Parallel Cost Analysis of Adaptive GMRES Implementations for Homotopy Methods

Sosonkina, Maria and Allison, S and Watson, Layne T (1997) Parallel Cost Analysis of Adaptive GMRES Implementations for Homotopy Methods. Technical Report ncstrl.vatech_cs//TR-97-22, Computer Science, Virginia Polytechnic Institute and State University.

Abstract

The success of homotopy methods in solving large-scale optimization problems and nonlinear systems of equations depends heavily on the solution of large sparse nonsymmetric linear systems on parallel architectures. Iterative solution techniques, such as GMRES(k), favor parallel implementations. However, their straightforward parallelization usually leads to a poor parallel performance because of global communication incurred by processors. One variation on GMRES(k) considered here is to adapt the restart value k for any given problem and use Householder reflections in the orthogonalization phase to achieve high accuracy and to reduce the communication overhead. The Householder transformations can be performed without global communications and modified to utilize an arbitrary row distribution of the coefficient matrix. The effect of this modification on the GMRES(k) performance is discussed here, as well as the abilities of parallel GMRES implementations using Householder reflections to maintain fixed efficiency with increase in problem size and number of processors. Theoretical communication cost and isoefficiency analyses are compared with experimental results on an Intel Paragon, Gray T3E, and IBM SP2.