in: SciCADE97, Grado, Italy, 15-19 september 1997.
It is well known that the performance of GMRES on a distributed memory
platform suffers from the abundant amount of communication required
in the modified Gram-Schmidt process. Some authors have tried to alleviate
this problem by removing synchronization points in the Gram-Schmidt
process. However, this approaches usually increases the amount of
computation and/or diminishes the numerical stability of the process.
In this talk we propose a generalization of GMRES in which the solution
space is decomposed in a set of orthogonal subspaces.
We show that the modified Gram-Schmidt processes can be performed without
any communication to obtain orthogonal bases for these subspaces.
Moreover, the optimal solution can be computed efficiently by solving a
(small) least squares problem using planar rotations. We show that the
method is feasible for linear systems obtained by applying domain
decomposition to PDEs. Experiments indicate that the method requires hardly any communication, apart from the matrix-vector multiplication. Moreover, in
most problems the method converges much faster than GMRES.