On inner iterations of Jacobi-Davidson type methods for large SVD computations

TL;DR

This paper analyzes the convergence of inexact Jacobi-Davidson SVD methods, showing that solving correction equations with modest accuracy suffices for effective computation of interior singular triplets in large matrices.

Contribution

It provides a convergence analysis for inexact JDSVD methods, establishing practical stopping criteria for inner iterations to ensure efficiency and accuracy.

Findings

Inexact JDSVD methods mimic exact methods when correction equations are solved with modest accuracy.

Proposed stopping criteria improve computational efficiency without sacrificing convergence.

Numerical experiments validate the theoretical results and effectiveness of the inexact algorithms.

Abstract

We make a convergence analysis of the harmonic and refined harmonic extraction versions of Jacobi-Davidson SVD (JDSVD) type methods for computing one or more interior singular triplets of a large matrix $A$. At each outer iteration of these methods, a correction equation, i.e., inner linear system, is solved approximately by using iterative methods, which leads to two inexact JDSVD type methods, as opposed to the exact methods where correction equations are solved exactly. Accuracy of inner iterations critically affects the convergence and overall efficiency of the inexact JDSVD methods. A central problem is how accurately the correction equations should be solved so as to ensure that both of the inexact JDSVD methods can mimic their exact counterparts well, that is, they use almost the same outer iterations to achieve the convergence. In this paper, similar to the available results on the JD type methods for large matrix eigenvalue problems, we prove that each inexact JDSVD method behaves like its exact counterpart if all the correction equations are solved with $low\ or\ modest$ accuracy during outer iterations. Based on the theory, we propose practical stopping criteria for inner iterations. Numerical experiments confirm our theory and the effectiveness of the inexact algorithms.