Cluster robustness of preconditioned gradient subspace iteration eigensolvers

Linear Algebra and Its Applications

Volumn 415, Issue 1, 2006, Pages 140-166

  Ovtchinnikov, E ^a  

^a UNIVERSITY OF WESTMINSTER   (United Kingdom)

Author keywords

Clustered eigenvalues; Conjugate gradient method; Convergence estimates; Preconditioning; Self adjoint eigenvalue problem; Steepest descent ascent method

Indexed keywords

APPROXIMATION THEORY; COMPUTATIONAL METHODS; GRADIENT METHODS; ITERATIVE METHODS; NUMERICAL METHODS; ROBUSTNESS (CONTROL SYSTEMS);

CLUSTERED EIGENVALUES; CONJUGATE GRADIENT METHOD; CONVERGENCE ESTIMATES; PRECONDITIONING; SELF-ADJOINT EIGENVALUE PROBLEM; STEEPEST DESCENT/ASCENT METHOD;

EIGENVALUES AND EIGENFUNCTIONS;

EID: 33645139975     PISSN: 00243795     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.laa.2005.06.039     Document Type: Article

References (28)

1. Z. Bai et al. (Eds.), Templates for the Solution of Algebraic Eigenvalue Problems. A Practical Guide, SIAM, 2000.
2. L. Bergamaschi, G. Gambolati, and G. Pini Asymptotic convergence of conjugate gradient methods for the partial symmetric eigenproblems Numer. Linear Algebra Appl. 4 1997 69 84
3. J.H. Bramble, J.E. Pasciak, and A.V. Knyazev A subspace preconditioning algorithm for eigenvector/eigenvalue computation Adv. Comput. Math. 6 1996 159 189
4. V.E. Bulgakov, M.V. Belyi, and K.M. Mathisen Multilevel aggregation method for solving large-scale generalized eigenvalue problems in structural dynamics Internat. J. Numer. Methods Engrg. 40 1997 453 471
5. E.G. D'yakonov Optimization in Solving Elliptic Problems 1996 CRC Press
6. E.G. D'yakonov, A.V. Knyazev, Group iterative method for finding lower-order eigenvalues, Moscow University, Ser. 15, Comput. Math. Cybernet. 2 (1982) 32-40.
7. Y.T. Feng, and D.R. Owen Conjugate gradient methods for solving the smallest eigenpair of large symmetric eigenvalue problems Internat. J. Numer. Methods Engrg. 39 1996 2209 2229
8. A.V. Knyazev Sharp a priori error estimates of the Rayleigh-Ritz method without assumptions of fixed sign or compactness Math. Notes 38 1986 998 1002
9. A.V. Knyazev Convergence rate estimates for iterative methods for mesh symmetric eigenvalue problem Sov. J. Numer. Anal. Math. Modelling 2 1987 371 396
10. A.V. Knyazev A preconditioned conjugate gradient method for eigenvalue problems and its implementation in a subspace Internat. Ser. Numer. Math. 96 1991 143 154
11. A.V. Knyazev Preconditioned eigensolvers-an oxymoron? Electron. Trans. Numer. Anal. 7 1998 104 123
12. A.V. Knyazev Toward the optimal preconditioned eigensolver: Locally optimal block preconditioned conjugate gradient method SIAM J. Sci. Comput. 2 2001 517 541
13. A.V. Knyazev, and K. Neymeyr A geometric theory for preconditioned inverse iteration. III: A short and sharp convergence estimate for generalized eigenvalue problems Linear Algebra Appl. 358 2003 95 114
14. A.V. Knyazev, K, Neymeyr, Efficient solution of symmetric eigenvalue problems using multigrid preconditioners in the locally optimal block conjugate gradient method, ETNA, 2001.
15. A.V. Knyazev, and A.L. Skorokhodov The preconditioned gradient-type iterative methods in a subspace for partial generalized symmetric eigenvalue problem SIAM J. Numer. Anal. 31 1994 1226 1239
16. K. Neymeyr A geometric theory for preconditioned inverse iteration applied to a subspace Math. Comp. 71 2002 197 216
17. K. Neymeyr A geometric theory for preconditioned inverse iteration. I: Extrema of the Rayleigh quotient Linear Algebra Appl. 322 2001 61 85
18. K. Neymeyr A geometric theory for preconditioned inverse iteration. II: Convergence estimates Linear Algebra Appl. 322 2001 87 104
19. K. Neymeyr, A geometric theory for preconditioned inverse iteration. IV: On the fastest convergence cases, Linear Algebra Appl., accepted for publication.
20. Y. Notay Combination of Jacobi-Davidson and conjugate gradients for the partial symmetric eigenproblem Numer. Linear Algebra Appl. 9 2001 21 44
21. S. Oliveira On the convergence rate of a preconditioned subspace eigensolver Computing 63 1999 219 231
22. E. Ovtchinnikov Convergence estimates for the generalized Davidson method for symmetric eigenvalue problems I. The preconditioning aspect SIAM J. Numer. Anal. 41 2003 258 271
23. E. Ovtchinnikov, Convergence estimates for preconditioned gradient subspace iteration eigensolvers, Technical Report 1244, Universitat Utrecht, June 2002.
24. E. Ovtchinnikov, Cluster robust estimates for the Rayleigh-Ritz approximation I: Estimates for invariant subspaces, Linear Algebra Appl., accepted for publication.
25. E. Ovtchinnikov, Cluster robust estimates for the Rayleigh-Ritz approximation II: Estimates for eigenvalues, Linear Algebra Appl., accepted for publication.
26. B.N. Parlett The Rayleigh quotient iteration and some generalizations Math. Comp. 28 1974 679 693
27. Y. Saad Numerical Methods for Large Eigenvalue Problems Algorithms and Architectures for Advanced Scientific Computing Series 1992 Manchester University Press and Halsted Press
28. B. Samokish The steepest descent method for an eigenvalue problem with semi-bounded operators Izv. Vyssh. Uchebn. Zaved. Mat. 5 1958 105 114