Deflated GMRES for systems with multiple shifts and multiple right-hand sides

Linear Algebra and Its Applications

Volumn 429, Issue 10, 2008, Pages 2415-2434

  Darnell, Dean ^a   Morgan, Ronald B ^a   Wilcox, Walter ^a  

^a BAYLOR UNIVERSITY   (United States)

Author keywords

BiCG; Deflation; Eigenvalues; GMRES; Linear equations; Multiple right hand sides; Multiple shifts; QCD

Indexed keywords

COMPUTER NETWORKS; LINEAR EQUATIONS; NUMERICAL ANALYSIS; QUANTUM THEORY;

BICG; DEFLATION; EIGENVALUES; GMRES; MULTIPLE RIGHT-HAND SIDES; MULTIPLE SHIFTS; QCD;

MATRIX ALGEBRA;

EID: 51349149780     PISSN: 00243795     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.laa.2008.04.019     Document Type: Article

References (47)

1. G. Arnold, N. Cundy, J. van den Eshof, A. Frommer, S. Krieg, Th. Lippert, K. Sch fer. Numerical methods for the QCD overlap operator: II. Sign-function and error bounds, 2003.
2. Baglama J., Calvetti D., Golub G.H., and Reichel L. Adaptively preconditioned GMRES algorithms. SIAM J. Sci. Comput. 20 (1998) 243-269
3. Burrage K., and Erhel J. On the performance of various adaptive preconditioned GMRES strategies. Numer. Linear Algebra Appl. 5 (1998) 101-121
4. Le Calvez C., and Molina B. Implicitly restarted and deflated GMRES. Numer. Algorithms 21 (1999) 261-285
5. Chan T.F., and Wan W. Analysis of projection methods for solving linear systems with multiple right-hand sides. SIAM J. Sci. Comput. 18 (1997) 1698-1721
6. Chapman A., and Saad Y. Deflated and augmented Krylov subspace techniques. Numer. Linear Algebra Appl. 4 (1997) 43-66
7. Darnell D., Morgan R.B., and Wilcox W. Deflation of eigenvalues for iterative methods in lattice QCD. Nucl. Phys. B (Proc. Suppl.) 129 (2004) 856-858
8. Datta B.N., and Saad Y. Sylvester-like observer matrix equations, and an associated algorithm for partial spectrum assignment. Linear Algebra Appl. 154-156 (1991) 225-244
9. de Forcrand P. Progress on lattice QCD algorithms. Nucl. Phys. B (Proc. Suppl.) 47 (1996) 228-235
10. De Sturler E. Truncation strategies for optimal Krylov subspace methods. SIAM J. Numer. Anal. 36 (1999) 864-889
11. Dong S.J., Lee F.X., Liu K.F., and Zhang J.B. Chiral symmetry, quark mass, and scaling of the overlap fermions. Phys. Rev. Lett. 85 (2000) 5051-5054
12. Edwards R.G., Heller U.M., and Narayanan R. Study of chiral symmetry in quenched QCD using the overlap Dirac operator. Phys. Rev. D 59 (1999) 0945101-0945108
13. Erhel J., Burrage K., and Pohl B. Restarted GMRES preconditioned by deflation. J. Comput. Appl. Math. 69 (1996) 303-318
14. Feriani A., Perotti F., and Simoncini V. Iterative system solvers for the frequency analysis of linear mechanical systems. Comput. Methods Appl. Mech. Engrg. 190 (2000) 1719-1739
15. Freund R.W. On conjugate gradient type methods and polynomial preconditioners for a class of complex non-Hermitian matrices. Numer. Math. 57 (1990) 285-312
16. Freund R.W. Quasi-kernel polynomials and their use in non-Hermitian matrix iterations. J. Comput. Appl. Math. 43 (1992) 135-158
17. R.W. Freund, Solution of shifted linear systems by quasi-minimal residual iterations, in: L. Reichel, A. Ruttan, R.S. Varga (Eds.), Numerical Linear Algebra, W. de Gruyter, 1993, pp. 101-121.
18. Freund R.W., and Malhotra M. A block QMR algorithm for non-Hermitian linear systems with multiple right-hand sides. Linear Algebra Appl. 254 (1997) 119-157
19. Frommer A. BiCGStab(l) for families of shifted linear systems. Computing 70 (2003) 87-109
20. Frommer A., and Glässner U. Restarted GMRES for shifted linear systems. SIAM J. Sci. Comput. 19 (1998) 15-26
21. Gu G.D., Zhang J.J., and Li Z.W. Restarted GMRES augmented with eigenvectors for shifted linear systems. Int. J. Comput. Math. 80 (2003) 1037-1047
22. S. Gugercin, A.C. Antoulas, C.A. Beattie, Rational Krylov methods for optimal H2 model reduction, 2006.
23. Kharchenko S.A., and Yeremin A.Y. Eigenvalue translation based preconditioners for the GMRES(k) method. Numer. Linear Algebra Appl. 2 (1995) 51-77
24. Morgan R.B. Computing interior eigenvalues of large matrices. Linear Algebra Appl. 154-156 (1991) 289-309
25. Morgan R.B. A restarted GMRES method augmented with eigenvectors. SIAM J. Matrix Anal. Appl. 16 (1995) 1154-1171
26. Morgan R.B. On restarting the Arnoldi method for large nonsymmetric eigenvalue problems. Math. Comp. 65 (1996) 1213-1230
27. Morgan R.B. Implicitly restarted GMRES and Arnoldi methods for nonsymmetric systems of equations. SIAM J. Matrix Anal. Appl. 21 (2000) 1112-1135
28. Morgan R.B. GMRES with deflated restarting. SIAM J. Sci. Comput. 24 (2002) 20-37
29. Morgan R.B. Restarted block-GMRES with deflation of eigenvalues. Appl. Numer. Math. 54 (2005) 222-236
30. Morgan R.B., and Wilcox W. Deflation of eigenvalues for GMRES in lattice QCD. Nucl. Phys. B (Proc. Suppl.) 106 (2002) 1067-1069
31. R.B. Morgan, W. Wilcox, Deflated iterative methods for linear equations with multiple right-hand sides, 2004. Available from: arXiv:math-ph/0405053v2.
32. Morgan R.B., and Zeng M. Harmonic projection methods for large non-symmetric eigenvalue problems. Numer. Linear Algebra Appl. 5 (1998) 33-55
33. Morgan R.B., and Zeng M. A harmonic restarted Arnoldi for calculating eigenvalues and determining multiplicity. Linear Algebra Appl. 415 (2006) 96-113
34. Narayanan R., and Neuberger H. An alternative to domain wall fermions. Phys. Rev. D 62 (2000) 074504
35. Neff H., Eicker N., Lippert Th., Negele J.W., and Schilling K. On the low fermionic eigenmode dominance in QCD on the lattice. Phys. Rev. D 64 (2001) 114509-1-114509-12
36. Paige C.C., Parlett B.N., and van der Vorst H.A. Approximate solutions and eigenvalue bounds from Krylov subspaces. Numer. Linear Algebra Appl. 2 (1995) 115-133
37. Parks M.L., de Sturler E., Mackey G., Johnson D.D., and Maiti S. Recycling Krylov subspaces for sequences of linear systems. SIAM J. Sci. Comput. 28 (2006) 1651-1674
38. Saad Y. Iterative Methods for Sparse Linear Systems (1996), PWS Publishing, Boston, MA
39. Saad Y. Analysis of augmented Krylov subspace techniques. SIAM J. Matrix Anal. Appl. 18 (1997) 435-449
40. Saad Y., and Schultz M.H. GMRES: a generalized minimum residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Statist. Comput. 7 (1986) 856-869
41. Simoncini V. Restarted full orthogonalization method for shifted linear systems. BIT 43 (2003) 459-466
42. Simoncini V., and Gallopoulos E. An iterative method for nonsymmetric systems with multiple right-hand sides. SIAM J. Sci. Comput. 16 (1995) 917-933
43. Simoncini V., and Gallopoulos E. A hybrid block GMRES method for nonsymmetric systems with multiple right-hand sides. J. Comput. Appl. Math. 66 (1996) 457-469
44. Sorensen D.C. Implicit application of polynomial filters in a k-step Arnoldi method. SIAM J. Matrix Anal. Appl. 13 (1992) 357-385
45. Stewart G.W. A Krylov-Schur algorithm for large eigenproblems. SIAM J. Matrix Anal. Appl. 23 (2001) 601-614
46. Wu K., and Simon H. Thick-restart Lanczos method for symmetric eigenvalue problems. SIAM J. Matrix Anal. Appl. 22 (2000) 602-616
47. Young D.M., and Vona B.R. On the use of rational iterative methods for solving large sparse linear systems. Appl. Numer. Math. 10 (1992) 261-278