Any nonincreasing convergence curves are simultaneously possible for GMRES and weighted GMRES, as well as for left and right preconditioned GMRES

TL;DR

This paper demonstrates that any convergence behavior can be achieved by GMRES and weighted GMRES through appropriate matrix choices, revealing the flexibility of convergence curves under different preconditioning strategies.

Contribution

It extends Greenbaum et al.'s result to weighted GMRES, showing that any convergence curve can be realized with a suitable weight matrix and characterizing its form.

Findings

Any two convergence curves are simultaneously possible for left and right preconditioned GMRES.

Existence of a weight matrix M for any convergence curve in weighted GMRES.

Necessary and sufficient conditions on M for prescribing two convergence curves.

Abstract

The convergence of the GMRES linear solver is notoriously hard to predict. A particularly enlightening result by [Greenbaum, Pt\'ak, Strako\v{s}, 1996] is that, given any convergence curve, one can build a linear system for which GMRES realizes that convergence curve. What is even more extraordinary is that the eigenvalues of the problem matrix can be chosen arbitrarily. We build upon this idea to derive novel results about weighted GMRES. We prove that for any linear system and any prescribed convergence curve, there exists a weight matrix M for which weighted GMRES (i.e., GMRES in the inner product induced by M) realizes that convergence curve, and we characterize the form of M. Additionally, we exhibit a necessary and sufficient condition on M for the simultaneous prescription of two convergence curves, one realized by GMRES in the Euclidean inner product, and the other in the inner product induced by M. These results are then applied to infer some properties of preconditioned GMRES when the preconditioner is applied either on the left or on the right. For instance, we show that any two convergence curves are simultaneously possible for left and right preconditioned GMRES.