Parallel Adaptive GMRES Implementations for Homotopy Methods

Abstract

The success of probability-one homotopy methods in solving large-scale optimization problems and nonlinear systems of equations on parallel architectures may be significantly enhanced by the accurate parallel solution of large sparse nonsymmetric linear systems. 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 of GMRES(k) considered here is to adapt the restart value k for any given problem and use Householder reflections in the orthogonalization phase, coupled with graph-based matrix partitioning, to achieve high accuracy and reduce the communication overhead. This particular GMRES implementation is tailored to the uniquely stringent requirements imposed on a linear system solver by probability-one homotopy algorithms: occasionally unusually high accuracy, ability to adapt to problems of widely varying difficulty, and parallelism.