Harmonic and Refined Harmonic Shift-Invert Residual Arnoldi and Jacobi--Davidson Methods for Interior Eigenvalue Problems

TL;DR

This paper introduces refined harmonic residual Arnoldi and Jacobi--Davidson methods for interior eigenvalue problems, demonstrating their efficiency and superiority over standard methods through theoretical analysis and numerical experiments.

Contribution

It develops and analyzes refined harmonic shift-invert residual Arnoldi and Jacobi--Davidson methods, including inexact variants, with practical stopping criteria and improved performance.

Findings

Refined harmonic methods outperform standard harmonic algorithms.

Inexact methods mimic exact counterparts with low to modest accuracy.

Numerical results show significant efficiency gains.

Abstract

This paper concerns the harmonic shift-invert residual Arnoldi (HSIRA) and Jacobi--Davidson (HJD) methods as well as their refined variants RHSIRA and RHJD for the interior eigenvalue problem. Each method needs to solve an inner linear system to expand the subspace successively. When the linear systems are solved only approximately, we are led to the inexact methods. We prove that the inexact HSIRA, RHSIRA, HJD and RHJD methods mimic their exact counterparts well when the inner linear systems are solved with only low or modest accuracy. We show that (i) the exact HSIRA and HJD expand subspaces better than the exact SIRA and JD and (ii) the exact RHSIRA and RHJD expand subspaces better than the exact HSIRA and HJD. Based on the theory, we design stopping criteria for inner solves. To be practical, we present restarted HSIRA, HJD, RHSIRA and RHJD algorithms. Numerical results demonstrate that these algorithms are much more efficient than the restarted standard SIRA and JD algorithms and furthermore the refined harmonic algorithms outperform the harmonic ones very substantially.