GPMR: An Iterative Method for Unsymmetric Partitioned Linear Systems

TL;DR

GPMR is a new iterative method for efficiently solving unsymmetric 2x2 block linear systems, showing faster convergence and better orthogonality resilience compared to GMRES variants.

Contribution

The paper introduces GPMR, a novel iterative approach that reduces two matrices simultaneously and improves efficiency over existing methods like GMRES.

Findings

GPMR terminates earlier than GMRES with 10-50% fewer iterations.

GPMR requires less storage and computational work per iteration.

GPMR demonstrates greater resilience to loss of orthogonality than Block-GMRES.

Abstract

We introduce an iterative method named GPMR for solving 2x2 block unsymmetric linear systems. GPMR is based on a new process that reduces simultaneously two rectangular matrices to upper Hessenberg form and that is closely related to the block-Arnoldi process. GPMR is tantamount to Block-GMRES with two right-hand sides in which the two approximate solutions are summed at each iteration, but requires less storage and work per iteration. We compare the performance of GPMR with GMRES and Block-GMRES on linear systems from the SuiteSparse Matrix Collection. In our experiments, GPMR terminates significantly earlier than GMRES on a residual-based stopping condition with an improvement ranging from around 10% up to 50% in terms of number of iterations. We also illustrate by experiment that GPMR appears more resilient to loss of orthogonality than Block-GMRES.