How to Make Simpler GMRES and GCR More Stable

Abstract

Abstract.In this paper we analyze the numerical behavior of several minimum residual methods which are mathematically equivalent to the GMRES method. Two main approaches are compared: one that computes the approximate solution in terms of a Krylov space basis from an upper triangular linear system for the coordinates, and one where the approximate solutions are updated with a simple recursion formula. We show that a different choice of the basis can significantly influence the numerical behavior of the resulting implementation. While Simpler GMRES and ORTHODIR are less stable due to the ill-conditioning of the basis used, the residual basis is well-conditioned as long as we have a reasonable residual norm decrease. These results lead to a new implementation, which is conditionally backward stable, and they explain the experimentally observed fact that the GCR method delivers very accurate approximate solutions when it converges fast enough without stagnation.

An Intelligent Method for Accelerating the Convergence of Different Versions of SGMRES Algorithm

“…Simpler GMRES (SGMRES) method is a new version of GMRES, which was proposed by Walker and Zhou [2] and analyzed by Jiránek et al [4]. where t m is the solution of the upper triangular system…”

“…In their implementation, they used ܼ = ሾr ‖r ‖ ⁄ , V ୫ିଵ ሿ. Note that different restarted simpler GMRES, called the residual-based restarted simpler GMRES (RB-SGMRES(m)), is proposed in [4]. RB-SGMRES(m) uses…”

“…The implementation of RB-SGMRES(m) is closely similar to that of SGMRES(m) except that ‫ݖ‬ = ‫ݎ‬ ିଵ /ฮ‫ݎ‬ ିଵ ฮ in Step 3.1 in Algorithm 1. In [4], it has been shown that SGMRES(m) is essentially unstable, but RB-SGMRES(m) is conditionally stable provided that we have some reasonable residual decrease.…”

“…Simpler GMRES (SGMRES) method is a new version of GMRES, which was proposed by Walker and Zhou [2] and analyzed by Jiránek et al [4]. SGMRES(m)) is a restarted version of SGMRES which restarts at the iteration when the Krylov subspace reaches dimension m, then the current solution is used as the new initial guess for the next m iterations.…”

Adaptive version of Simpler GMRES

“…However, it was shown in [15,8] that the conditioning of [ r 0 , V n−1 ] is in fact proportional to the inverse of the relative residual norm, i.e., it grows as the residual norm decreases. Therefore the original implementation of the Simpler GM-RES method can suffer from numerical instability due to the ill-conditioning of the basis which moreover leads to the severe ill-conditioning of the upper triangular factor U n in (1.4) possibly affected also by ill-conditioning of A; see the numerical experiments in [8,7]. On the other hand, if the minimum residual method (nearly) stagnates the Simpler GMRES basis [ r 0 , V n−1 ] remains well-conditioned.…”