A Note on Approximate Inverse Iteration

TL;DR

This paper extends classical convergence theory of approximate inverse iteration to operators with essential spectrum on infinite-dimensional spaces, accommodating small perturbations, and complements existing matrix case theories.

Contribution

It generalizes convergence results for approximate inverse iteration to infinite-dimensional operators with essential spectrum, including perturbations, suitable for educational purposes.

Findings

Extended convergence theory to infinite-dimensional Hilbert spaces.

Included perturbations in the convergence analysis.

Complemented existing matrix case theories.

Abstract

Different variants of approximate inverse iteration like the locally optimal block preconditioned conjugate gradient method became in recent years increasingly popular for the solution of the large matrix eigenvalue problems arising from the discretization of selfadjoint elliptic partial differential equations, in particular for the calculation of the minimum eigenvalue. We extend in this little note the classical convergence theory of D'yakonov and Orekhov [Math. Notes 27 (1980)] to the case of operators with an essential spectrum on infinite dimensional Hilbert spaces and allow for arbitrary, sufficiently small perturbations of the solutions of the equation that links the iterates. The note complements the much more elaborate convergence theory of Neymeyr and Knyazev and Neymeyr for the matrix case (see [Knyazev and Neymeyr, SIAM J. Matrix Anal. Appl. 31 (2009)] and the references therein) and is suitable for classroom presentation.