H-矩阵线性方程组的一类预条件并行多分裂SOR迭代法

基于并行多分裂算法的思想及SOR迭代格式,本文提出一种求解H-矩阵线性方程组新的并行多分裂SOR迭代法,新方法某种程度上避免了SOR迭代法中选取最优参数的困难.同时,选取Kohno等(1997)提出的预条件子P=I+S_α对原始线性方程组进行预处理,进而给出了一种实用的预条件并行多分裂SOR迭代法.理论分析和数值实验均表明,新算法是实用而有效的.     

奇异线性方程组的并行多分裂迭代法

改进了奇异M-矩阵的线性方程组的并行多分裂法的一些最近结果,给出了并行多分裂迭代方法的一些收敛性的理论结果。     

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。     

广义异步矩阵多分裂向前向后松驰算法

本文建立了一类广义异步矩阵多分裂向前向后松驰算法，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。     

矩阵分裂的单调收敛性

本文在非负矩阵分裂条件下证明了迭代算法(3)的单调收敛性,它不仅推广了[1]～[5]中的相应结果,而且在比[7]中定理较弱的条件下,得到了广义AOR迭代法的单调收敛性。本文最后还给出了一个数值例子。     

广义并行矩阵多分裂松弛算法

求解大型线性代数方程组的并行矩阵多分裂算法讨论的大多为系数矩阵是非奇日矩阵的情况，[2]提出了当系数矩阵是非奇H矩阵时的广义矩阵多分裂松弛算法．对系数矩阵是奇异日矩阵的情况研究较少，本文给出了当系数矩阵G是不可约奇异H矩阵时的齐次线性方程组Gx=0的广义矩阵多分裂松弛算法并讨论其收敛性。     

研究简报：并行矩阵多分裂多参数松弛算法

求解大型稀疏线性方程组Ax=b，A∈L(R^n)，x，b∈R^n的并行矩阵多分裂算法最早由[1]提出，[2]提出了当系数矩阵是非奇H—矩阵时的多分裂多参数松弛算法，但是对于奇异H—矩阵的理论及算法的研究结果都很少，为此，     

一类并行数值Schwarz格式的收敛性及比较定理

本文对［１］给出的一类求解椭圆型偏微分方程的并行数值Ｓｃｈｗａｒｚ格式作了进一步的分析，给出了一个新的收敛定理，根据该收敛定理，可以把文［１］中的主要收敛快慢进行了比较，得出了一些比较结果。     

求解线性系统的广义二阶段多分裂迭代法的收敛性

一般二阶段多分裂迭代法的权矩阵都是预先给出的,在迭代过程中并不知道它的优劣.提出了广义的二阶段多分裂迭代法,它的加权矩阵不必预先给出,而是在迭代过程中通过求超平面上的最优解而得出的随迭代步数变化的动态的权矩阵.这样,动态的权矩阵能使得第k步的近似解更加逼近问题的真解.文中建立了新方法的收敛性理论,并以数值实验验证新方法的有效性.     

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.     

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。     

On preconditioned Uzawa methods and SOR methods for saddle-point problems

This paper studies convergence analysis of a preconditioned inexact Uzawa method for nondifferentiable saddle-point problems. The SOR-Newton method and the SOR-BFGS method are special cases of this method. We relax the Bramble-Pasciak-Vassilev condition on preconditioners for convergence of the inexact Uzawa method for linear saddle-point problems. The relaxed condition is used to determine the relaxation parameters in the SOR-Newton method and the SOR-BFGS method. Furthermore, we study global convergence of the multistep inexact Uzawa method for nondifferentiable saddle-point problems.     

M-矩阵线性互补问题模系多分裂迭代方法的收敛性

首先证明了M-矩阵的H-相容分裂都是正则分裂,反之不成立.这表明对于M-矩阵而言,其正则分裂包含H-相容分裂.然后针对系数矩阵为M-矩阵的线性互补问题,建立了两个收敛定理:一是模系多分裂迭代方法关于正则分裂的收敛定理;二是模系二级多分裂迭代方法关于外迭代为正则分裂和内迭代为弱正则分裂的收敛定理.     

Convergence properties of ART and SOR algorithms

Summary ART algorithms with relaxation parameters are studied for general (consistent or inconsistent) linear algebraic systemsRx=f, and a general convergence theorem is formulated. The advantage of severe underrelaxation is reexamined and clarified. The relationship to solutions obtained by applying SOR methods to the equationRR ^T y=f is investigated.The work of this autor was supported by a research grant from the Natural Sciences and Engineering Research Council of Canada     

m步非线性多重分裂Newton法的收敛性

本文讨论了多重分裂算法在求解一类非线性方程组的全局收敛性和单侧收敛性．当用研步Newton法来代替求得每个非线性多重分裂子问题的近似解时，同样给出相应收敛性结论．数值算例证实了算法的有效性．     

Convergence of Non-stationary Parallel Multisplitting Methods for Hermitian Positive Definite Matrices

Non-stationary multisplitting algorithms for the solution of linear systems are studied. Convergence of these algorithms is analyzed when the coefficient matrix of the linear system is hermitian positive definite. Asynchronous versions of these algorithms are considered and their convergence investigated.

     

SOR和AOR迭代收敛的新判别准则

充分条件1和2,对于SOR迭代(0<ω≤1)和AOR迭代(0≤r≤1,0<ω≤1)也是适用的。那么,充分条件3对于SOR迭代和AOR迭代是否适用呢?迄今为止尚没有讨论过。这里我们给予肯定的回答。我们的结论基于如下两个引理。 引理1 如果A的主对角元全不为零,且满足条件(4),则detA≠0。 引理2 对线性方程组(1)的迭代法     

Nonstationary monotone iterative methods for nonlinear partial differential equations

A simple technique is given in this paper for the construction and analysis of monotone iterative methods for a class of nonlinear partial differential equations. With the help of the special nonlinear property we can construct nonstationary parameters which can speed up the iterative process in solving the nonlinear system. Picard, Gauss–Seidel, and Jacobi monotone iterative methods are presented and analyzed for the adaptive solutions. The adaptive meshes are generated by the 1-irregular mesh refinement scheme which together with the M-matrix of the finite element stiffness matrix lead to existence–uniqueness–comparison theorems with simple upper and lower solutions as initial iterates. Some numerical examples, including a test problem with known analytical solution, are presented to demonstrate the accuracy and efficiency of the adaptive and monotone properties. Numerical results of simulations on a MOSFET with the gate length down to 34 nm are also given.     

Convergence of nonstationary multisplitting methods using ILU factorizations

In this paper, we first study convergence of nonstationary multisplitting methods associated with a multisplitting which is obtained from the ILU factorizations for solving a linear system whose coefficient matrix is a large sparse H-matrix. We next study a parallel implementation of the relaxed nonstationary two-stage multisplitting method (called Algorithm 2 in this paper) using ILU factorizations as inner splittings and an application of Algorithm 2 to parallel preconditioner of Krylov subspace methods. Lastly, we provide parallel performance results of both Algorithm 2 using ILU factorizations as inner splittings and the BiCGSTAB with a parallel preconditioner which is derived from Algorithm 2 on the IBM p690 supercomputer.