A Robust and Efficient Implementation of LOBPCG

doi:10.1137/17M1129830

Title: A Robust and Efficient Implementation of LOBPCG

Abstract

Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) is widely used to compute eigenvalues of large sparse symmetric matrices. The algorithm can suffer from numerical instability if it is not implemented with care. This is especially problematic when the number of eigenpairs to be computed is relatively large. In this paper we propose an improved basis selection strategy based on earlier work by Hetmaniuk and Lehoucq as well as a robust convergence criterion which is backward stable to enhance the robustness. Here, we also suggest several algorithmic optimizations that improve performance of practical LOBPCG implementations. Numerical examples confirm that our approach consistently and significantly outperforms previous competing approaches in both stability and speed.

Authors:

Duersch, Jed A. ^[1];

^[2]; Yang, Chao ^[2]; Gu, Ming ^[3]

Univ. of California, Berkeley, CA (United States). Dept. of Mathematics
Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States). Computational Research Div.
Univ. of California, Berkeley, CA (United States). Dept. of Mathematics; Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States). Computational Research Div.

Publication Date:: 2018-10-04

Research Org.:: Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)

Sponsoring Org.:: USDOE Office of Science (SC)

OSTI Identifier:: 1525270

Grant/Contract Number:: AC02-05CH11231

Resource Type:: Accepted Manuscript

Journal Name:: SIAM Journal on Scientific Computing

Additional Journal Information:: Journal Volume: 40; Journal Issue: 5; Journal ID: ISSN 1064-8275

Publisher:: SIAM

Country of Publication:: United States

Language:: English

Subject:: 97 MATHEMATICS AND COMPUTING; symmetric eigenvalue problem; LOBPCG; numerical stability

Citation Formats


                    Duersch, Jed A., Shao, Meiyue, Yang, Chao, and Gu, Ming. A Robust and Efficient Implementation of LOBPCG.  United States: N. p., 2018. 
        Web.  doi:10.1137/17M1129830.

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                    Duersch, Jed A., Shao, Meiyue, Yang, Chao, & Gu, Ming. A Robust and Efficient Implementation of LOBPCG.  United States.  doi:10.1137/17M1129830.

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                    Duersch, Jed A., Shao, Meiyue, Yang, Chao, and Gu, Ming. Thu .  
        "A Robust and Efficient Implementation of LOBPCG".  United States.  doi:10.1137/17M1129830.  https://www.osti.gov/servlets/purl/1525270.

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@article{osti_1525270,

  title        = {A Robust and Efficient Implementation of LOBPCG},

  author       = {Duersch, Jed A. and Shao, Meiyue and Yang, Chao and Gu, Ming},

  abstractNote = {Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) is widely used to compute eigenvalues of large sparse symmetric matrices. The algorithm can suffer from numerical instability if it is not implemented with care. This is especially problematic when the number of eigenpairs to be computed is relatively large. In this paper we propose an improved basis selection strategy based on earlier work by Hetmaniuk and Lehoucq as well as a robust convergence criterion which is backward stable to enhance the robustness. Here, we also suggest several algorithmic optimizations that improve performance of practical LOBPCG implementations. Numerical examples confirm that our approach consistently and significantly outperforms previous competing approaches in both stability and speed.},

  doi          = {10.1137/17M1129830},

  journal      = {SIAM Journal on Scientific Computing},

  number       = 5,

  volume       = 40,

  place        = {United States},

  year         = {2018},

  month        = {10}

}

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Journal Article:

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DOI: 10.1137/17M1129830

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Figures / Tables:

Table 1: A list of test matrices.

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