Parallel Krylov Solvers for the Polynomial Eigenvalue Problem in SLEPc
Abstract
Polynomial eigenvalue problems are often found in scientific computing applications. When the coefficient matrices of the polynomial are large and sparse, usually only a few eigenpairs are required and projection methods are the best choice. We focus on Krylov methods that operate on the companion linearization of the polynomial but exploit the block structure with the aim of being memory-efficient in the representation of the Krylov subspace basis. The problem may appear in the form of a low-degree polynomial (quartic or quintic, say) expressed in the monomial basis, or a high-degree polynomial (coming from interpolation of a nonlinear eigenproblem) expressed in a nonmonomial basis. We have implemented a parallel solver in SLEPc covering both cases that is able to compute exterior as well as interior eigenvalues via spectral transformation. We discuss important issues such as scaling and restart and illustrate the robustness and performance of the solver with some numerical experiments.
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References
1.
P. R. Amestoy, I. S. Duff, J.-Y. L'Excellent, and J. Koster, A fully asynchronous multifrontal solver using distributed dynamic scheduling, SIAM J. Matrix Anal. Appl., 23 (2001), pp. 15--41.
2.
A. Amiraslani, D. A. Aruliah, and R. M. Corless, Block LU factors of generalized companion matrix pencils, Theoret. Comput. Sci., 381 (2007), pp. 134--147.
3.
A. Amiraslani, R. M. Corless, and P. Lancaster, Linearization of matrix polynomials expressed in polynomial bases, IMA J. Numer. Anal., 29 (2009), pp. 141--157.
4.
Z. Bai, J. Demmel, J. Dongarra, A. Ruhe, and H. van der Vorst, eds., Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide, SIAM, Philadelphia, 2000.
5.
Z. Bai and Y. Su, SOAR: A second-order Arnoldi method for the solution of the quadratic eigenvalue problem, SIAM J. Matrix Anal. Appl., 26 (2005), pp. 640--659.
6.
S. Balay, S. Abhyankar, M. Adams, J. Brown, P. Brune, K. Buschelman, L. Dalcin, V. Eijkhout, W. Gropp, D. Kaushik, M. Knepley, L. Curfman McInnes, K. Rupp, B. Smith, S. Zampini, and H. Zhang, PETSc Users Manual, Tech. Report ANL-95/11--Revision 3.6, Argonne National Laboratory, 2015.
7.
T. Betcke, Optimal scaling of generalized and polynomial eigenvalue problems, SIAM J. Matrix Anal. Appl., 30 (2008), pp. 1320--1338.
8.
T. Betcke, N. J. Higham, V. Mehrmann, C. Schröder, and F. Tisseur, NLEVP: A collection of nonlinear eigenvalue problems, ACM Trans. Math. Software, 39 (2013), pp. 7:1--7:28.
9.
T. Betcke and D. Kressner, Perturbation, extraction and refinement of invariant pairs for matrix polynomials, Linear Algebra Appl., 435 (2011), pp. 514--536.
10.
C. Campos and J. E. Roman, Parallel iterative refinement in polynomial eigenvalue problems, Numer. Linear Algebra Appl. (2016).
11.
C. Effenberger and D. Kressner, Chebyshev interpolation for nonlinear eigenvalue problems, BIT, 52 (2012), pp. 933--951.
12.
H. Fan, W. Lin, and P. Van Dooren, Normwise scaling of second order polynomial matrices, SIAM J. Matrix Anal. Appl., 26 (2004), pp. 252--256.
13.
I. Gohberg, P. Lancaster, and L. Rodman, Matrix Polynomials, Academic Press, New York, 1982.
14.
S. Hammarling, C. J. Munro, and F. Tisseur, An algorithm for the complete solution of quadratic eigenvalue problems, ACM Trans. Math. Software, 39 (2013), pp. 18:1--18:19.
15.
V. Hernandez, J. E. Roman, and A. Tomas, Parallel Arnoldi eigensolvers with enhanced scalability via global communications rearrangement, Parallel Comput., 33 (2007), pp. 521--540.
16.
V. Hernandez, J. E. Roman, and V. Vidal, SLEPc: A scalable and flexible toolkit for the solution of eigenvalue problems, ACM Trans. Math. Software, 31 (2005), pp. 351--362.
17.
N. J. Higham, R.-C. Li, and F. Tisseur, Backward error of polynomial eigenproblems solved by linearization, SIAM J. Matrix Anal. Appl., 29 (2007), pp. 1218--1241.
18.
M. Hochbruck and D. Lochel, A multilevel Jacobi-Davidson method for polynomial PDE eigenvalue problems arising in plasma physics, SIAM J. Sci. Comput., 32 (2010), pp. 3151--3169.
19.
F.-N. Hwang, Z.-H. Wei, T.-M. Huang, and W. Wang, A parallel additive Schwarz preconditioned Jacobi-Davidson algorithm for polynomial eigenvalue problems in quantum dot simulation, J. Comput. Phys., 229 (2010), pp. 2932--2947.
20.
D. Kressner, A block Newton method for nonlinear eigenvalue problems, Numer. Math., 114 (2009), pp. 355--372.
21.
D. Kressner and J. E. Roman, Memory-efficient Arnoldi algorithms for linearizations of matrix polynomials in Chebyshev basis, Numer. Linear Algebra Appl., 21 (2014), pp. 569--588.
22.
D. Lu and Y. Su, Two-Level Orthogonal Arnoldi Process for the Solution of Quadratic Eigenvalue Problems, manuscript, 2012.
23.
D. S. Mackey, N. Mackey, C. Mehl, and V. Mehrmann, Vector spaces of linearizations for matrix polynomials, SIAM J. Matrix Anal. Appl., 28 (2006), pp. 971--1004.
24.
K. Meerbergen, The Quadratic Arnoldi method for the solution of the quadratic eigenvalue problem, SIAM J. Matrix Anal. Appl., 30 (2008), pp. 1463--1482.
25.
J. E. Roman, C. Campos, E. Romero, and A. Tomas, SLEPc Users Manual, Tech. report DSIC-II/24/02--Revision 3.6, D. Sistemes Informàtics i Computació, Universitat Politècnica de València, 2015.
26.
G. W. Stewart, A Krylov--Schur algorithm for large eigenproblems, SIAM J. Matrix Anal. Appl., 23 (2001), pp. 601--614.
27.
Y. Su, J. Zhang, and Z. Bai, A compact Arnoldi algorithm for polynomial eigenvalue problems, \newblock presented at RANMEP, 2008.
28.
F. Tisseur, Backward error and condition of polynomial eigenvalue problems, Linear Algebra Appl., 309 (2000), pp. 339--361.
29.
F. Tisseur and K. Meerbergen, The quadratic eigenvalue problem, SIAM Rev., 43 (2001), pp. 235--286.
Information & Authors
Information
Published In
SIAM Journal on Scientific Computing
Pages: S385 - S411
DOI: 10.1137/15M1022458
ISSN (online): 1095-7197
Copyright
© 2016, Society for Industrial and Applied Mathematics.
History
Submitted: 20 May 2015
Accepted: 24 March 2016
Published online: 27 October 2016
Keywords
MSC codes
Authors
Funding Information
Ministerio de EconomÃa y Competitividadhttp://dx.doi.org/10.13039/501100003329: TIN2013-41049-P
Funding Information
Ministerio de Educación, Cultura y Deportehttp://dx.doi.org/10.13039/501100003176: AP2012-0608
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