Convergence analysis of a locally accelerated preconditioned steepest descent method for Hermitian-definite generalized eigenvalue problems

TL;DR

This paper proves the convergence and superlinear rate of a preconditioned steepest descent method with implicit deflation for solving Hermitian-definite generalized eigenvalue problems, supported by numerical verification.

Contribution

It extends classical analysis techniques to establish convergence and derive convergence rates for the PSD-id method, highlighting the effectiveness of a locally accelerated preconditioner.

Findings

Proves convergence of PSD-id for Hermitian problems.

Derives a nonasymptotic convergence rate estimate.

Numerical examples confirm theoretical convergence behavior.

Abstract

By extending the classical analysis techniques due to Samokish, Faddeev and Faddeeva, and Longsine and McCormick among others, we prove the convergence of preconditioned steepest descent with implicit deflation (PSD-id) method for solving Hermitian-definite generalized eigenvalue problems. Furthermore, we derive a nonasymptotic estimate of the rate of convergence of the \psdid method. We show that with the proper choice of the shift, the indefinite shift-and-invert preconditioner is a locally accelerated preconditioner, and is asymptotically optimal that leads to superlinear convergence. Numerical examples are presented to verify the theoretical results on the convergence behavior of the \psdid method for solving ill-conditioned Hermitian-definite generalized eigenvalue problems arising from electronic structure calculations. While rigorous and full-scale convergence proofs of preconditioned block steepest descent methods in practical use still largely eludes us, we believe the theoretical results presented in this paper sheds light on an improved understanding of the convergence behavior of these block methods.