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Mathematics of Computation

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Cluster robust estimates for block gradient-type eigensolvers


Authors: Ming Zhou and Klaus Neymeyr
Journal: Math. Comp. 88 (2019), 2737-2765
MSC (2010): Primary 65F15, 65N12, 65N25
DOI: https://doi.org/10.1090/mcom/3446
Published electronically: May 15, 2019
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Abstract: Sharp convergence estimates have been derived in recent years for gradient-type eigensolvers for large and sparse symmetric matrices or matrix pairs. An extension of these estimates to the corresponding block iterative methods can be achieved by applying a similar analysis to an embedded vector iteration. Although the resulting estimates are also sharp in the sense that they are not improvable without further assumptions, they cannot reflect the well-known cluster robustness of block eigensolvers. In the present paper, we analyze the cluster robustness of the preconditioned inverse subspace iteration. The main estimate has a weaker assumption and a simpler form compared to some known cluster robust estimates. In addition, it is applicable to further block gradient-type eigensolvers such as the locally optimal block preconditioned conjugate gradient method. The analysis is based on an orthogonal splitting for the block power method and a geometric interpretation of preconditioning. As a by-product, a cluster robust Ritz value estimate for the block power method is improved.


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Additional Information

Ming Zhou
Affiliation: Institut für Mathematik, Universität Rostock, Ulmenstraße 69, 18055 Rostock, Germany
Email: ming.zhou@uni-rostock.de

Klaus Neymeyr
Affiliation: Institut für Mathematik, Universität Rostock, Ulmenstraße 69, 18055 Rostock, Germany
Email: klaus.neymeyr@uni-rostock.de

DOI: https://doi.org/10.1090/mcom/3446
Keywords: Rayleigh quotient, gradient iterations, block eigensolvers, cluster robustness, multigrid, elliptic eigenvalue problem
Received by editor(s): September 5, 2018
Received by editor(s) in revised form: January 30, 2019
Published electronically: May 15, 2019
Article copyright: © Copyright 2019 American Mathematical Society