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On the spectral equivalence of hierarchical matrix preconditioners for elliptic problems


Authors: M. Bebendorf, M. Bollhöfer and M. Bratsch
Journal: Math. Comp. 85 (2016), 2839-2861
MSC (2010): Primary 65F08, 65F50, 65N30
DOI: https://doi.org/10.1090/mcom/3086
Published electronically: March 28, 2016
MathSciNet review: 3522972
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Abstract: We will discuss the spectral equivalence of hierarchical matrix approximations for second order elliptic problems. Our theory will show that a modified variant of the hierarchical matrix Cholesky decomposition which preserves test vectors while truncating blocks to lower rank will lead to a spectrally equivalent approximation when using an adapted truncation threshold. Our theory also covers the usual hierarchical Cholesky decomposition which does not preserve test vectors but expects a significantly more restrictive threshold adaption to obtain a spectrally equivalent approximation. Numerical experiments indicate that the adaption of the truncation parameter seems to be necessary for the traditional hierarchical Cholesky preconditioner to obtain mesh-independent convergence while the variant which preserves test vectors works in practice quite well even with a fixed parameter.


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Additional Information

M. Bebendorf
Affiliation: Department of Mathematics, University of Bayreuth, Germany
Email: mario.bebendorf@uni-bayreuth.de

M. Bollhöfer
Affiliation: Institute for Computational Mathematics, TU Brunswick, Germany

M. Bratsch
Affiliation: formerly Institute for Numerical Simulation, University of Bonn, Germany

DOI: https://doi.org/10.1090/mcom/3086
Keywords: Approximate LU decomposition, preconditioning, hierarchical matrices
Received by editor(s): August 13, 2014
Received by editor(s) in revised form: April 28, 2015
Published electronically: March 28, 2016
Additional Notes: This work was supported by DFG collaborative research center SFB 611
Article copyright: © Copyright 2016 American Mathematical Society