Volume 36, Issue 5
Convergence Analysis of a Locally Accelerated Preconditioned Steepest Descent Method for Hermitian-Definite Generalized Eigenvalue Problems

Yunfeng Cai, Zhaojun Bai, John E. Pask & N. Sukumar

J. Comp. Math., 36 (2018), pp. 739-760.

Published online: 2018-06

Preview Full PDF 8 1146
Export citation
  • 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 (PSD-id) method for solving Hermitian-definite generalized eigenvalue problems. Furthermore, we derive a nonasymptotic estimate of the rate of convergence of the PSD-id method. We show that with a proper choice of the shift, the indefinite shift-and-invert 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 ill-conditioned Hermitian-definite generalized eigenvalue problems arising from electronic structure calculations. While rigorous and full-scale convergence proofs of preconditioned block steepest descent methods in practical use still largely eludes us, we believe the theoretical results presented in this paper sheds light on an improved understanding of the convergence behavior of these block methods.

  • Keywords

Eigenvalue problem Steepest descent method Preconditioning Superlinear convergence

  • AMS Subject Headings

65F08 65F15 65Z05 15A12

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

yfcai@math.pku.edu.cn (Yunfeng Cai)

bai@cs.ucdavis.edu (Zhaojun Bai)

pask1@llnl.gov (John E. Pask)

nsukumar@ucdavis.edu (N. Sukumar)

  • References
  • Hide All
    View All

@Article{JCM-36-739, author = {Cai , Yunfeng and Bai , Zhaojun and Pask , John E. and Sukumar , N. }, title = {Convergence Analysis of a Locally Accelerated Preconditioned Steepest Descent Method for Hermitian-Definite Generalized Eigenvalue Problems}, journal = {Journal of Computational Mathematics}, year = {2018}, volume = {36}, number = {5}, pages = {739--760}, 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 (PSD-id) method for solving Hermitian-definite generalized eigenvalue problems. Furthermore, we derive a nonasymptotic estimate of the rate of convergence of the PSD-id method. We show that with a proper choice of the shift, the indefinite shift-and-invert 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 ill-conditioned Hermitian-definite generalized eigenvalue problems arising from electronic structure calculations. While rigorous and full-scale convergence proofs of preconditioned block steepest descent methods in practical use still largely eludes us, we believe the theoretical results presented in this paper sheds light on an improved understanding of the convergence behavior of these block methods.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1703-m2016-0580}, url = {http://global-sci.org/intro/article_detail/jcm/12455.html} }
Copy to clipboard
The citation has been copied to your clipboard