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:
-
[2];
- 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:
- 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}
}
Free Publicly Available Full Text
Publisher's Version of Record
Other availability
Search WorldCat to find libraries that may hold this journal
Cited by: 4 works
Citation information provided by
Web of Science
Web of Science
Figures / Tables:
Table 1: A list of test matrices.
All figures and tables
(7 total)
Save to My Library
You must Sign In or Create an Account in order to save documents to your library.
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
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
Communication lower bounds and optimal algorithms for numerical linear algebra
journal, May 2014
- Ballard, G.; Carson, E.; Demmel, J.
- Acta Numerica, Vol. 23
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
Basis selection in LOBPCG
journal, October 2006
- Hetmaniuk, U.; Lehoucq, R.
- Journal of Computational Physics, Vol. 218, Issue 1
Toward the Optimal Preconditioned Eigensolver: Locally Optimal Block Preconditioned Conjugate Gradient Method
journal, January 2001
- Knyazev, Andrew V.
- SIAM Journal on Scientific Computing, Vol. 23, Issue 2
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
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
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
Accelerating nuclear configuration interaction calculations through a preconditioned block iterative eigensolver
journal, January 2018
- Shao, Meiyue; Aktulga, H. Metin; Yang, Chao
- Computer Physics Communications, Vol. 222
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
A Block Orthogonalization Procedure with Constant Synchronization Requirements
journal, January 2002
- Stathopoulos, Andreas; Wu, Kesheng
- SIAM Journal on Scientific Computing, Vol. 23, Issue 6
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
Figures / Tables found in this record:
Figures/Tables have been extracted from DOE-funded journal article accepted manuscripts.