On Inner Iterations in the Shift-Invert Residual Arnoldi Method and the Jacobi--Davidson Method

TL;DR

This paper develops a convergence theory for the Shift-Invert Residual Arnoldi (SIRA) and Jacobi--Davidson methods, showing that inexact inner solves with modest accuracy do not significantly affect outer iteration convergence.

Contribution

It introduces a new convergence analysis for SIRA and JD methods, establishing that low-accuracy inner solves suffice for effective outer iteration convergence.

Findings

Inexact SIRA mimics exact SIRA with low/modest inner solve accuracy.

Practical stopping criteria for inner solves are proposed.

Numerical experiments confirm the effectiveness of inexact SIRA and JD methods.

Abstract

Using a new analysis approach, we establish a general convergence theory of the Shift-Invert Residual Arnoldi (SIRA) method for computing a simple eigenvalue nearest to a given target $\sigma$ and the associated eigenvector. In SIRA, a subspace expansion vector at each step is obtained by solving a certain inner linear system. We prove that the inexact SIRA method mimics the exact SIRA well, that is, the former uses almost the same outer iterations to achieve the convergence as the latter does if all the inner linear systems are iteratively solved with {\em low} or {\em modest} accuracy during outer iterations. Based on the theory, we design practical stopping criteria for inner solves. Our analysis is on one step expansion of subspace and the approach applies to the Jacobi--Davidson (JD) method with the fixed target $\sigma$ as well, and a similar general convergence theory is obtained for it. Numerical experiments confirm our theory and demonstrate that the inexact SIRA and JD are similarly effective and are considerably superior to the inexact SIA.