Domain decomposition approaches for accelerating contour integration eigenvalue solvers for symmetric eigenvalue problems
Abstract
Summary This paper discusses techniques for computing a few selected eigenvalue–eigenvector pairs of large and sparse symmetric matrices. A recently developed class of techniques to solve this type of problems is based on integrating the matrix resolvent operator along a complex contour that encloses the interval containing the eigenvalues of interest. This paper considers such contour integration techniques from a domain decomposition viewpoint and proposes two schemes. The first scheme can be seen as an extension of domain decomposition linear system solvers in the framework of contour integration methods for eigenvalue problems, such as FEAST. The second scheme focuses on integrating the resolvent operator primarily along the interface region defined by adjacent subdomains. A parallel implementation of the proposed schemes is described, and results on distributed computing environments are reported. These results show that domain decomposition approaches can lead to reduced run times and improved scalability.
- Authors:
-
[1];
- Univ. of Minnesota, Minneapolis, MN (United States)
- Univ. of Massachusetts, Amherst, MA (United States)
- Publication Date:
- Research Org.:
- Univ. of Texas, Austin, TX (United States)
- Sponsoring Org.:
- USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR). Scientific Discovery through Advanced Computing (SciDAC); USDOE Office of Science (SC), Basic Energy Sciences (BES); National Science Foundation (NSF); Gerondelis Foundation Fellowship
- OSTI Identifier:
- 1611165
- Alternate Identifier(s):
- OSTI ID: 1421773
- Grant/Contract Number:
- SC0008877; CCF-1505970; CCF-1510010
- Resource Type:
- Accepted Manuscript
- Journal Name:
- Numerical Linear Algebra with Applications
- Additional Journal Information:
- Journal Volume: 25; Journal Issue: 5; Journal ID: ISSN 1070-5325
- Publisher:
- Wiley
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 97 MATHEMATICS AND COMPUTING; mathematics; Cauchy integral formula; domain decomposition; FEAST; parallel computing; symmetric eigenvalue problem
Citation Formats
Kalantzis, Vassilis, Kestyn, James, Polizzi, Eric, and Saad, Yousef. Domain decomposition approaches for accelerating contour integration eigenvalue solvers for symmetric eigenvalue problems. United States: N. p., 2018.
Web. doi:10.1002/nla.2154.
Kalantzis, Vassilis, Kestyn, James, Polizzi, Eric, & Saad, Yousef. Domain decomposition approaches for accelerating contour integration eigenvalue solvers for symmetric eigenvalue problems. United States. https://doi.org/10.1002/nla.2154
Kalantzis, Vassilis, Kestyn, James, Polizzi, Eric, and Saad, Yousef. Mon .
"Domain decomposition approaches for accelerating contour integration eigenvalue solvers for symmetric eigenvalue problems". United States. https://doi.org/10.1002/nla.2154. https://www.osti.gov/servlets/purl/1611165.
@article{osti_1611165,
title = {Domain decomposition approaches for accelerating contour integration eigenvalue solvers for symmetric eigenvalue problems},
author = {Kalantzis, Vassilis and Kestyn, James and Polizzi, Eric and Saad, Yousef},
abstractNote = {Summary This paper discusses techniques for computing a few selected eigenvalue–eigenvector pairs of large and sparse symmetric matrices. A recently developed class of techniques to solve this type of problems is based on integrating the matrix resolvent operator along a complex contour that encloses the interval containing the eigenvalues of interest. This paper considers such contour integration techniques from a domain decomposition viewpoint and proposes two schemes. The first scheme can be seen as an extension of domain decomposition linear system solvers in the framework of contour integration methods for eigenvalue problems, such as FEAST. The second scheme focuses on integrating the resolvent operator primarily along the interface region defined by adjacent subdomains. A parallel implementation of the proposed schemes is described, and results on distributed computing environments are reported. These results show that domain decomposition approaches can lead to reduced run times and improved scalability.},
doi = {10.1002/nla.2154},
journal = {Numerical Linear Algebra with Applications},
number = 5,
volume = 25,
place = {United States},
year = {Mon Feb 19 00:00:00 EST 2018},
month = {Mon Feb 19 00:00:00 EST 2018}
}
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Works referenced in this record:
A Fast and High Quality Multilevel Scheme for Partitioning Irregular Graphs
journal, January 1998
- Karypis, George; Kumar, Vipin
- SIAM Journal on Scientific Computing, Vol. 20, Issue 1
pARMS: a parallel version of the algebraic recursive multilevel solver
journal, January 2003
- Li, Zhongze; Saad, Yousef; Sosonkina, Masha
- Numerical Linear Algebra with Applications, Vol. 10, Issue 5-6
ShyLU: A Hybrid-Hybrid Solver for Multicore Platforms
conference, May 2012
- Rajamanickam, Sivasankaran; Boman, Erik G.; Heroux, Michael A.
- 2012 IEEE International Symposium on Parallel & Distributed Processing (IPDPS), 2012 IEEE 26th International Parallel and Distributed Processing Symposium
Applied Numerical Linear Algebra
book, January 1997
- Demmel, James W.
- Society for Industrial and Applied Mathematics
A FEAST algorithm with oblique projection for generalized eigenvalue problems: A non-Hermitian FEAST algorithm for generalized eigenproblems
journal, March 2017
- Yin, Guojian; Chan, Raymond H.; Yeung, Man-Chung
- Numerical Linear Algebra with Applications, Vol. 24, Issue 4
A Thick-Restart Lanczos Algorithm with Polynomial Filtering for Hermitian Eigenvalue Problems
journal, January 2016
- Li, Ruipeng; Xi, Yuanzhe; Vecharynski, Eugene
- SIAM Journal on Scientific Computing, Vol. 38, Issue 4
SuperLU_DIST: A scalable distributed-memory sparse direct solver for unsymmetric linear systems
journal, June 2003
- Li, Xiaoye S.; Demmel, James W.
- ACM Transactions on Mathematical Software, Vol. 29, Issue 2
On correction equations and domain decomposition for computing invariant subspaces
journal, January 2007
- Philippe, Bernard; Saad, Yousef
- Computer Methods in Applied Mechanics and Engineering, Vol. 196, Issue 8
CIRR: a Rayleigh-Ritz type method with contour integral for generalized eigenvalue problems
journal, November 2007
- Sakurai, Tetsuya; Tadano, Hiroto
- Hokkaido Mathematical Journal, Vol. 36, Issue 4
An integral method for solving nonlinear eigenvalue problems
journal, May 2012
- Beyn, Wolf-Jürgen
- Linear Algebra and its Applications, Vol. 436, Issue 10
Computation of Smallest Eigenvalues using Spectral Schur Complements
journal, January 2005
- Bekas, Constantine; Saad, Yousef
- SIAM Journal on Scientific Computing, Vol. 27, Issue 2
Computing Partial Spectra with Least-Squares Rational Filters
journal, January 2016
- Xi, Yuanzhe; Saad, Yousef
- SIAM Journal on Scientific Computing, Vol. 38, Issue 5
Distributed Schur Complement Techniques for General Sparse Linear Systems
journal, January 1999
- Saad, Yousef; Sosonkina, Maria
- SIAM Journal on Scientific Computing, Vol. 21, Issue 4
Sparse Approximations of the Schur Complement for Parallel Algebraic Hybrid Solvers in 3D
journal, June 2010
- Giraud, L.
- Numerical Mathematics: Theory, Methods and Applications, Vol. 3, Issue 3
A Parallel Geometric Multifrontal Solver Using Hierarchically Semiseparable Structure
journal, May 2016
- Wang, Shen; Li, Xiaoye S.; Rouet, François-Henry
- ACM Transactions on Mathematical Software, Vol. 42, Issue 3
PFEAST: A High Performance Sparse Eigenvalue Solver Using Distributed-Memory Linear Solvers
conference, November 2016
- Kestyn, James; Kalantzis, Vasileios; Polizzi, Eric
- SC16: International Conference for High Performance Computing, Networking, Storage and Analysis
Designing rational filter functions for solving eigenvalue problems by contour integration
journal, August 2016
- Van Barel, Marc
- Linear Algebra and its Applications, Vol. 502
A Fully Asynchronous Multifrontal Solver Using Distributed Dynamic Scheduling
journal, January 2001
- Amestoy, Patrick R.; Duff, Iain S.; L'Excellent, Jean-Yves
- SIAM Journal on Matrix Analysis and Applications, Vol. 23, Issue 1
Convergence properties of block GMRES and matrix polynomials
journal, November 1996
- Simoncini, V.; Gallopoulos, E.
- Linear Algebra and its Applications, Vol. 247
Algebraic Two-Level Preconditioners for the Schur Complement Method
journal, January 2001
- Carvalho, L. M.; Giraud, L.; Le Tallec, P.
- SIAM Journal on Scientific Computing, Vol. 22, Issue 6
Computing Eigenvalues of Real Symmetric Matrices with Rational Filters in Real Arithmetic
journal, January 2015
- Austin, Anthony P.; Trefethen, Lloyd N.
- SIAM Journal on Scientific Computing, Vol. 37, Issue 3
ARMS: an algebraic recursive multilevel solver for general sparse linear systems
journal, January 2002
- Saad, Y.; Suchomel, B.
- Numerical Linear Algebra with Applications, Vol. 9, 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
A projection method for generalized eigenvalue problems using numerical integration
journal, October 2003
- Sakurai, Tetsuya; Sugiura, Hiroshi
- Journal of Computational and Applied Mathematics, Vol. 159, Issue 1
An Automated Multilevel Substructuring Method for Eigenspace Computation in Linear Elastodynamics
journal, January 2004
- Bennighof, Jeffrey K.; Lehoucq, R. B.
- SIAM Journal on Scientific Computing, Vol. 25, Issue 6
GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems
journal, July 1986
- Saad, Youcef; Schultz, Martin H.
- SIAM Journal on Scientific and Statistical Computing, Vol. 7, Issue 3
Accelerating data uncertainty quantification by solving linear systems with multiple right-hand sides
journal, January 2013
- Kalantzis, V.; Bekas, C.; Curioni, A.
- Numerical Algorithms, Vol. 62, Issue 4
Feast Eigensolver for Non-Hermitian Problems
journal, January 2016
- Kestyn, James; Polizzi, Eric; Peter Tang, Ping Tak
- SIAM Journal on Scientific Computing, Vol. 38, Issue 5
Efficient Management of Parallelism in Object-Oriented Numerical Software Libraries
book, January 1997
- Balay, Satish; Gropp, William D.; McInnes, Lois Curfman
- Modern Software Tools for Scientific Computing
FEAST As A Subspace Iteration Eigensolver Accelerated By Approximate Spectral Projection
journal, January 2014
- Peter Tang, Ping Tak; Polizzi, Eric
- SIAM Journal on Matrix Analysis and Applications, Vol. 35, Issue 2
An Augmented Incomplete Factorization Approach for Computing the Schur Complement in Stochastic Optimization
journal, January 2014
- Petra, Cosmin G.; Schenk, Olaf; Lubin, Miles
- SIAM Journal on Scientific Computing, Vol. 36, Issue 2
Density-matrix-based algorithm for solving eigenvalue problems
journal, March 2009
- Polizzi, Eric
- Physical Review B, Vol. 79, Issue 11
On the parallel iterative solution of linear systems arising in the FEAST algorithm for computing inner eigenvalues
journal, November 2015
- Galgon, Martin; Krämer, Lukas; Thies, Jonas
- Parallel Computing, Vol. 49
Algorithm 832: UMFPACK V4.3---an unsymmetric-pattern multifrontal method
journal, June 2004
- Davis, Timothy A.
- ACM Transactions on Mathematical Software, Vol. 30, Issue 2
A numerical method for nonlinear eigenvalue problems using contour integrals
journal, January 2009
- Asakura, Junko; Sakurai, Tetsuya; Tadano, Hiroto
- JSIAM Letters, Vol. 1, Issue 0
Nonlinear eigenvalue problems and contour integrals
journal, January 2016
- Van Barel, Marc; Kravanja, Peter
- Journal of Computational and Applied Mathematics, Vol. 292
Domain decomposition methods for eigenvalue problems
journal, May 2000
- Lui, S. H.
- Journal of Computational and Applied Mathematics, Vol. 117, Issue 1
Zolotarev Quadrature Rules and Load Balancing for the FEAST Eigensolver
journal, January 2015
- Güttel, Stefan; Polizzi, Eric; Tang, Ping Tak Peter
- SIAM Journal on Scientific Computing, Vol. 37, Issue 4
Parallel eigenvalue calculation based on multiple shift–invert Lanczos and contour integral based spectral projection method
journal, July 2014
- Aktulga, Hasan Metin; Lin, Lin; Haine, Christopher
- Parallel Computing, Vol. 40, Issue 7
Dissecting the FEAST algorithm for generalized eigenproblems
journal, May 2013
- Krämer, Lukas; Di Napoli, Edoardo; Galgon, Martin
- Journal of Computational and Applied Mathematics, Vol. 244
Numerical methods for large eigenvalue problems
journal, January 2002
- Sorensen, Danny C.
- Acta Numerica, Vol. 11
A Density Matrix-based Algorithm for Solving Eigenvalue Problems
text, January 2009
- Polizzi, Eric
- arXiv
An integral method for solving nonlinear eigenvalue problems
text, January 2010
- Beyn, Wolf-Jürgen
- arXiv
Dissecting the FEAST algorithm for generalized eigenproblems
preprint, January 2012
- Krämer, Lukas; Di Napoli, Edoardo; Galgon, Martin
- arXiv
FEAST Eigensolver for non-Hermitian Problems
preprint, January 2015
- Kestyn, James; Polizzi, Eric; Tang, Ping Tak Peter
- arXiv
Works referencing / citing this record:
Recursive integral method with Cayley transformation: RIM with Cayley transformation
journal, July 2018
- Huang, R.; Sun, J.; Yang, C.
- Numerical Linear Algebra with Applications, Vol. 25, Issue 6
Fast randomized non-Hermitian eigensolver based on rational filtering and matrix partitioning
preprint, January 2021
- Kalantzis, Vassilis; Xi, Yuanzhe; Horesh, Lior
- arXiv