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Subspace Methods with Local Refinements for Eigenvalue Computation Using Low-Rank Tensor-Train Format

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Abstract

Computing a few eigenpairs from large-scale symmetric eigenvalue problems is far beyond the tractability of classic eigensolvers when the storage of the eigenvectors in the classical way is impossible. We consider a tractable case in which both the coefficient matrix and its eigenvectors can be represented in the low-rank tensor train formats. We propose a subspace optimization method combined with some suitable truncation steps to the given low-rank Tensor Train formats. Its performance can be further improved if the alternating minimization method is used to refine the intermediate solutions locally. Preliminary numerical experiments show that our algorithm is competitive to the state-of-the-art methods on problems arising from the discretization of the stationary Schrödinger equation.

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Notes

  1. Downloadable from http://anchp.epfl.ch/TTeMPS.

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Acknowledgments

We thank D. Kressner, M. Steinlechner and A. Uschmajew for sharing online their matlab codes on EVAMEn and the TT/MPS tensor toolbox TTeMPS. The authors would like to thank the associate editor Prof. Wotao Yin and two anonymous referees for their detailed and valuable comments and suggestions.

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Authors and Affiliations

  1. Department of Industrial and Systems Engineering, University of Minnesota, Minneapolis, MN, USA

    Junyu Zhang

  2. Beijing International Center for Mathematical Research, Peking University, Beijing, China

    Zaiwen Wen

  3. Department of Computational and Applied Mathematics, Rice University, Houston, TX, USA

    Yin Zhang

Authors
  1. Junyu Zhang
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  2. Zaiwen Wen
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  3. Yin Zhang
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Corresponding author

Correspondence to Zaiwen Wen.

Additional information

J. Zhang Research supported in part by NSF Grant CMMI-1462408. Z. Wen Research supported in part by NSFC Grants 11322109 and 91330202, and by the National Basic Research Project under the Grant 2015CB856002. Y. Zhang Research supported in part by NSF DMS-1115950 and NSF DMS-1418724.

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Zhang, J., Wen, Z. & Zhang, Y. Subspace Methods with Local Refinements for Eigenvalue Computation Using Low-Rank Tensor-Train Format. J Sci Comput 70, 478–499 (2017). https://doi.org/10.1007/s10915-016-0255-0

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  • DOI: https://doi.org/10.1007/s10915-016-0255-0

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