Asynchronous multisplitting iteration with different weighting schemes

Abstract

The multisplitting iteration method was presented by O’Leary and White [5] for solving large sparse linear systems on parallel multiprocessor system. In this paper, we further set up an asynchronous variant for the multisplitting iteration method with different weighting schemes studied by White [8]. Moreover, we establish a general convergence criterion for asynchronous iteration framework, and then prove the convergence of the new asynchronous multisplitting iteration method with different weighting schemes by making use of this general criterion.

Introduction

A multisplitting of a nonsingular matrix $A\in L(R^n)$, as introduced in [5], is a collection of triples of $n\times n$ matrices $(B_i\text{,}{C}_i\text{,}{E}_i)(i=1\text{,}2\text{,}\dots \text{,}\alpha )$ ($\alpha ⩽n$, an integer) with

• $A=B_i-C_i\text{,}i=1\text{,}2\text{,}\dots \text{,}\alpha$;

• $B_i$ nonsingular, $i=1\text{,}2\text{,}\dots \text{,}\alpha$;

• for $i=1\text{,}2\text{,}\dots \text{,}\alpha$, the matrices $E_i=\mathit{diag}(e_1^{(i)}\text{,}{e}_2^{(i)}\text{,}\dots \text{,}{e}_n^{(i)})$ being diagonal with nonnegative entries

$$e_m^{(i)}=\left\{\begin{array}{cc}{e}_m^{(i)}>0\text{,}\hfill & m\in J_i\text{,}\hfill \\ 0\text{,}\hfill & m\notin J_i\text{,}\hfill \end{array}\right.m=1\text{,}2\text{,}\dots \text{,}n\text{,}$$

such that ${\sum }_{i=1}^{\alpha }{E}_i=I$ (the $n\times n$ identity matrix), where $J_i(i=1\text{,}2\text{,}\dots \text{,}\alpha )$ are nonempty subsets of $\{1\text{,}2\text{,}\dots \text{,}n\}$, satisfying

$$J_i\subseteq \{1\text{,}2\text{,}\dots \text{,}n\}\text{,}{\cup }_{i=1}^{\alpha }{J}_i=\{1\text{,}2\text{,}\dots \text{,}n\}\text{;}$$

see also [3].

Given a parameter $\lambda \in [0\text{,}1]$ and a starting vector $x^0\in R^n$, the corresponding multisplitting iteration (depending on $\lambda$) defined in [8] for solving the system of linear equations

$$\mathit{Ax}=b$$

is described in the following.

Method 1.1 Given an initial guess $x^0\in R^n$, for $p=0\text{,}1\text{,}2\text{,}\dots$ compute

$$x^{p+1}=H_{\lambda }{x}^p+G_{\lambda }b\text{,}$$

where

$$G_{\lambda }=\sum _{i=1}^{\alpha }{E}_i^{\lambda }{B}_i^{-1}{E}_i^{1-\lambda }\text{,}{H}_{\lambda }=I-G_{\lambda }A\text{.}$$

For a kind of particularly structured coefficient matrix $A\in L(R^n)$ and its specially chosen multisplitting, White [8] and Frommer and Mayer [4] discussed the convergence properties of Method 1.1. Furthermore, Bai [2] gave a convergence theorem for this method; see also [3].

The paper is organized as follows. In Section 2, we propose an asynchronous multisplitting iteration method with different weighting schemes. In Section 3, some preliminaries are given, and a convergence criterion about a general asynchronous iteration framework is established. The convergence of the new method is analyzed in Section 4.