GMRES convergence bounds for eigenvalue problems

TL;DR

This paper derives new, sharper convergence bounds for GMRES when solving eigenvalue problems, emphasizing the influence of the right hand side and proposing tailored preconditioning strategies.

Contribution

It provides detailed bounds for (block) GMRES considering the structure of the right hand side, leading to improved preconditioning methods for eigenvalue computations.

Findings

New GMRES bounds are significantly sharper than traditional ones.

Preconditioned subspace iteration with tuned or polynomial preconditioners performs well.

The bounds explain the initial rapid decrease in GMRES residuals.

Abstract

The convergence of GMRES for solving linear systems can be influenced heavily by the structure of the right hand side. Within the solution of eigenvalue problems via inverse iteration or subspace iteration, the right hand side is generally related to an approximate invariant subspace of the linear system. We give detailed and new bounds on (block) GMRES that take the special behavior of the right hand side into account and explain the initial sharp decrease of the GMRES residual. The bounds give rise to adapted preconditioners applied to the eigenvalue problems, e.g. tuned and polynomial preconditioners. The numerical results show that the new (block) GMRES bounds are much sharper than conventional bounds and that preconditioned subspace iteration with either a tuned or polynomial preconditioner should be used in practice.