Angle-free cluster robust Ritz value bounds for restarted block eigensolvers

Convergence rates of block iterations for solving eigenvalue problems typically measure errors of Ritz values approximating eigenvalues. The errors of the Ritz values are commonly bounded in terms of principal angles between the initial or iterative subspace and the invariant subspace associated with the target eigenvalues. Such bounds thus cannot be applied repeatedly as needed for restarted block eigensolvers, since the left- and right-hand sides of the bounds use different terms. They must be combined with additional bounds which could cause an overestimation. Alternative repeatable bounds that are angle-free and depend only on the errors of the Ritz values have been pioneered for Hermitian eigenvalue problems in doi:10.1515/rnam.1987.2.5.371 but only for a single extreme Ritz value. We extend this result to all Ritz values and achieve robustness for clustered eigenvalues by utilizing nonconsecutive eigenvalues. Our new bounds cover the restarted block Lanczos method and its modifications with shift-and-invert and deflation, and are numerically advantageous.Comment: 24 pages, 4 figure

A priori convergence analysis for Krylov subspace eigensolvers

This thesis contributes to the convergence theory of Krylov subspace eigensolvers for discretized self-adjoint elliptic differential operators. A central topic refers to a priori convergence estimates with weak assumptions and concise bounds, which can reasonably predict the convergence rate, in particular for clustered eigenvalues. By avoiding the dependence on current approximate eigenvalues, such estimates significantly improve certain state-of-the-art estimates with regard to their sharpness and applicability.Diese Arbeit widmet sich der Konvergenztheorie Krylovraum-basierter Lösungsverfahren für Eigenwertprobleme diskretisierter selbstadjungierter elliptischer Differentialoperatoren. Ein zentrales Thema bezieht sich auf A-priori-Konvergenzabschätzungen mit schwachen Voraussetzungen und prägnanten Schranken, welche die Konvergenzrate vernünftig vorhersagen können, insbesondere bei dicht aneinanderliegenden Eigenwerten. Durch Vermeidung der Abhängigkeit von aktuellen Näherungseigenwerten lassen sich einige State-of-the-art-Abschätzungen hinsichtlich Schärfe und Anwendbarkeit deutlich verbessern