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。     

ON MONOTONE CONVERGENCE OF NONLINEARMULTISPLITTING RELAXATION METHODS

ＯＮＭＯＮＯＴＯＮＥＣＯＮＶＥＲＧＥＮＣＥＯＦＮＯＮＬＩＮＥＡＲＭＵＬＴＩＳＰＬＩＴＴＩＮＧＲＥＬＡＸＡＴＩＯＮＭＥＴＨＯＤＳ￥ＷＡＮＧＤＥＲＥＮ；ＢＡＩＺＨＯＮＧＺＨＩ（ＤｅｐａｚａｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＳｈｓｌｌｇｈａｉＵｎｉｖｅ...     

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…     

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。     

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.     

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...     

ON THE CONVERGENCE OF THE RELAXATION METHODS FOR POSITIVE DEFINITE LINEAR SYSTEMS

1.IntroductionTheclassicaliterativemethods,suchastheJacobimethod,theGauss-SeidelmethodandtheSORmethod,aswellastheirsymmetrizedvariants,playanimportantroleforsolvingthelargesparsesystemoflinearequationsInaccordancewiththebasicextrapolationprincipleofthelineariterativemethod,Hadjidimos[1]furtherproposedaclassofacceleratedoverrelaxation(AOR)methodforsolyingthelinearsystem(1.1)in1978.Thismethodincludestwoarbitraryparameters,andtheirsuitablechoicesnotonlycannaturallyrecovertheJacobi,theGauss-S…     

ON THE CONVERGENCE OF WAVEFORM RELAXATION METHODS FOR LINEAR INITIAL VALUE PROBLEMS

We study a class of blockwise waveform relaxation methods,and investigate its con-vergence properties in both asymptotic and monotone senses.In addition,the monotoneconvergence rates between different pointwise/blockwise waveform relaxation methods re-sulted from different matrix splittings,and those between the pointwise and blockwisewaveform relaxation methods are discussed in depth.     

广义预条件迭代方法

§1.引言和新方法的提出 设线性代数方程组 Ax=b,(1.1)这里A是n阶非奇异矩阵,x,b是n维向量且b是已知向量,x是未知向量.对于(1.1)的数值解,我们考虑如下的分裂:     

A NEW GENERALIZED ASYNCHRONOUS PARALLEL MULTISPLITTING ITERATION METHOD

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

二阶锥线性互补问题的广义模系矩阵分裂迭代算法

通过将二阶锥线性互补问题转化为等价的不动点方程,介绍了一种广义模系矩阵分裂迭代算法,并研究了该算法的收敛性.进一步,数值结果表明广义模系矩阵分裂迭代算法能够有效地求解二阶锥线性互补问题.     

ON SOLVABILITY AND WAVEFORM RELAXATION METHODS FOR LINEAR VARIABLE-COEFFICIENT DIFFERENTIAL-ALGEBRAIC EQUATIONS

This paper is concerned with the solvability and waveform relaxation methods of linear variable-coefficient differential-algebraic equations （DAEs）. Most of the previous works have been focused on linear variable-coefficient DAEs with smooth coefficients and data, yet no results related to the convergence rate of the corresponding waveform relaxation methods has been obtained. In this paper, we develope the solvability theory for the linear variable-coefficient DAEs on Legesgue square-integrable function space in both traditional and least squares senses, and determine the convergence rate of the waveform relaxation methods for solving linear variable-coefficient DAEs.     

波形松弛算法及其在计算流体力学中的应用

本文介绍求解线性常系数微分代数方程组的波形松弛算法, 基于Laplace积分变换得到该算法新的收敛理论. 进一步将波形松弛算法应用于求解非定常Stokes方程, 介绍并讨论了连续时间波形松弛算法CABSOR算法和离散时间波形松弛算法DABSOR算法.     

ALMOST CONVERGENCE AND DOUBLE SEQUENTIAL BAND MATRIX

The class f of almost convergent sequences was introduced by G.G. Lorentz,using the idea of the anach limits [A contribution to the theory of divergent sequences,Acta Math. 80(1948), 167–190]. Let f0( ) and f( ) be the domain of the double sequential band matrix ( r, s) in the sequence spaces f0 and f. In this article, the β-and γ-duals of the space f( ) are determined. Additionally, we give some inclusion theorems concerning with the spaces f0( ) and f( ). Moreover, the classes(f( ) : μ) and(μ : f( )) of infinite matrices are characterized, and the characterizations of some other classes are also given as an application of those main results, where μ is an arbitrary sequence space.     

WAVEFORM RELAXATION METHODS AND ACCURACY INCREASE

ＷＡＶＥＦＯＲＭ　ＲＥＬＡＸＡＴＩＯＮ　ＭＥＴＨＯＤＳ　ＡＮＤ　ＡＣＣＵＲＡＣＹ　ＩＮＣＲＥＡＳＥＷＡＶＥＦＯＲＭＲＥＬＡＸＡＴＩＯＮＭＥＴＨＯＤＳＡＮＤＡＣＣＵＲＡＣＹＩＮＣＲＥＡＳＥ￥ＳｏｎｇＹｏｎｇｚｈｏｎｇ（ＮａｎｊｉｎｇＮｏｒｍａｌＵｎｉｖ...     

WAVEFORM RELAXATION METHODS OF NONLINEAR INTEGRAL-DIFFERENTIAL-ALGEBRAIC EQUATIONS

In this paper we consider continuous-time and discrete-time waveform relaxation meth-ods for general nonlinear integral-differential-algebraic equations. For the continuous-time case we derive the convergence condition of the iterative methods by invoking the spec-tral theory on the resulting iterative operators. By use of the implicit difference forms,namely the backward-differentiation formulae, we also yield the convergence condition of the discrete waveforms. Numerical experiments are provided to illustrate the theoretical work reported here.     

ON THE CONVERGENCE DOMAIN OF THEMATRIX MULTISPLITTING RELAXATION METHODS FOR LINEAR SYSTEMS

The convergence of the parallel matrix multisplitting relaxation methods presented by Wang (Linear Algebra and Its Applications 154/156 (1991) 473-486) is further investigated.The investigations show that these relaxation methods really have considerably larger convergence domains.     

COMPONENTWISE CONDITION NUMBERS FOR GENERALIZED MATRIX INVERSION AND LINEAR LEAST SQUARES

We present componentwise condition numbers for the problems of Moore-Penrose generalized matrix inversion and linear least squares. Also, the condition numbers for these condition numbers are given.     

THE MONOTONICITY OF CONVERGENCE RATE FOR MGS METHODS

In this paper we prove that the asymptotic rate of convergence of the modified Gauss-Seidel method of a non-singular M-matrix is a monotonic function for 1 precondition parameters 0 ≤ αi≤1/2, （i = 1,2,... ,n - 1）.     

ON SPECTRAL METHODS FOR VOLTERRA INTEGRAL EQUATIONS AND THE CONVERGENCE ANALYSIS

The main purpose of this work is to provide a novel numerical approach for the Volterra integral equations based on a spectral approach. A Legendre-collocation method is proposed to solve the Volterra integral equations of the second kind. We provide a rigorous error analysis for the proposed method, which indicates that the numerical errors decay exponentially provided that the kernel function and the source function are sufficiently smooth. Numerical results confirm the theoretical prediction of the exponential rate of convergence. The result in this work seems to be the first successful spectral approach （with theoretical justification） for the Volterra type equations.