Spectral Schur complement techniques for symmetric eigenvalue problems
Abstract
This paper presents a domain decomposition-type method for solving real symmetric (Hermitian) eigenvalue problems in which we seek all eigenpairs in an interval [α; β] or a few eigenpairs next to a given real shift. Here, a Newton-based scheme is described whereby the problem is converted to one that deals with the interface nodes of the computational domain. This approach relies on the fact that the inner solves related to each local subdomain are relatively inexpensive. This Newton scheme exploits spectral Schur complements, and these lead to so-called eigenbranches, which are rational functions whose roots are eigenvalues of the original matrix. Theoretical and practical aspects of domain decomposition techniques for computing eigenvalues and eigenvectors are discussed. A parallel implementation is presented and its performance on distributed computing environments is illustrated by means of a few numerical examples.
- Authors:
-
- Univ. of Minnesota, Minneapolis, MN (United States). Computer Science & Engineering
- Publication Date:
- Research Org.:
- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
- Sponsoring Org.:
- USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR); National Science Foundation (NSF)
- OSTI Identifier:
- 1438784
- Report Number(s):
- LLNL-JRNL-691697
Journal ID: ISSN 1068-9613
- Grant/Contract Number:
- AC52-07NA27344; SC0008877; NSF/DMS-1216366
- Resource Type:
- Accepted Manuscript
- Journal Name:
- Electronic Transactions on Numerical Analysis
- Additional Journal Information:
- Journal Volume: 45; Journal ID: ISSN 1068-9613
- Publisher:
- Kent State University - Johann Radon Institute for Computational and Applied Mathematics (RICAM)
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 97 MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; Domain decomposition; spectral Schur complements; eigenvalue problems; Newton’s method; parallel computing
Citation Formats
Li, Ruipeng, Kalantzis, Vassilis, and Saad, Yousef. Spectral Schur complement techniques for symmetric eigenvalue problems. United States: N. p., 2016.
Web.
Li, Ruipeng, Kalantzis, Vassilis, & Saad, Yousef. Spectral Schur complement techniques for symmetric eigenvalue problems. United States.
Li, Ruipeng, Kalantzis, Vassilis, and Saad, Yousef. Fri .
"Spectral Schur complement techniques for symmetric eigenvalue problems". United States. https://www.osti.gov/servlets/purl/1438784.
@article{osti_1438784,
title = {Spectral Schur complement techniques for symmetric eigenvalue problems},
author = {Li, Ruipeng and Kalantzis, Vassilis and Saad, Yousef},
abstractNote = {This paper presents a domain decomposition-type method for solving real symmetric (Hermitian) eigenvalue problems in which we seek all eigenpairs in an interval [α; β] or a few eigenpairs next to a given real shift. Here, a Newton-based scheme is described whereby the problem is converted to one that deals with the interface nodes of the computational domain. This approach relies on the fact that the inner solves related to each local subdomain are relatively inexpensive. This Newton scheme exploits spectral Schur complements, and these lead to so-called eigenbranches, which are rational functions whose roots are eigenvalues of the original matrix. Theoretical and practical aspects of domain decomposition techniques for computing eigenvalues and eigenvectors are discussed. A parallel implementation is presented and its performance on distributed computing environments is illustrated by means of a few numerical examples.},
doi = {},
journal = {Electronic Transactions on Numerical Analysis},
number = ,
volume = 45,
place = {United States},
year = {Fri Sep 02 00:00:00 EDT 2016},
month = {Fri Sep 02 00:00:00 EDT 2016}
}