Overlapping domain decomposition preconditioners for the generalized Davidson method for the eigenvalue problem
Abstract
The solution of the large, sparse, symmetric eigenvalue problem, Ax = {lambda}x, is central to many scientific applications. Among many iterative methods that attempt to solve this problem, the Lanczos and the Generalized Davidson (GD) are the most widely used methods. The Lanczos method builds an orthogonal basis for the Krylov subspace, from which the required eigenvectors are approximated through a Rayleigh-Ritz procedure. Each Lanczos iteration is economical to compute but the number of iterations may grow significantly for difficult problems. The GD method can be considered a preconditioned version of Lanczos. In each step the Rayleigh-Ritz procedure is solved and explicit orthogonalization of the preconditioned residual ((M {minus} {lambda}I){sup {minus}1}(A {minus} {lambda}I)x) is performed. Therefore, the GD method attempts to improve convergence and robustness at the expense of a more complicated step.
- Authors:
-
- Vanderbilt Univ., Nashville, TN (United States)
- Publication Date:
- Research Org.:
- Front Range Scientific Computations, Inc., Boulder, CO (United States); US Department of Energy (USDOE), Washington DC (United States); National Science Foundation, Washington, DC (United States)
- OSTI Identifier:
- 223845
- Report Number(s):
- CONF-9404305-Vol.1
ON: DE96005735; CNN: Grant ASC-9005687;Grant DMR-9217287;Grant DAAL03-89-C-0038; TRN: 96:002320-0019
- Resource Type:
- Conference
- Resource Relation:
- Conference: Colorado conference on iterative methods, Breckenridge, CO (United States), 5-9 Apr 1994; Other Information: PBD: [1994]; Related Information: Is Part Of Colorado Conference on iterative methods. Volume 1; PB: 203 p.
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 99 MATHEMATICS, COMPUTERS, INFORMATION SCIENCE, MANAGEMENT, LAW, MISCELLANEOUS; MATRICES; EIGENVALUES; ITERATIVE METHODS; ALGEBRA; SYMMETRY; RITZ METHOD; COMPUTER CALCULATIONS
Citation Formats
Stathopoulos, A, Fischer, C F, and Saad, Y. Overlapping domain decomposition preconditioners for the generalized Davidson method for the eigenvalue problem. United States: N. p., 1994.
Web.
Stathopoulos, A, Fischer, C F, & Saad, Y. Overlapping domain decomposition preconditioners for the generalized Davidson method for the eigenvalue problem. United States.
Stathopoulos, A, Fischer, C F, and Saad, Y. 1994.
"Overlapping domain decomposition preconditioners for the generalized Davidson method for the eigenvalue problem". United States. https://www.osti.gov/servlets/purl/223845.
@article{osti_223845,
title = {Overlapping domain decomposition preconditioners for the generalized Davidson method for the eigenvalue problem},
author = {Stathopoulos, A and Fischer, C F and Saad, Y},
abstractNote = {The solution of the large, sparse, symmetric eigenvalue problem, Ax = {lambda}x, is central to many scientific applications. Among many iterative methods that attempt to solve this problem, the Lanczos and the Generalized Davidson (GD) are the most widely used methods. The Lanczos method builds an orthogonal basis for the Krylov subspace, from which the required eigenvectors are approximated through a Rayleigh-Ritz procedure. Each Lanczos iteration is economical to compute but the number of iterations may grow significantly for difficult problems. The GD method can be considered a preconditioned version of Lanczos. In each step the Rayleigh-Ritz procedure is solved and explicit orthogonalization of the preconditioned residual ((M {minus} {lambda}I){sup {minus}1}(A {minus} {lambda}I)x) is performed. Therefore, the GD method attempts to improve convergence and robustness at the expense of a more complicated step.},
doi = {},
url = {https://www.osti.gov/biblio/223845},
journal = {},
number = ,
volume = ,
place = {United States},
year = {Sat Dec 31 00:00:00 EST 1994},
month = {Sat Dec 31 00:00:00 EST 1994}
}