Convergence Analysis of a Locally Accelerated Preconditioned Steepest Descent Method for HermitianDefinite Generalized Eigenvalue Problems
Abstract
By extending the classical analysis techniques due to Samokish, Faddeev and Faddeeva, and Longsine and McCormick among others, we prove the convergence of preconditioned steepest descent with implicit deflation (PSDid) method for solving Hermitiandefinite generalized eigenvalue problems. Moreover, we derive a nonasymptotic estimate of the rate of convergence of the PSDid method. We show that with a proper choice of the shift, the indefinite shiftandinvert preconditioner is a locally accelerated preconditioner, and is asymptotically optimal that leads to superlinear convergence. Numerical examples are presented to verify the theoretical results on the convergence behavior of the PSDid method for solving illconditioned Hermitiandefinite generalized eigenvalue problems arising from electronic structure calculations. While rigorous and fullscale convergence proofs of preconditioned block steepest descent methods in practical use still largely eludes us, we postulate that the theoretical results introduced in this paper sheds light on an improved understanding of the convergence behavior of these block methods.
 Authors:

 Peking Univ., Beijing (China)
 Univ. of California, Davis, CA (United States)
 Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
 Publication Date:
 Research Org.:
 Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
 Sponsoring Org.:
 USDOE National Nuclear Security Administration (NNSA)
 OSTI Identifier:
 1526870
 Report Number(s):
 LLNLJRNL776239
Journal ID: ISSN 02549409; 968787
 Grant/Contract Number:
 AC5207NA27344; DMS1522697; CCF1527091
 Resource Type:
 Accepted Manuscript
 Journal Name:
 Journal of Computational Mathematics
 Additional Journal Information:
 Journal Volume: 36; Journal Issue: 5; Journal ID: ISSN 02549409
 Country of Publication:
 United States
 Language:
 English
 Subject:
 97 MATHEMATICS AND COMPUTING; Eigenvalue problem; Steepest descent method; Preconditioning; Superlinear convergence
Citation Formats
Cai, Yunfeng, Bai, Zhaojun, Pask, John E., and Sukumar, N. Convergence Analysis of a Locally Accelerated Preconditioned Steepest Descent Method for HermitianDefinite Generalized Eigenvalue Problems. United States: N. p., 2018.
Web. https://doi.org/10.4208/jcm.1703m20160580.
Cai, Yunfeng, Bai, Zhaojun, Pask, John E., & Sukumar, N. Convergence Analysis of a Locally Accelerated Preconditioned Steepest Descent Method for HermitianDefinite Generalized Eigenvalue Problems. United States. https://doi.org/10.4208/jcm.1703m20160580
Cai, Yunfeng, Bai, Zhaojun, Pask, John E., and Sukumar, N. Fri .
"Convergence Analysis of a Locally Accelerated Preconditioned Steepest Descent Method for HermitianDefinite Generalized Eigenvalue Problems". United States. https://doi.org/10.4208/jcm.1703m20160580. https://www.osti.gov/servlets/purl/1526870.
@article{osti_1526870,
title = {Convergence Analysis of a Locally Accelerated Preconditioned Steepest Descent Method for HermitianDefinite Generalized Eigenvalue Problems},
author = {Cai, Yunfeng and Bai, Zhaojun and Pask, John E. and Sukumar, N.},
abstractNote = {By extending the classical analysis techniques due to Samokish, Faddeev and Faddeeva, and Longsine and McCormick among others, we prove the convergence of preconditioned steepest descent with implicit deflation (PSDid) method for solving Hermitiandefinite generalized eigenvalue problems. Moreover, we derive a nonasymptotic estimate of the rate of convergence of the PSDid method. We show that with a proper choice of the shift, the indefinite shiftandinvert preconditioner is a locally accelerated preconditioner, and is asymptotically optimal that leads to superlinear convergence. Numerical examples are presented to verify the theoretical results on the convergence behavior of the PSDid method for solving illconditioned Hermitiandefinite generalized eigenvalue problems arising from electronic structure calculations. While rigorous and fullscale convergence proofs of preconditioned block steepest descent methods in practical use still largely eludes us, we postulate that the theoretical results introduced in this paper sheds light on an improved understanding of the convergence behavior of these block methods.},
doi = {10.4208/jcm.1703m20160580},
journal = {Journal of Computational Mathematics},
number = 5,
volume = 36,
place = {United States},
year = {2018},
month = {6}
}
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