Flexible GMRES converges in two phases

TL;DR

This paper derives a precise upper bound on the residuals of the flexible GMRES method, revealing a two-phase convergence behavior influenced by the inner preconditioner's tolerance.

Contribution

It provides a sharp, proven bound on FGMRES residuals and characterizes its two-phase convergence depending on preconditioner tolerance.

Findings

FGMRES exhibits two distinct convergence phases.

The residual bound is proven to be tight and cannot be improved.

Convergence behavior varies with the inner preconditioner's tolerance.

Abstract

We derive a sharp upper bound on the residuals produced by the flexible GMRES (FGMRES) method. The bound shows that FGMRES exhibits two phases of convergence depending on the residual tolerance of the inner preconditioner. For small tolerances, the convergence of FGMRES is practically geometric with a constant rate throughout, while for looser tolerances the two-phase behavior becomes more pronounced. We also show that the derived bound cannot be improved and construct an example for which it becomes an equality.