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Numerical simulation of Richtmyer-Meshkov instability

Science China Mathematics 2004年第z1期47卷 234-244页

作者： FU Dexun, MA Yanwen, ZHANG Linbo & TIAN BaolinState key laboratory of Nonlinear Mechanics, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100080, China State key laboratory of scientific and engineering computing, Institute of Computational Mathematics, Chinese Academy of Sciences, Beijing 100080, China 1. State Key Laboratory of Nonlinear Mechanics Institute of Mechanics Chinese Academy of Sciences 100080 Beijing China2. State Key Laboratory of Scientific and Engineering Computing Institute of Computational Mathematics Chinese Academy of Sciences 100080 Beijing China

The compressible Navier-Stokes equations discretized with a fourth order accurate compact finite difference scheme with group velocity control are used to simulate the Richtmyer-Meshkov (R-M) instability problem produced by cylindrical shock-cylindrical material interface with shock Mach number Ms=1.2 and density ratio 1:20 (interior density/outer density). Effect of shock refraction, reflection, interaction of the reflected shock with the material interface, and effect of initial perturbation modes on R-M instability are investigated numerically. It is noted that the shock refraction is a main physical mechanism of the initial phase changing of the material surface. The multiple interactions of the reflected shock from the origin with the interface and the R-M instability near the material interface are the reason for formation of the spike-bubble structures. Different viscosities lead to different spike-bubble structure characteristics. The vortex pairing phenomenon is found in the initial double mode simulation. The mode interaction is the main factor of small structures production near the interface.

关键词： R-M instability, direct numerical simulation, shock-interface interaction.

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Construction and Analysis of Structured Preconditioners for Block Two-by-Two Matrices

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Journal of Shanghai University(English Edition) 2004年第4期8卷 397-405页

作者：白中治 State Key Laboratory of Scientific/Engineering Computing Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and System Sciences Chinese Academy of Sciences Beijing 100080 P.R. China

For the large sparse block two-by-two real nonsingular matrices, we establish a general framework of structured preconditioners through matrix transformation and matrix approximations. For the specific versions such as modified block Jacobi-type, modified block Gauss-Seidel-type, and modified block unsymmetric (symmetric) Gauss-Seidel-type preconditioners, we precisely describe their concrete expressions and deliberately analyze eigenvalue distributions and positive definiteness of the preconditioned matrices. Also, we show that when these structured preconditioners are employed to precondition the Krylov subspace methods such as GMRES and restarted GMRES, fast and effective iteration solvers can be obtained for the large sparse systems of linear equations with block two-by-two coefficient matrices. In particular, these structured preconditioners can lead to high-quality preconditioning matrices for some typical matrices from the real-world applications.

关键词： block two-by-two matrix preconditioner modified block relaxation iteration eigenvalue distribution positive definiteness.

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Teracluster LSSC-II--Its designing principles and applications in large scale numerical simulations

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中国科学A辑（英文版） 2004年第z1期47卷 53-68页

作者： Zhang Linbo Chen Shiyi LI Xinliang CAO Jianwen ZHANG Wensheng GONG Xingao State Key Laboratory of Scientific and Engineering Computing Academy of Mathematics and System Sciences Chinese Academy of Sciences Beijing 100080 China Department of Mechanical Engineering The Johns Hopkins University and Center for Computational Science and Engineering Peking University Beijing 100875 China Institute of Mechanics Chinese Academy of SciencesBeijing 100080 China Parallel Computing Laboratory Institute of Software Chinese Academy of Sciences Beijing 100080 China Academy of Mathematics and System Sciences Chinese Academy of Sciences Beijing 100080 China Surface Physics Laboratory National Key Laboratory Fudan University Shanghai 200433 Institute of Solid State Physics Chinese Academy of Sciences Hefei 230031 China

The teracluster LSSC-II installed at the State key laboratory of scientific and engineering computing, Chinese Academy of Sciences is one of the most powerful PC clusters in China. It has a peek performance of 2Tflops. With a Linpack performance of 1.04Tflops, it is ranked at the 43rd place in the 20th TOP500 List (November 2002), 51st place in the 21st TOP500 List (June 2003), and the 82nd place in the 22nd TOP500 List (November 2003) with a new Linpack performance of 1.3Tflops. In this paper, we present some design principles of this cluster, as well as its applications in some large-scale numerical simulations.

关键词： high performance computing PC cluster large scale numerical simulations

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A class of globally convergent conjugate gradient methods

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Science China Mathematics 2003年第2期46卷 251-261页

作者：戴彧虹袁亚湘 State Key Laboratory of Scientific and Engineering Computing Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and System Sciences

Conjugate gradient methods are very important ones for solving nonlinear optimization problems,especially for large scale problems. However, unlike quasi-Newton methods, conjugate gradient methods wereusually analyzed individually. In this paper, we propose a class of conjugate gradient methods, which can beregarded as some kind of convex combination of the Fletcher-Reeves method and the method proposed byDai et al. To analyze this class of methods, we introduce some unified tools that concern a general methodwith the scalarβk having the form of φk/φk-1. Consequently, the class of conjugate gradient methods canuniformly be analyzed.

关键词： unconstrained optimization, conjugate gradient, line search, global convergence.

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A NEW FAMILY OF TRUST REGION ALGORITHMS FOR UNCONSTRAINED OPTIMIZATION

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Journal of Computational Mathematics 2003年第2期21卷 221-228页

作者： Yuhong Dai Dachuan Xu(State key laboratory of scientific/engineering computing, Institute of Computational Mathematicsand scientific/engineering computing, Academy of Mathematics and System Sciences, ChineseAcademy of Sciences, P.O. Box 2719, Beijing 100080, China) Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and System Sciences Chinese Academy of Sciences State Key Laboratory of Scientific/Engineering Computing 北京 100080

Trust region (TR) algorithms are a class of recently developed algorithms for nonlinear optimization. A new family of TR algorithms for unconstrained optimization, which is the extension of the usual TR method, is presented in this paper. When the objective function is bounded below and continuously, differentiable, and the norm of the Hesse approximations increases at most linearly with the iteration number, we prove the global convergence of the algorithms. Limited numerical results are reported, which indicate that our new TR algorithm is competitive.

关键词： trust region method global convergence quasi-Newton method unconstrained optimization nonlinear programming.

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LINEAR SYSTEMS ASSOCIATED WITH NUMERICAL METHODS FOR CONSTRAINED OPITMIZATION

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Journal of Computational Mathematics 2003年第1期21卷 71-84页

作者： Y. Yuan(State key laboratory of scientific and engineering computing, Institute of Computational Mathematics and scientific/engineering computing, Chinese Academy of Sciences, P. O. Box 2719,Beijing 100080, China.) Institute of Computational Mathematics and Scientific/Engineering Computing Chinese Academy of Sciences P. 0. Box 2719 State Key Laboratory of Scientific and Engineering Computing

Linear systems associated with numerical methods for constrained optimization are discussed in this paper. It is shown that the corresponding subproblems arise in most well-known methods, no matter line search methods or trust region methods for constrained optimization can be expressed as similar systems of linear equations. All these linear systems can be viewed as some kinds of approximation to the linear system derived by the Lagrange-Newton method. Some properties of these linear systems are analyzed.

关键词： Constrained optimization Linear equations Lagrange-Newton method Trust region Line search

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On the regularity of trust region-cg algorithm: With application to deconvolution problem

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Science China Mathematics 2003年第3期46卷 312-325页

作者：王彦飞 Laboratory of Remote Sensing Information Sciences Institute of Remote Sensing ApplicationsChinese Academy of Sciences State Key Laboratory of Scientific and Engineering Computing Institute of Computational Mathematicsand Scientific/Engineering ComputingAcademy of Mathematics and System SciencesChinese Academy of SciencesBeijing 100080China

Deconvolution problem is a main topic in signal processing. Many practical applications are re-quired to solve deconvolution problems. An important example is image reconstruction. Usually, researcherslike to use regularization method to deal with this problem. But the cost of computation is high due to thefact that direct methods are used. This paper develops a trust region-cg method, a kind of iterative methodsto solve this kind of problem. The regularity of the method is proved. Based on the special structure of thediscrete matrix, FFT can be used for calculation. Hence combining trust region-cg method with FFT is suitablefor solving large scale problems in signal processing.

关键词： ill-posed problems, efficient implementation, trust region-cg method, regularity.

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A Restarted Conjugate Gradient Method for Ill-posed Problems

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Acta Mathematicae Applicatae Sinica 2003年第1期19卷 31-40页

作者： Yan-fei Wanglaboratory of Remote Sensing Information Sciences, Institute of Remote Sensing Applications, Chinese Academy of Sciences, P.O. Box 9718, Beijing 100101, ChinaState key laboratory of scientific and engineering computing, Institute of Computational Mathematics and scientific/engineering computing, Chinese Academy of Sciences, P.O. Box 2719, Beijing 100080, China Laboratory of Remote Sensing Information Sciences Institute of Remote Sensing Applications Chinese Academy of Sciences Beijing China State Key Laboratory of Scientific and Engineering Computing Institute of Computational Mathematics and Scientific/Engineering computing Chinese Academy of Sciences Beijing China

Abstract This paper presents a restarted conjugate gradient iterative algorithm for solving ill-posed problems. The damped Morozov's discrepancy principle is used as a stopping rule. Numerical experiments are give... 详细信息

关键词： keywords Ill-posed problems restarted CG damped discrepancy principle.

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Backlund Transformations for the Initial Problems of Nizhnich and Nizhnich-Novikov-Veselov Equations

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理论物理通讯(英文版) 2003年第7期40卷 15-18页

作者： SU Hong-Liang WANG Ming-Liang QIN Meng-Zhao State Key Laboratory of Scientific and Engineering Computing Institute of Computational Mathematics and Scien-tific/Engineering Computing Academy of Mathematics and System Science the Chinese Academy of Sciences Beijing 100080 China Mathematical Department Luoyang Institute of Technology Luoyang 471039 China

The homogeneous balance method is a method for solving general partial differential equations (PDEs). Inthis paper we solve a kind of initial problems of the PDEs by using the special Backlund transformations of the initialproblem. The basic Fourier transformation method and some variable-separation skill are used as auxiliaries. Two initialproblems of Nizhnich and the Nizhnich-Novikov-Veselov equations are solved by using this approach.

关键词： Backlund transformation initial problem Nizhnick equation Nizhnich Novikov-Veselov equation homogeneous balance method

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A hybrid preconditioner of banded matrix approximation and alternating direction implicit iteration for symmetric Sinc-Galerkin linear systems

A hybrid preconditioner of banded matrix approximation and a...

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作者： Ng, Michael K. Bai, Zhong-Zhi Department of Mathematics University of Hong Kong Pokfulam Road Hong Kong Hong Kong State Key Laboratory of Scientific/Engineering Computing Institute of Computational Mathematics and Scientific/Engineering Computing Chinese Academy of Sciences P.O. Box 2719 Beijing 100080 China

The symmetric Sinc-Galerkin method applied to a sparable second-order self-adjoint elliptic boundary value problem gives rise to a system of linear equations(Ψx⊗Dy+Dx⊗Ψ y)u=g,where⊗ is the Kronecker product symbol, Ψx and Ψy are Toeplitz-plus-diagonal matrices, and Dx and Dy are diagonal matrices. The main contribution of this paper is to present and analyze a two-step preconditioning strategy based on the banded matrix approximation (BMA) and the alternating direction implicit (ADI) iteration for these Sinc-Galerkin systems. In particular, we show that the two-step preconditioner is symmetric positive definite, and the condition number of the preconditioned matrix is bounded by the convergence factor of the involved ADI iteration. Numerical examples show that the new preconditioner is practical and efficient to precondition the conjugate gradient method for solving the above symmetric Sinc-Galerkin linear system. © 2003 Elsevier Science Inc.

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