A new class of parallel alternating-type iterative methods

This paper is concerned with parallel alternating-type iterative methods for solving large sparse linear systems of the form Au = b arising in the numerical solution of partial differential equations by finite difference methods. Examples of alternating-type methods include the alternating direction implicit (ADI) method and the unsymmetric SOR (USSOR) method. Each iteration of an alternating-type method involves the use of two parameters, say ϱ and ϱ′. We consider parallel alternating-type methods where, given an initial vector u(0), the positive integer m, and two sets of m parameters “ϱi” and “ϱ′j”, one carries out m2 single iterations in parallel, each involving one pair (ϱi,ϱ′j) of the parameters. It is shown that in some cases a linear combination v∗ of the vectors thus obtained is the same as the vector v∗∗ which would be obtained by a sequential process involving m iterations based on the successive use of the parameter pairs (ϱ1, ϱ′1), (ϱ2, ϱ′2),…, (ϱm, ϱ′m). Thus, the parallel procedure offers the potential of reducing the wall-clock time by a factor of m as compared with the sequential procedure. Preliminary numerical results based on the use of a virtual parallel system of sequential computers confirm the expected reductions in the number of iterations.