A preconditioned inverse iteration with an improved convergence guarantee

TL;DR

This paper introduces a new convergence analysis for a variant of preconditioned inverse iteration, demonstrating improved global convergence guarantees with less restrictive initial vector conditions, supported by theoretical bounds and numerical experiments.

Contribution

It provides a novel non-asymptotic convergence proof for a PINVIT variant, using Riemannian steepest descent analysis to achieve better global convergence guarantees.

Findings

Convergence rate nearly matches traditional PINVIT.

Requires less restrictive initial vector conditions.

Validated through theoretical bounds and numerical experiments.

Abstract

Preconditioned eigenvalue solvers offer the possibility to incorporate preconditioners for the solution of large-scale eigenvalue problems, as they arise from the discretization of partial differential equations. The convergence analysis of such methods is intricate. Even for the relatively simple preconditioned inverse iteration (PINVIT), which targets the smallest eigenvalue of a symmetric positive definite matrix, the celebrated analysis by Neymeyr is highly nontrivial and only yields convergence if the starting vector is fairly close to the desired eigenvector. In this work, we prove a new non-asymptotic convergence result for a variant of PINVIT. Our proof proceeds by analyzing an equivalent Riemannian steepest descent method and leveraging convexity-like properties. We show a convergence rate that nearly matches the one of PINVIT. As a major benefit, we require a condition on the starting vector that tends to be less stringent. This improved global convergence property is demonstrated for two classes of preconditioners with theoretical bounds and a range of numerical experiments.