Convergence analysis of a block preconditioned steepest descent eigensolver with implicit deflation

TL;DR

This paper provides a new convergence analysis for the block preconditioned steepest descent eigensolver with implicit deflation, demonstrating superlinear convergence and robustness through theoretical proofs and numerical validation.

Contribution

It introduces an alternative convergence analysis for PSD-id under weaker assumptions and extends it to a block version, BPSD-id, with proven cluster robustness.

Findings

Superlinear convergence under weaker assumptions

Extension to block version BPSD-id

Numerical validation of theoretical estimates

Abstract

Gradient-type iterative methods for solving Hermitian eigenvalue problems can be accelerated by using preconditioning and deflation techniques. A preconditioned steepest descent iteration with implicit deflation (PSD-id) is one of such methods. The convergence behavior of the PSD-id is recently investigated based on the pioneering work of Samokish on the preconditioned steepest descent method (PSD). The resulting non-asymptotic estimates indicate a superlinear convergence of the PSD-id under strong assumptions on the initial guess. The present paper utilizes an alternative convergence analysis of the PSD by Neymeyr under much weaker assumptions. We embed Neymeyr's approach into the analysis of the PSD-id using a restricted formulation of the PSD-id. More importantly, we extend the new convergence analysis of the PSD-id to a practically preferred block version of the PSD-id, or BPSD-id, and show the cluster robustness of the BPSD-id. Numerical examples are provided to validate the theoretical estimates.