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.     

Generalizations of the nonstationary multisplitting iterative method for symmetric positive definite linear systems

In this paper, we generalize the nonstationary parallel multisplitting iterative method for solving the symmetric positive definite linear systems. With several choices of variable weighting matrices, the convergence properties of these generalized methods can be improved. Finally, the numerical comparison of several nonstationary parallel multisplitting methods are shown.     

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.     

A NEW GENERALIZED ASYNCHRONOUS PARALLEL MULTISPLITTING ITERATION METHOD

1.IntroductionTosolvelargesparsesystemsoflinearandnonlinearequationsonthemultiprocessorsystems,manyauthorspresentedandstudiedvariousparalleliterativemethodsinthesenseofmultisplittinginrecentyears.FOrdetailsonecanreferto[1]-[9]andreferencestherein.Amo...     

Parallel Newton two‐stage methods based on ILU factorizations for nonlinear systems

Parallel iterative algorithms based on the Newton method and on two of its variants, the Shamanskii method and the Chord method, for solving nonlinear systems are proposed. These algorithms are based on two‐stage multisplitting methods where incomplete LU factorizations are considered as a mean of constructing the inner splittings. Convergence properties of these parallel methods are studied for H‐matrices. Computational results of these methods on two parallel computing systems are discussed. The reported experiments show the effectiveness of these methods. Copyright © 2006 John Wiley & Sons, Ltd.     

PARALLEL CHAOTIC MULTISPLITTING ITERATIVE METHODS FOR THE LARGE SPARSE LINEAR COMPLEMENTARITY PROBLEM

1. IntroductionWe consider the linear complementarity problem LCP(M,q): Find a z E m such thatwhere M = (mij) E boxs and q ~ (qi) 6 m are given real matriX and vector, respectively.This problem axises in various scientific computing areas such as the Nash equilibritun poillt ofa bimatrir game (e.g., Cottle and Dantzig[4] and Lelnke[12j) and the free boundary problems offluid mechedcs (e.g., Cryer[8]). There have been a lot of researches on the approximate solutionof the linear complemeat…     

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.     

A CLASS OF ASYNCHRONOUS MATRIX MULTI-SPLITTING MULTI-PARAMETER RELAXATION ITERATIONS

1.IntroductionMultisplittingmethodsforgettingthesolutionoflargesparsesystemoflinearequationsAx=b,A=(and)6L(Rn)nonsingular,x=(x.),b=(b.)eR"(1.1)areefficientparalleliterativemethodswhicharebasedonseveralsplittingsofthecoefficientmatrixAEL(R").Following[11th…     

Parallel multisplitting two-stage iterative methods for large sparse systems of weakly nonlinear equations

The finite difference or the finite element discretizations of many differential or integral equations often result in a class of systems of weakly nonlinear equations. In this paper, by reasonably applying both the multisplitting and the two-stage iteration techniques, and in accordance with the special properties of this system of weakly nonlinear equations, we first propose a general multisplitting two-stage iteration method through the two-stage multiple splittings of the system matrix. Then, by applying the accelerated overrelaxation (AOR) technique of the linear iterative methods, we present a multisplitting two-stage AOR method, which particularly uses the AOR-like iteration as inner iteration and is substantially a relaxed variant of the afore-presented method. These two methods have a forceful parallel computing function and are much more suitable to the high-speed multiprocessor systems. For these two classes of methods, we establish their local convergence theories, and precisely estimate their asymptotic convergence factors under some suitable assumptions when the involved nonlinear mapping is only directionally differentiable. When the system matrix is either an H-matrix or a monotone matrix, and the nonlinear mapping is a P-bounded mapping, we thoroughly set up the global convergence theories of these new methods. Moreover, under the assumptions that the system matrix is monotone and the nonlinear mapping is isotone, we discuss the monotone convergence properties of the new multisplitting two-stage iteration methods, and investigate the influence of the multiple splittings as well as the relaxation parameters upon the convergence behaviours of these methods. Numerical computations show that our new methods are feasible and efficient for parallel solving of the system of weakly nonlinear equations. This revised version was published online in August 2006 with corrections to the Cover Date.     

Convergence analysis of the parallel multisplitting TOR method

In this paper, we propose the parallel multisplitting TOR method, for solving a large nonsingular systems of linear equations Ax = b. These new methods are a generalization and an improvement of the relaxed parallel multisplitting method (Formmer and Mager, 1989) and parallel multisplitting AOR Algorithm (Wang Deren, 1991). The convergence theorem of this new algorithm is established under the condition that the coefficient matrix A of linear systems is an H-matrix. Some results also yield new convergence theorem for TOR method.     

并行多分裂迭代收敛速率的估计

本文给出了并多分裂迭代（PMI）收敛速度的一个估计式,利用此估计式可以简化和统一PMI方法的收敛性证明。     

A CLASS OF ASYNCHRONOUS PARALLEL MULTISPLITTING RELAXATION METHODS FOR LARGE SPARSE LINEAR COMPLEMENTARITY PROBLEMS

Asynchronous parallel multisplitting relaxation methods for solving large sparse linear complementarity problems are presented, and their convergence is proved when the system matrices are H-matrices having positive diagonal elements. Moreover, block and multi-parameter variants of the new methods, together with their convergence properties,are investigated in detail. Numerical results show that these new methods can achieve high parallel efficiency for solving the large sparse linear complementarity problems on multiprocessor systems.     

并行多分裂AOR迭代法的收敛定理

对解非奇异线性方程组的并行多分裂ＡＯＲ方法,本文给出了该方法的收敛性定理,同时也给出了该方法的迭代矩阵的谱半径的上界估计式。     

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.     

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     

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...     

Synchronous and asynchronous solution of a 3D transport model in a grid computing environment

Numerical simulation is a common approach to understand many phenomena, usually yielding a computationally intensive problem. To overcome insufficient computer capacity and computational speed, a grid computing environment is a suitable approach. In this paper we focus on the development of parallel algorithms to solve a 3D transport model in such a context. The solver is based on the multisplitting Newton method that provides a coarse-grained scheme. Algorithms are implemented using JACE, a grid-enabled Java Asynchronous Computing Environment. This programming environment allows users to design synchronous and asynchronous parallel iterative algorithms as well. Experiments are carried out on a heterogeneous grid environment in which the behaviour of both parallel iterative algorithms is analysed. The results allow us to draw some conclusions about the use of the programming library JACE and the design of parallel iterative algorithms in a grid computing environment.     

On M–multisplittings of singular M–matrices with application to Markov chains

Given a singular M–matrix of a linear system, convergent conditions under which iterative schemes based on M–multisplittings are studied. Two of those conditions, the index of the iteration matrix and its spectral radius are investigated and related to those of the M-matrix. Furthermore, a parallel multisplitting iteration scheme for solving singular linear systems is suggested which can be applied to practical problems such as Poisson and elasticity problems under certain boundary conditions, the Neumann problem, and in Markov chains. A discussion of that multisplitting scheme, based on Gauss–Seidel type splittings is given for computing the stationary distribution vector of Markov chains. In this case a computational viable algorithm can be constructed, since only the nonsingularity of one weighting matrix of the multisplitting is needed. © 1998 John Wiley & Sons, Ltd.     

阶梯矩阵及其一般化在迭代法中的应用

Lu Hao首先给出了阶梯矩阵及其一般性的定义和性质．这类矩阵为迭代法提供了新矩阵分裂的基础．基于此新矩阵类的迭代方法的显著特征是它对于并行计算很容易被实现．应用这一新的分解方法,给出了一般的加速松弛方法（GAOR）,而关于AOR方法的一些性质可以被延伸到该新方法中,并针对Hermite正定矩阵进行了新方法收敛性的分析．最后,给出了一些例子来表明新方法的优越性．     

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。