Minimal residual rational Krylov subspace method for sequences of shifted linear systems

TL;DR

This paper introduces a new rational Krylov subspace method with minimal residuals and a specialized pole selection strategy to efficiently solve challenging sequences of shifted linear systems, especially nonsymmetric cases with complex shifts.

Contribution

The paper presents a novel projection strategy and pole selection procedure for rational Krylov methods, improving convergence on difficult shifted linear systems.

Findings

Outperforms existing methods on challenging nonsymmetric problems

Faster convergence due to tailored pole selection

Effective for complex, non-conjugate shift scenarios

Abstract

The solution of sequences of shifted linear systems is a classic problem in numerical linear algebra, and a variety of efficient methods have been proposed over the years. Nevertheless, there still exist challenging scenarios witnessing a lack of performing solvers. For instance, state-of-the-art procedures struggle to handle nonsymmetric problems where the shifts are complex numbers that do not come as conjugate pairs. We design a novel projection strategy based on the rational Krylov subspace equipped with a minimal residual condition. We also devise a novel pole selection procedure, tailored to our problem, providing poles for the rational Krylov basis construction that yield faster convergence than those computed by available general-purpose schemes. A panel of diverse numerical experiments shows that our novel approach performs better than state-of-the-art techniques, especially on the very challenging problems mentioned above.