Approximation of matrix operators applied to multiple vectors

Mathematics and Computers in Simulation

Volumn 79, Issue 4, 2008, Pages 1270-1283

  Hochbruck, Marlis ^a   Niehoff, Jörg ^a  

^a HEINRICH HEINE UNIVERSITY   (Germany)

Author keywords

Exponential integrators; Krylov subspace methods; Matrix functions; Multiple right hand sides; Q R decomposition

Indexed keywords

FUNCTION EVALUATION; NUMERICAL METHODS; PROBABILITY DENSITY FUNCTION; VECTORS;

EXPONENTIAL INTEGRATORS; KRYLOV SUBSPACE METHODS; MATRIX FUNCTIONS; MULTIPLE RIGHT-HAND SIDES; Q R DECOMPOSITION;

NUMBER THEORY;

EID: 56949090299     PISSN: 03784754     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.matcom.2008.03.016     Document Type: Article

References (39)

1. In: Bai Z., Demmel J., Dongarra J., and Ruhe A. (Eds). Templates for the Solution of Algebraic Eigenvalue Problems, Software, Environments, and Tools (2000), SIAM, Philadelphia
2. Bergamaschi L., Caliari M., Martínez A., and Vianello M. Comparing Leja and Krylov approximations of large scale matrix exponentials. In: Alexandrov V.N., van Albada G.D., Sloot P.M.A., and Dongarra J. (Eds). Computational Science-ICCS 2006, vol. 3994 of Lecture Notes in Computer Science (2006), Springer, Berlin/Heidelberg
3. Bergamaschi L., Caliari M., and Vianello M. The ReLPM exponential integrator for FE discretizations of advection-diffusion equations. In: Bubak M., van Albada G.D., Sloot P.M.A., and Dongarra J. (Eds). Computational Science-ICCS 2004, vol. 3039 of Lecture Notes in Computer Science (2004), Springer, Berlin/Heidelberg
4. Bergamaschi L., and Vianello M. Efficient computation of the exponential operator for large, sparse, symmetric matrices. Numer. Linear Algebra Appl. 7 (2000) 27-45
5. Berland H., Owren B., and Skaflestad B. B-series and order conditions for exponential integrators. SIAM J. Numer. Anal. 43 4 (2005) 1715-1727
6. H. Berland, B. Skaflestad, Solving the nonlinear Schrödinger equation using exponential integrators, Tech. Rep., Norwegian University of Science and Technology, Trondheim, 2005.
7. Calvo M.P., and Palencia C. A class of explicit multistep exponential integrators for semilinear problems. Numer. Math. 102 3 (2005) 367-381
8. Celledoni E., Marthinsen A., and Owren B. Commutator-free lie group methods. Future Generation Comput. Syst. 19 3 (2003) 341-352
9. Cox S.M., and Matthews P.C. Exponential time differencing for stiff systems. J. Comput. Phys. 176 2 (2002) 430-455
10. Daniel J., Gragg W., Kaufman L., and Stewart G. Reorthogonalization and stable algorithms for updating the Gram-Schmidt QR factorization. Math. Comput. 30 (1976) 772-795
11. Druskin V., Greenbaum A., and Knizhnerman L. Using nonorthogonal Lanczos vectors in the computation of matrix functions. SIAM J. Sci. Comput. 19 1 (1998) 38-54
12. Druskin V., and Knizhnerman L. Krylov subspace approximation of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Linear Algebra Appl. 2 3 (1995) 205-217
13. Eiermann M., and Ernst O. A restarted Krylov subspace method for the evaluation of matrix functions. SIAM J. Numer. Anal. 44 6 (2006) 2481-2504
14. Fischer P. Projection techniques for iterative solution of A under(x, _) = under(b, _) with successive right-hand sides. Comput. Methods Appl. Mech. Eng. 163 (1998) 193-204
15. A. Frommer, V. Simoncini, Matrix functions, Tech. rep., Dipartimento di Matematica, Bologna (2006).
16. Gallopoulos E., and Saad Y. Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Stat. Comput. 13 (1992) 1236-1264
17. Golub G.H., and Loan C.F.V. Matrix Computation. Third Edition (1996), John Hopkins University Press
18. Hairer E., and Wanner G. Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems. 2nd ed. (1996), Springer-Verlag, New York
19. Hochbruck M., and Lubich C. On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34 5 (1997) 1911-1925
20. Hochbruck M., and Ostermann A. Explicit exponential Runge-Kutta methods for semilinear parabolic problems. SIAM J. Numer. Anal. 43 3 (2005) 1069-1090
21. Hochbruck M., and Ostermann A. Exponential Runge-Kutta methods for parabolic problems. Appl. Num. Math. 53 2-4 (2005) 323-339
22. M. Hochbruck, A. Ostermann, J. Schweitzer, Exponential integrators of Rosenbrock-type, in: Oberwolfach Reports, vol. 18/2006, Mathematisches Forschungsinstitut Oberwolfach, 2006.
23. A.-K. Kassam, L. Trefethen, Fourth-order time stepping for stiff PDEs, SIAM J. Sci. Comput. 26 (4) (2005) 1214-1233.
24. Knizhnerman L.A. Calculation of functions of unsymmetric matrices using Arnoldi's method. Comput. Math. Math. Phys. 31 (1992) 1-9
25. Knizhnerman L.A. Error bounds in Arnoldi's method: The case of a normal matrix. Comput. Math. Math. Phys. 32 (1992) 1199-1211
26. Krogstad S. Generalized integrating factor methods for stiff PDEs. J. Comput. Phys. 203 1 (2005) 72-88
27. Lopez L., and Simoncini V. Analysis of projection methods for rational function approximation to the matrix exponential. SIAM J. Numer. Anal. 44 (2006) 613-635
28. Lötstedt P., and Nilsson M. A minimal residual interpolation method for linear equations with multiple right hand sides. SIAM J. Sci. Comput. 25 6 (2004) 2126-2144
29. B. Minchev, Exponential integrators for semilinear problems, Ph.D. Thesis, Tech. Rep., University of Bergen, Norway, 2004.
30. Moret I., and Novati P. RD rational approximations of the matrix exponential. BIT 44 (2004) 595-615
31. Moret I., and Novati P. Interpolation functions of matrices on zeros of quasi-kernel polynomials. Numer. Linear Algebra Appl. 11 (2005) 337-353
32. J. Niehoff, Projektionsverfahren zur Approximation von Matrixfunktionen mit Anwendungen auf die Implementierung exponentieller Integratoren, Ph.D. Thesis, Mathematisches Institut, Heinrich-Heine-Universität Düsseldorf, 2006.
33. Nour-Omid B. Applications of the Lanczos algorithm. Comput. Phys. Commun. 53 (1989) 157-168
34. Ostermann A., Thalhammer M., and Wright W.M. A class of explicit exponential general linear methods. BIT 46 2 (2006) 409-432
35. Saad Y. Analysis of some Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 29 1 (1992) 209-228
36. Sommeijer B., Shampine L., and Verwer J. RKC: an explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88 (1991) 315-326
37. Stewart D.E., and Leyk T.S. Error estimates for Krylov subspace approximations of matrix exponentials. J. Comput. Appl. Math. 72 (1996) 359-406
38. Trefethen L.N., Weidemann J.A.C., and Schmelzer T. Talbot quadratures and rational approximations. BIT 46 (2006) 653-670
39. van den Eshof J., and Hochbruck M. Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27 4 (2006) 1438-1457