given properly.In addition,we present a feasible and efficient approach to get suitable time step sizes and report some numerical experiments to validate our theory.

关键词： Gradient flow based model density functional theory orthogonality preserving scheme convergence temporal discretization

3D HYBRID DEPTH MIGRATION AND FOUR-WAY SPLITTING SCHEMES

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Journal of Computational Mathematics 2006年第4期24卷 463-474页

作者： Wen-sheng zhang Guan-quan zhang lsec ICMSEC Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing 100080 China

The alternately directional implicit （ADI） scheme is usually used in 3D depth migration. It splits the 3D square-root operator along crossline and inline directions alternately. In this paper, based on the ideal of data line, the four-way splitting schemes and their splitting errors for the finite-difference （FD） method and the hybrid method are investigated. The wavefield extrapolation of four-way splitting scheme is accomplished on a data line and is stable unconditionally. Numerical analysis of splitting errors show that the two-way FD migration have visible numerical anisotropic errors, and that four-way FD migration has much less splitting errors than two-way FD migration has. For the hybrid method, the differences of numerical anisotropic errors between two-way scheme and four-way scheme are small in the case of lower lateral velocity variations. The schemes presented in this paper can be used in 3D post-stack or prestack depth migration. Two numerical calculations of 3D depth migration are completed. One is the four-way FD and hybrid 3D post-stack depth migration for an impulse response, which shows that the anisotropic errors can be eliminated effectively in the cases of constant and variable velocity variations. The other is the 3D shot-profile prestack depth migration for SEG/EAEG benchmark model with two-way hybrid splitting scheme, which presents good imaging results. The Message Passing Interface （MPI） programme based on shot number is adopted.

关键词： 3D depth migration Multiway splitting Data line Wavefield computation Finite-difference Hybrid method.

A SET OF SYMMETRIC QUADRATURE RULES ON TRIANGLES AND TETRAHEDRA

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Journal of Computational Mathematics 2009年第1期27卷 89-96页

作者： Linbo zhang Tao Cui Hui Liu lsec ICMSEC Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing 100190 China

We present a program for computing symmetric quadrature rules on triangles and tetrahedra. A set of rules are obtained by using this program. Quadrature rules up to order 21 on triangles and up to order 14 on tetrahedra have been obtained which are useful for use in finite element computations. All rules presented here have positive weights with points lying within the integration domain.

关键词： Finite element Numerical integration Quadrature Cubature Triangle Tetrahedron.

Transition pathways in Cylinder-Gyroid interface

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Communications in Computational Physics 2022年第8期32卷 810-828页

作者： Xiaomei Yao Jie Xu Lei zhang Beijing International Center for Mathematical Research Peking UniversityBeijing 100871China lsec&NCMIS Institute of Computational Mathematics and Scientific/Engineering Computing(ICMSEC)Academy of Mathematics and Systems Science(AMSS)Chinese Academy of SciencesBeijingChina

When two distinct ordered phases contact,the interface may exhibit rich and fascinating structures.Focusing on the Cylinder-Gyroid interface system,transition pathways connecting various interface morphologies are studied armed with the Landau–Brazovskii model.Specifically,minimum energy paths are obtained by computing transition states with the saddle dynamics.We present four primary transition pathways connecting different local minima,representing four different mechanisms of the formation of the Cylinder-Gyroid interface.The connection of Cylinder and Gyroid can be either direct or indirect via Fddd with three different orientations.Under different displacements,each of the four pathways may have the lowest energy.

关键词： Landau-Brazovskii model Cylinder-Gyroid interface saddle dynamics transition state transition pathway

Optimal L_∞ Estimates for Galerkin Methods for Nonlinear Singular Two-point Boundary Value Problems

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Acta Mathematicae Applicatae Sinica 2015年第3期31卷 719-728页

作者： Xu zhang Zhong-ci SHI Nanjing University of Aeronautics and Astronautics lsec ICMSEC Academy of Mathematics and Systems Science CAS

In this paper, we apply the symmetric Galerkin methods to the numerical solutions of a kind of singular linear two-point boundary value problems. We estimate the error in the maximum norm. For the sake of obtaining full superconvergence uniformly at all nodal points, we introduce local mesh refinements. Then we extend these results to a class of nonlinear problems. Finally, we present some numerical results which confirm our theoretical conclusions.

关键词： singular two-point boundary value problems symmetric Galerkin method maximum norm error estimate superconvergence local mesh refinement

Convergence analysis of the formal energies of symplectic methods for Hamiltonian systems

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Science China Mathematics 2016年第2期59卷 379-396页

作者： zhang RuiLi TANG YiFa ZHU BeiBei TU XiongBiao ZHAO Yue lsec icmsecacademy of mathematics and systems sciencechinese academy of sciencesBeijing 100190China

Based on Feng's theory of formal vector fields and formal flows, we study the convergence problem of the formal energies of symplectic methods for Hamiltonian systems and give the clear growth of the coefficients in the formal energies. With the help of B-series and Bernoulli functions, we prove that in the formal energy of the mid-point rule, the coefficient sequence of the merging products of an arbitrarily given rooted tree and the bushy trees of height 1(whose subtrees are vertices), approaches 0 as the number of branches goes to ∞; in the opposite direction, the coefficient sequence of the bushy trees of height m(m ≥ 2), whose subtrees are all tall trees, approaches ∞ at large speed as the number of branches goes to +∞. The conclusion extends successfully to the modified differential equations of other Runge-Kutta methods. This disproves a conjecture given by Tang et al.(2002), and implies:(1) in the inequality of estimate given by Benettin and Giorgilli(1994) for the terms of the modified formal vector fields, the high order of the upper bound is reached in numerous cases;(2) the formal energies/formal vector fields are nonconvergent in general case.

关键词：收敛性分析哈密顿系统辛方法能量形式向量场树木高度微分方程场理论

AN ADAPTIVE HYBRID STRESS TRANSITION QUADRILATERAL FINITE ELEMENT METHOD FOR LINEAR ELASTICITY

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Journal of Computational Mathematics 2016年第4期34卷 339-364页

作者： Feiteng Huang Xiaoping Xie Chensong zhang School of Mathematics Sichuan University Chengdu 610064 China lsec Academy of Mathematics and System Sciences Beijing 100190 China

In this paper, we discuss an adaptive hybrid stress finite element method on quadri- lateral meshes for linear elasticity problems. To deal with hanging nodes arising in the adaptive mesh refinement, we propose new transition types of hybrid stress quadrilateral elements with 5 to 7 nodes. In particular, we derive a priori error estimation for the 5- node transition hybrid stress element to show that it is free from Poisson-locking, in the sense that the error bound in the a priori estimate is independent of the Lam~ constant A. We introduce~ for quadrilateral meshes, refinement/coarsening algorithms, which do not require storing the refinement tree explicitly, and give an adaptive algorithm. Finally, we provide some numerical results.

关键词： Hybrid stress element Transition element Adaptive method Quadrilateralmesh Poisson-locking Plane elasticity.