Volume 5, pp. 48-61, 1997.

Asynchronous weighted additive Schwarz methods

Andreas Frommer, Hartmut Schwandt, and Daniel B. Szyld


A class of asynchronous Schwarz methods for the parallel solution of nonsingular linear systems of the form $Ax=f$ is investigated. This class includes, in particular, an asynchronous algebraic Schwarz method as well as asynchronous multisplitting. Theorems are obtained demonstrating convergence for the cases when $A^{-1}$ is nonnegative and when $A$ is an $H$-matrix. The results shown are for both the situations with or without overlap between the domains in which an underlying mesh is divided, if such a mesh exists. Numerical experiments on systems of up to over ten million variables on up to 256 processors are presented. They illustrate the convergence properties of the method, as well as the fact that when the domains are not all of the same size, the asynchronous method can be up to 50% faster than the corresponding synchronous one.

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Key words

Asynchronous methods, monotone matrices, H-matrices, linear system, parallel algorithms, multisplittings, additive Schwartz.

AMS subject classifications

65F10, 65Y05.

Links to the cited ETNA articles

[5]Vol. 3 (1995), pp. 24-38 Rafael Bru, Violeta Migallón, José Penadés, and Daniel B. Szyld: Parallel, synchronous and asynchronous two-stage multisplitting methods

ETNA articles which cite this article

Vol. 13 (2002), pp. 38-55 Pierre Spiteri, Jean-Claude Miellou, and Didier El Baz: Perturbation of parallel asynchronous linear iterations by floating point errors
Vol. 30 (2008), pp. 323-345 Christian Mense and Reinhard Nabben: On algebraic multilevel methods for non-symmetric systems - convergence results

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