ON THE CONVERGENCE OF ASYNCHRONOUS NESTED MATRIX MULTISPLITTING METHODS FOR LINEAR SYSTEMS

1.IntroductionTherehasbeenalotofliterature(see[1]--[61and[12])ontheparalleliterativemethodsforthelarge--scalesystemoflinearequationsinthesenseofmatrixmultisplittingsincethepioneeringworkofO'LearyandWhite(see[l])waspublishedin1985.Oneofthemostrecentre...     

广义异步并行多分裂松弛算法

在本文中，我们设计了求解大型线性代数方程组的适用于ＭＩＭＤ系统的异步并行多分裂松弛算法的一般模型，并在系数矩阵是Ｈ－矩阵的条件下，建立了该一般模型的收敛性理论。     

Modified parallel multisplitting iterative methods for non-Hermitian positive definite systems

In this paper we present three modified parallel multisplitting iterative methods for solving non-Hermitian positive definite systems Ax?=?b. The first is a direct generalization of the standard parallel multisplitting iterative method for solving this class of systems. The other two are the iterative methods obtained by optimizing the weighting matrices based on the sparsity of the coefficient matrix A. In our multisplitting there is only one that is required to be convergent (in a standard method all the splittings must be convergent), which not only decreases the difficulty of constructing the multisplitting of the coefficient matrix A, but also releases the constraints to the weighting matrices (unlike the standard methods, they are not necessarily be known or given in advance). We then prove the convergence and derive the convergent rates of the algorithms by making use of the standard quadratic optimization technique. Finally, our numerical computations indicate that the methods derived are feasible and efficient.     

广义异步矩阵多分裂向前向后松弛算法

本文建立了一类广义异步矩阵多分裂向前向后松弛算法，并在系数矩阵是Ｈ－矩阵的条件下，证明了这类算法的收敛性．     

对称正定线性方程组具有任意权矩阵的多分裂迭代求解

本文利用优化模型研究求解对称正定线性方程组Ax=6的多分裂并行算法的权矩阵.在我们的多分裂并行算法中,m个分裂仅要求其中之一为P-正则分裂而其余的则可以任意构造,这不仅大大降低了构造多分裂的难度,而且也放宽了对权矩阵的限制(不像标准的多分裂迭代方法中要求权矩阵为预先给定的非负数量矩阵).并且证明了新的多分裂迭代法是收敛的.最后,通过数值例子展示了新算法的有效性.     

异步并行矩阵多分裂块松弛迭代算法

1 引言 众所周知,许多微分方程经过差分或有限元离散,即可归结为线性代数方程组 Ax=b,A∈L(R~n)非奇异,x,b∈R~n.(1.1)缘于原问题的物理特性,系数矩阵A∈L(R~n)通常是大型稀疏的,并且具有规则的分块结构。鉴此,文[1]基于矩阵多重分裂的概念,并运用线性迭代法的松弛加速技巧,提出了求解这类大型稀疏分块线性代数方程组的并行矩阵多分裂块松弛迭代算法,并在适当的条件下建立了算法的收敛理论。对于SIMD多处理机系统,这类算法是颇为适用和行之有效的。     

A class of asynchronous parallel nonlinear accelerated overrelaxation methods for the nonlinear complementarity problems

In accordance with the principle of using sufficiently the delayed information, and by making use of the nonlinear multisplitting and the nonlinear relaxation techniques, we present in this paper a class of asynchronous parallel nonlinear multisplitting accelerated overrelaxation (AOR) methods for solving the large sparse nonlinear complementarity problems on the high-speed MIMD multiprocessor systems. These new methods, in particular, include the so-called asynchronous parallel nonlinear multisplitting AOR-Newton method, the asynchronous parallel nonlinear multisplitting AOR-chord method and the asynchronous parallel nonlinear multisplitting AOR-Steffensen method. Under suitable constraints on the nonlinear multisplitting and the relaxation parameters, we establish the local convergence theory of this class of new methods when the Jacobi matrix of the involved nonlinear mapping at the solution point of the nonlinear complementarity problem is an H-matrix.     

并行区间矩阵多分裂AOR算法及其收敛性

本文提出了一类求解大型区间线性方程组的并行区间矩阵多分裂松弛算法,并在系数矩阵是区间H-矩阵的条件下,建立了这类算法的收敛理论。     

Comparison theorems for splittings of matrices

Summary. This paper investigates the comparisons of asymptotic rates of convergence of two iteration matrices. On the basis of nonnegative matrix theory, comparisons between two nonnegative splittings and between two parallel multisplitting methods are derived. When the coefficient matrix A is Hermitian positive (semi)definite, comparison theorems about two P-regular splittings and two parallel multisplitting methods are proved. Received April 4, 1998 / Revised version received October 18, 1999 / Published online November 15, 2001     

矩阵多分裂迭代法的收敛条件

本文给出了求解非奇异线性方程组的矩阵多分裂并行迭代法的一些新的收敛结果.当系数矩阵单调和多分裂序列为弱正则分裂时,得到了几个与已有的收敛准则等价的条件,并且证明了异步迭代法在较弱条件下的收敛性.对于同步迭代,给出了与异步迭代不同且较为宽松的收敛条件.     

并行矩阵多分裂块松弛迭代算法

并行矩阵多分裂块松弛迭代算法白中治（复旦大学数学研究所）ＰＡＲＡＬＬＥＬＭＡＴＲＩＸＭＵＬＴＩＳＰＬＩＴＴＩＮＧＢＬＯＣＫＲＥＬＡＸＡＴＩＯＮＩＴＥＲＡＴＩＯＮＭＥＴＨＯＤＳ￥ＢａｔＺｈｏｎｇ－ｚｈｉ（ＩｎｓｔｉｔｕｔｅｏｆＭａｔｈｅｍａｔｉｃｓ，Ｍ...     

并行二级多分裂迭代方法

1．二级多分裂迭代法本义考虑求解线性代数方程组的几种同步与异步二级多分裂迭代法，其中A为nXn非奇异矩阵．多分别选代法考虑A的多种分裂用IF负对角权矩阵EI（ZEI一川进行组合，可得l＝1．多分裂迭代法任给刘始向量。0对k—1，2，…，直到收敛如果（1．2）中对所有l，MI＝Dilg（Al；…；A。。），EI＝（O，…，O，I，O，…，O），则多分裂迭代法退化成块Jacobi迭代法，同样，多分裂迭代法具有自然的并行性，若EI的某个对角元为0，则从的对应分量无需计算．当然，（1．3）可用直接法精确求解，如Gauss消去法，LU分解法等，但有…     

On the convergence of nonstationary multisplitting two-stage iteration methods for hermitian positive definite linear systems

Convergence properties of the nonstationary multisplitting two-stage iteration methods for solving large sparse system of linear equations are further studied when the coefficient matrices are hermitian positive definite matrices.     

A class of asynchronous multisplitting two-stage iterations for large sparse block systems of weakly nonlinear equations

For the block system of weakly nonlinear equations Ax=G(x), where is a large sparse block matrix and is a block nonlinear mapping having certain smoothness properties, we present a class of asynchronous parallel multisplitting block two-stage iteration methods in this paper. These methods are actually the block variants and generalizations of the asynchronous multisplitting two-stage iteration methods studied by Bai and Huang (Journal of Computational and Applied Mathematics 93(1) (1998) 13–33), and they can achieve high parallel efficiency of the multiprocessor system, especially, when there is load imbalance. Under quite general conditions that is a block H-matrix of different types and is a block P-bounded mapping, we establish convergence theories of these asynchronous multisplitting block two-stage iteration methods. Numerical computations show that these new methods are very efficient for solving the block system of weakly nonlinear equations in the asynchronous parallel computing environment.     

线性互补问题的异步并行多分裂松弛迭代算法

将求解线性方程组的异步并行多分裂松弛迭代算法推广到线性互补问题.当问题的系数矩阵为H-矩阵类时,证明了算法的全局收敛性.     

Asynchronous Multisplitting GAOR Method and Asynchronous Multisplitting SSOR Method for Systems of Weakly Nonlinear Equations

In this article, we introduce two new asynchronous multisplitting methods for solving the system of weakly nonlinear equations Ax = G(x) in which A is an n × n real matrix and G(x) = (g ₁(x), g ₂(x), . . . , g _n (x)) ^T is a P-bounded mapping. First, by generalized accelerated overrelaxation (GAOR) technique, we introduce the asynchronous parallel multisplitting GAOR method (including the synchronous parallel multisplitting AOR method as a special case) for solving the system of weakly nonlinear equations. Second, asynchronous parallel multisplitting method based on symmetric successive overrelaxation (SSOR) multisplitting is introduced, which is called asynchronous parallel multisplitting SSOR method. Then under suitable conditions, we establish the convergence of the two introduced methods. The given results contain synchronous multisplitting iterations as a special case.     

Modulus-based synchronous two-stage multisplitting iteration methods for linear complementarity problems

In order to solve large sparse linear complementarity problems on parallel multiprocessor systems, we construct modulus-based synchronous two-stage multisplitting iteration methods based on two-stage multisplittings of the system matrices. These iteration methods include the multisplitting relaxation methods such as Jacobi, Gauss–Seidel, SOR and AOR of the modulus type as special cases. We establish the convergence theory of these modulus-based synchronous two-stage multisplitting iteration methods and their relaxed variants when the system matrix is an H _?+?-matrix. Numerical results show that in terms of computing time the modulus-based synchronous two-stage multisplitting relaxation methods are more efficient than the modulus-based synchronous multisplitting relaxation methods in actual implementations.     

EXPERIMENTAL STUDY OF THE ASYNCHRONOUS MULTISPLITTING RELAXATION METHODS FOR THE LINEAR COMPLEMENTARITY PROBLEMS

We study the numerical behaviours of the relaxed asynchronous multisplitting methods for the linear complementarity problems by solving some typical problems from practical applications on a real multiprocessor system. Numerical results show that the parallel multisplitting relaxation methods always perform much better than the corresponding sequential alternatives, and that the asynchronous multisplitting relaxation methods often outperform their corresponding synchronous counterparts. Moreover, the two-sweep relaxed multisplitting methods have better convergence properties than their corresponding one-sweep relaxed ones in the sense that they have larger convergence domains and faster convergence speeds. Hence, the asynchronous multisplitting unsymmetric relaxation iterations should be the methods of choice for solving the large sparse linear complementarity problems in the parallel computing environments.     

RELAXED ASYNCHRONOUS ITERATIONS FOR THE LINEAR COMPLEMENTARITY PROBLEM

1. IntroductionConsider the large sparse Linear Complementarity Problem (LCP):where .1I = (mkj) 6 L(R") is a gitren real matrix and q = (qk) E R" a given real vector. Thisproblem arises in many areas of scientific computing. FOr example, it arises from problemsin (linear and) contrex quadratic programming, the prob1em of finding a Nash equilibriumpoint of a bimatrix game (e.g., Cottle and Dantzig[5] and Lemke[13]), a11d also a number of freeboundary problems of fluid mechanics (e.g., Cr…     

Two-Stage Multisplitting Iteration Methods Using Modulus-Based Matrix Splitting as Inner Iteration for Linear Complementarity Problems

The matrix multisplitting iteration method is an effective tool for solving large sparse linear complementarity problems. However, at each iteration step we have to solve a sequence of linear complementarity sub-problems exactly. In this paper, we present a two-stage multisplitting iteration method, in which the modulus-based matrix splitting iteration and its relaxed variants are employed as inner iterations to solve the linear complementarity sub-problems approximately. The convergence theorems of these two-stage multisplitting iteration methods are established. Numerical experiments show that the two-stage multisplitting relaxation methods are superior to the matrix multisplitting iteration methods in computing time, and can achieve a satisfactory parallel efficiency.