DOE PAGES title logo U.S. Department of Energy
Office of Scientific and Technical Information

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:
 [1]; ORCiD logo [2];  [2];  [3]
  1. Univ. of California, Berkeley, CA (United States). Dept. of Mathematics
  2. Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States). Computational Research Div.
  3. Univ. of California, Berkeley, CA (United States). Dept. of Mathematics; Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States). Computational Research Div.
Publication Date:
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.
Duersch, Jed A., Shao, Meiyue, Yang, Chao, & Gu, Ming. A Robust and Efficient Implementation of LOBPCG. United States. https://doi.org/10.1137/17M1129830
Duersch, Jed A., Shao, Meiyue, Yang, Chao, and Gu, Ming. Thu . "A Robust and Efficient Implementation of LOBPCG". United States. https://doi.org/10.1137/17M1129830. https://www.osti.gov/servlets/purl/1525270.
@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}
}

Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record

Citation Metrics:
Cited by: 4 works
Citation information provided by
Web of Science

Figures / Tables:

Table 1 Table 1: A list of test matrices.

Save / Share:

Works referenced in this record:

A High Performance Block Eigensolver for Nuclear Configuration Interaction Calculations
journal, June 2017

  • Aktulga, Hasan Metin; Afibuzzaman, Md.; Williams, Samuel
  • IEEE Transactions on Parallel and Distributed Systems, Vol. 28, Issue 6
  • DOI: 10.1109/TPDS.2016.2630699

Anasazi software for the numerical solution of large-scale eigenvalue problems
journal, July 2009

  • Baker, C. G.; Hetmaniuk, U. L.; Lehoucq, R. B.
  • ACM Transactions on Mathematical Software, Vol. 36, Issue 3
  • DOI: 10.1145/1527286.1527287

Communication lower bounds and optimal algorithms for numerical linear algebra
journal, May 2014


An overview of the sparse basic linear algebra subprograms: The new standard from the BLAS technical forum
journal, June 2002

  • Duff, Iain S.; Heroux, Michael A.; Pozo, Roldan
  • ACM Transactions on Mathematical Software, Vol. 28, Issue 2
  • DOI: 10.1145/567806.567810

Basis selection in LOBPCG
journal, October 2006


Toward the Optimal Preconditioned Eigensolver: Locally Optimal Block Preconditioned Conjugate Gradient Method
journal, January 2001


Block Locally Optimal Preconditioned Eigenvalue Xolvers (BLOPEX) in Hypre and PETSc
journal, January 2007

  • Knyazev, A. V.; Argentati, M. E.; Lashuk, I.
  • SIAM Journal on Scientific Computing, Vol. 29, Issue 5
  • DOI: 10.1137/060661624

PARSEC – the pseudopotential algorithm for real-space electronic structure calculations: recent advances and novel applications to nano-structures
journal, April 2006

  • Kronik, Leeor; Makmal, Adi; Tiago, Murilo L.
  • physica status solidi (b), Vol. 243, Issue 5
  • DOI: 10.1002/pssb.200541463

Basic Linear Algebra Subprograms for Fortran Usage
journal, September 1979

  • Lawson, C. L.; Hanson, R. J.; Kincaid, D. R.
  • ACM Transactions on Mathematical Software, Vol. 5, Issue 3
  • DOI: 10.1145/355841.355847

Accelerating nuclear configuration interaction calculations through a preconditioned block iterative eigensolver
journal, January 2018


The SIESTA method for ab initio order- N materials simulation
journal, March 2002

  • Soler, José M.; Artacho, Emilio; Gale, Julian D.
  • Journal of Physics: Condensed Matter, Vol. 14, Issue 11
  • DOI: 10.1088/0953-8984/14/11/302

A Block Orthogonalization Procedure with Constant Synchronization Requirements
journal, January 2002


Works referencing / citing this record:

A Scalable Matrix-Free Iterative Eigensolver for Studying Many-Body Localization
conference, January 2020

  • Van Beeumen, Roel; Kahanamoku-Meyer, Gregory D.; Yao, Norman Y.
  • HPCAsia2020: International Conference on High Performance Computing in Asia-Pacific Region, Proceedings of the International Conference on High Performance Computing in Asia-Pacific Region
  • DOI: 10.1145/3368474.3368497

Figures/Tables have been extracted from DOE-funded journal article accepted manuscripts.