The Effect of Symmetric Permutations on the Convergence of a Restarted GMRES Solver with ILU-Type Preconditioners

Abstract

This paper is concerned with applying heuristics for bandwidth reduction as a preprocessing step of a restarted Generalized Minimal Residual (GMRES for short) solver preconditioned by ILU-type preconditioners. Hundreds of heuristics have been proposed to solve the problem of bandwidth reduction since the mid-1960s. Previous publications have reviewed several heuristics for bandwidth reduction. Based on this experience, this paper evaluates nine low-cost symmetric permutations. Numerical simulations are presented to investigate the influence of these orderings on the convergence of the preconditioned GMRES solver restarted every 50 steps when applied to large-scale nonsymmetric and not positive definite matrices. This paper shows the most promising combination of preconditioner and ordering for each linear system used.