A geometric convergence theory for the preconditioned steepest descent iteration

TL;DR

This paper develops a new geometric convergence theory for the preconditioned steepest descent method in large eigenvalue problems, allowing for arbitrarily scaled preconditioners and improving convergence estimates over fixed step size methods.

Contribution

It introduces a sharp convergence estimate for the preconditioned steepest descent iteration that incorporates Rayleigh-Ritz acceleration, broadening applicability to scaled preconditioners.

Findings

New convergence estimate improves upon fixed step size iteration

Arbitrarily scaled preconditioners can be used effectively

Rayleigh-Ritz procedure implicitly finds optimal scaling

Abstract

Preconditioned gradient iterations for very large eigenvalue problems are efficient solvers with growing popularity. However, only for the simplest preconditioned eigensolver, namely the preconditioned gradient iteration (or preconditioned inverse iteration) with fixed step size, sharp non-asymptotic convergence estimates are known and these estimates require an ideally scaled preconditioner. In this paper a new sharp convergence estimate is derived for the preconditioned steepest descent iteration which combines the preconditioned gradient iteration with the Rayleigh-Ritz procedure for optimal line search convergence acceleration. The new estimate always improves that of the fixed step size iteration. The practical importance of this new estimate is that arbitrarily scaled preconditioners can be used. The Rayleigh-Ritz procedure implicitly computes the optimal scaling.