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A subspace preconditioning algorithm for eigenvector/eigenvalue computation

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Abstract

We consider the problem of computing a modest number of the smallest eigenvalues along with orthogonal bases for the corresponding eigenspaces of a symmetric positive definite operatorA defined on a finite dimensional real Hilbert spaceV. In our applications, the dimension ofV is large and the cost of invertingA is prohibitive. In this paper, we shall develop an effective parallelizable technique for computing these eigenvalues and eigenvectors utilizing subspace iteration and preconditioning forA. Estimates will be provided which show that the preconditioned method converges linearly when used with a uniform preconditioner under the assumption that the approximating subspace is close enough to the span of desired eigenvectors.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Texas A&M University, 77843, College Station, TX, USA

    James H. Bramble & Joseph E. Pasciak

  2. Department of Mathematics, University of Colorado at Denver, P.O. Box 173364, Campus Box 170, 80217-3364, Denver, CO, USA

    Andrew V. Knyazev

Authors
  1. James H. Bramble
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  2. Joseph E. Pasciak
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  3. Andrew V. Knyazev
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Additional information

Communicated by D.N. Arnold

This manuscript has been authored under contract number DE-AC02-76CH00016 with the U.S. Department of Energy. Accordingly, the U.S. Government retains a non-exclusive, royalty-free license to publish or reproduce the published form of this contribution, or allow others to do so, for U.S. Government purposes. This work was also supported in part under the National Science Foundation Grants No. DMS-9007185, DMS-9501507 and NSF-CCR-9204255, and by the U.S. Army Research Office through the Mathematical Sciences Institute, Cornell University.

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Bramble, J.H., Pasciak, J.E. & Knyazev, A.V. A subspace preconditioning algorithm for eigenvector/eigenvalue computation. Adv Comput Math 6, 159–189 (1996). https://doi.org/10.1007/BF02127702

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  • DOI: https://doi.org/10.1007/BF02127702

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