On parallel multisplitting methods for non-Hermitian positive definite linear systems

TL;DR

This paper develops and analyzes parallel multisplitting methods for solving non-Hermitian positive definite linear systems, extending existing methods and establishing their convergence properties on parallel computing platforms.

Contribution

It introduces convergence results for parallel multisplitting methods and extends the positive-definite and skew-Hermitian splitting methods to parallel algorithms.

Findings

Convergence of parallel multisplitting methods established.

Extension of PSS methods to parallel frameworks.

Generalization of results from Hermitian to non-Hermitian cases.

Abstract

To solve non-Hermitian linear system Ax=b on parallel and vector machines, some paralell multisplitting methods are considered. In this work, in particular: i) We establish the convergence results of the paralell multisplitting methods, together with its relaxed version, some of which can be regarded as generalizations of analogous results for the Hermitian positive definite case; ii) We extend the positive-definite and skew-Hermitian splitting (PSS) method methods in [{\em SIAM J. Sci. Comput.}, 26:844--863, 2005] to the parallel PSS methods and propose the corresponding convergence results.