Efficient preconditioning of sequences of nonsymmetric linear systems

SIAM Journal on Scientific Computing

Volumn 29, Issue 5, 2007, Pages 1918-1941

  Tebbens, Jurjen Duintjer ^a   Tůma, Miroslav ^a  

^a INSTITUTE OF COMPUTER SCIENCE   (Czech Republic)

Author keywords

Factorization updates; Gauss Jordan transformations; Incomplete factorizations; Minimum spanning tree; Preconditioned iterative methods; Sequences of linear algebraic systems; Sparse matrices

Indexed keywords

ITERATIVE METHODS; LINEAR SYSTEMS; LINEAR TRANSFORMATIONS; MATRIX ALGEBRA;

FACTORIZATION UPDATE; GAUSS-JORDAN TRANSFORMATION; INCOMPLETE FACTORIZATION; LINEAR ALGEBRAIC SYSTEMS; MINIMUMS PANNING TREE; NONSYMMETRIC LINEAR SYSTEMS; PRECONDITIONED ITERATIVE METHODS; PRECONDITIONERS; SEQUENCE OF LINEAR ALGEBRAIC SYSTEM; SPARSE MATRICES;

FACTORIZATION;

EID: 52149094645     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/06066151X     Document Type: Conference Paper

References (62)

1. M. A. AJIZ AND A. JENNINGS, A robust incomplete Choleski-conjugate gradient algorithm, Internat. J. Numer. Methods Engrg., 20 (1984), pp. 949-966.
2. J. BAGLAMA, D. CALVETTI, G. H. GOLUB, AND L. REICHEL, Adoptively preconditioned GMRES algorithms, SIAM J. Sci. Comput., 20 (1998), pp. 243-269.
3. M. BENZI AND D. BERTACCINI, Approximate inverse preconditioning for shifted linear systems, BIT, 43 (2003), pp. 231-244.
4. M. BENZI, J. K. CULLUM, AND M. TŮMA, Robust approximate inverse preconditioning for the conjugate gradient method, SIAM J. Sci. Comput., 22 (2000), pp. 1318-1332.
5. M. BENZI, J. C. HAWS, AND M. TŮMA, Preconditioning highly indefinite and nonsymrnetric matrices, SIAM J. Sci. Comput., 22 (2000), pp. 1333-1353.
6. M. BENZI, C. D. MEYER, AND M. TŮMA, A sparse approximate inverse preconditioner for the conjugate gradient method, SIAM J. Sci. Comput., 17 (1996), pp. 1135-1149.
7. M. BENZI AND M. TŮMA, Orderings for factorized sparse approximate inverse preconditioners, SIAM J. Sci. Comput., 21 (2000), pp. 1851-1868.
8. L. BERGAMASCHI, R. BRU, A. MARTÍNEZ, AND M. PUTTI, Quasi-Newton preconditioners for the inexact Newton method, Electron. Trans. Numer. Anal., 23 (2006), pp. 76-87.
9. D. BERTACCINI, Efficient preconditioning for sequences of parametric complex symmetric linear systems, Electron. Trans. Numer. Anal., 18 (2004), pp. 49-64.
10. P. BIRKEN, Numerical Simulation of Flows at Low Mach Numbers with Heat Sources, Ph.D. thesis, University of Kassel, Kassel, Germany, 2005.
11. R. P. BRENT, Some efficient algorithms for solving systems of nonlinear equations, SIAM J. Numer. Anal., 10 (1973), pp. 327-344.
12. P. N. BROWN AND Y. SAAD, Hybrid Krylov methods for nonlinear systems of equations, SIAM J. Sci. Statist. Comput., 11 (1990), pp. 450-481.
13. X.-C. CAI AND D. E. KEYES, Nonlinearly preconditioned inexact Newton algorithms, SIAM J. Sci. Comput., 24 (2002), pp. 183-200.
14. K. CHEN, Matrix Preconditioning Techniques and Applications, Cambridge Monogr. Appl. Comput. Math. 19, Cambridge University Press, Cambridge, UK, 2005.
15. E. CHOW AND Y. SAAD, Experimental study of ILU preconditioners for indefinite matrices, J. Comput. Appl. Math., 86 (1997), pp. 387-414.
16. P. CONCUS AND G. H. GOLUB, Use of fast direct methods for the efficient numerical solution of nonseparable elliptic equations, SIAM J. Numer. Anal., 10 (1973), pp. 1103-1120.
17. P. CONCUS AND G. H. GOLUB, A generalized conjugate gradient method for nonsymmetric systems of linear equations, in Computing Methods in Applied Sciences and Engineering, Lecture Notes in Econom. Math. Systems 134, R. Glowinski and J. L. Lions, eds., Springer-Verlag, Berlin, New York, 1976, pp. 56-65.
18. A. R. CURTIS, M. J. D. POWELL, AND J. K. REID, On the estimation of sparse Jacobian matrices, J. Inst. Math. Appl., 13 (1974), pp. 117-119.
19. T. A. DAVIS AND W. W. HAGER, Modifying a sparse Cholesky factorization, SIAM J. Matrix Anal. Appl., 20 (1999), pp. 606-627.
20. T. A. DAVIS ANDW. W. HAGER, Multiple-rank modifications of a sparse Cholesky factorization, SIAM J. Matrix Anal. Appl., 22 (2001), pp. 997-1013.
21. T. A. DAVIS AND W. W. HAGER, Row modifications of a sparse Cholesky factorization, SIAM J. Matrix Anal. Appl., 26 (2005), pp. 621-639.
22. I. S. DUFF AND J. KOSTER, The design and use of algorithms for permuting large entries to the diagonal of sparse matrices, SIAM J. Matrix Anal. Appl., 20 (1999), pp. 889-901.
23. I. S. DUFF AND J. KOSTER, On algorithms for permuting large entries to the diagonal of a sparse matrix, SIAM J. Matrix Anal. Appl., 22 (2001), pp. 973-996.
24. H. C. ELMAN AND M. H. SCHULTZ, Preconditioning by fast direct methods for nonself-adjoint nonseparable elliptic equations, SIAM J. Numer. Anal., 23 (1986), pp. 44-57.
25. P. F. FISCHER, Projection techniques for iterative solution of Ax = b with successive right-hand sides, Comput. Methods Appl. Mech. Engrg., 163 (1998), pp. 193-204.
26. R. W. FREUND, A transpose-free quasi-minimal residual algorithm for non-Hermitian linear systems, SIAM J. Sci. Comput., 14 (1993), pp. 470-482.
27. P. E. GILL, W. MURRAY, AND M. A. SAUNDERS, Methods for computing and modifying the LDV factors of a matrix, Math. Comp., 29 (1975), pp. 1051-1077.
28. G. H. GOLUB AND C. F. VAN LOAN, Matrix Computations, 3rd ed., The Johns Hopkins University Press, Baltimore, London, 1996.
29. D. HYSOM AND A. POTHEN, A scalable parallel algorithm for incomplete factor preconditioning, SIAM J. Sci. Comput., 22 (2001), pp. 2194-2215.
30. I. E. KAPORIN AND O. AXELSSON, On a class of nonlinear equation solvers based on the residual norm reduction over a sequence of affine subspaces, SIAM J. Sci. Comput., 16 (1995), pp. 228-249.
31. C. T. KELLEY, Iterative Methods for Linear and Nonlinear Equations, Frontiers Appl. Math. 16, SIAM, Philadelphia, 1995.
32. C. T. KELLEY, Solving Nonlinear Equations with Newton's Method, Fundam. Algorithms 1, SIAM, Philadelphia, 2003.
33. D. S. KERSHAW, The incomplete Cholesky-conjugate gradient method for the iterative solution of systems of linear equations, J. Comput. Phys., 26 (1978), pp. 43-65.
34. D. KEYES, Terascale implicit methods for partial differential equations, in Contemp. Math. 306, AMS, Providence, RI, 2001, pp. 29-84.
35. S. A. KHARCHENKO, L. YU. KOLOTILINA, A. A. NIKISHIN, AND A. YU. YEREMIN, A reliable AINV-type preconditioning method for constructing sparse approximate inverse preconditioned in factored form, Numer. Linear Algebra Appl., 8 (2001), pp. 165-179.
36. D. A. KNOLL AND D. E. KEYES, Jacobian-free Newton-Krylov methods: A survey of approaches and applications, J. Comput. Phys., 193 (2004), +pp. 357-397.
37. D. A. KNOLL AND P. R. MCHUGH, Newton-Krylov methods applied to a system of convectionreaction-diffusion equations, Comput. Phys. Comm., 88 (1995), pp. 141-160.
38. D. A. KNOLL, P. R. MCHUGH, AND D. E. KEYES, Newton-Krylov methods for low Mach number compressible combustion, AIAA J., 34 (1996), pp. 961-967.
39. J. B. KRUSKAL, On the shortest spanning subtree of a graph and the traveling salesman problem, Proc. Amer. Math. Soc, 2 (1956), pp. 48-50.
40. I. LEE, P. RAGHAVAN, AND E. G. NG, Effective preconditioning through ordering interleaved with incomplete factorization, SIAM J. Matrix Anal. Appl., 27 (2006), pp. 1069-1088.
41. D. LOGHIN, D. RUIZ, AND A. TOUHAMI, Adaptive preconditioners for nonlinear systems of equations, J. Comput. Appl. Math., 189 (2006), pp. 326-374.
42. L. LUKŠAN, M. TŮMA, J. VLČEK, N. RAMEŠOVÁ, M. ŠIŠKA, J. HARTMAN, AND C. MATONOHA, UFO 2006. Interactive System for Universal Functional Optimization, Technical report V-977, ICS AS CR, Prague, 2006.
43. T. A. MANTEUFFEL, An incomplete factorization technique for positive definite linear systems, Math. Comp., 34 (1980), pp. 473-497.
44. J. L. MORALES AND J. NOCEDAL, Automatic preconditioning by limited memory quasi-Newton updating, SIAM J. Optim., 10 (2000), pp. 1079-1096.
45. D. P. O'LEARY, The block conjugate gradient algorithm and related methods, Linear Algebra Appl., 29 (1980), pp. 293-322.
46. M. OLSCHOWKA AND A. NEUMAIER, A new pivoting strategy for Gaussian elimination, Linear Algebra Appl., 240 (1996), pp. 131-151.
47. M. L. PARKS, E. DE STURLER. G. MACKEY, D. D. JOHNSON, AND S. MAITI, Recycling Krylov subspaces for sequences of linear systems, SIAM J. Sci. Comput., 28 (2006), pp. 1651-1674.
48. D. W. PEACEMAN AND H. H. RACHFORD, JR., The numerical solution of parabolic and elliptic differential equations, SIAM J. Appl. Math., 3 (1955), pp. 28-41.
49. A. POTHEN, F. MANNE, AND A. H. GEBREMEDHIN, What color is your Jacobian? Graph coloring for computing derivatives, SIAM Rev., 47 (2005), pp. 629-705.
50. R. C. PRIM, Shortest connection networks and some generalizations, Bell System Tech. J., 36 (1957), pp. 1389-1401.
51. Y. SAAD, A flexible inner-outer preconditioned GMRES algorithm, SIAM J. Sci. Comput., 14 (1993), pp. 461-469.
52. Y. SAAD, ILUT: A dual threshold incomplete ILU decomposition, Numer. Linear Algebra Appl., 1 (1994), pp. 387-402.
53. Y. SAAD AND M. H. SCHULTZ, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Statist. Comput., 7 (1986), pp. 856-869.
54. V. E. SHAMANSKII, On a modification of the Newton method, Ukrain. Mat. Ž., 19 (1967), pp. 133-138.
55. V. SIMONCINI AND E. GALLOPOULOS, An iterative method for nonsymmeric systems with multiple right-hand sides, SIAM J. Sci. Comput., 16 (1995), pp. 917-933.
56. V. SIMONCINI AND D. B. SZYLD, Flexible inner-outer Krylov subspace methods, SIAM J. Numer. Anal., 40 (2003), pp. 2219-2239.
57. M. TISMENETSKY, A new preconditioning technique for solving large sparse linear systems, Linear Algebra Appl., 154/156 (1991), pp. 331-353.
58. H. A. vAN DER VORST, Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems, SIAM J. Sci. Statist. Comput., 13 (1992), pp. 631-644.
59. P. VASSILEVSKI, Preconditioning nonsymmetric and indefinite finite element matrices, Numer. Linear Algebra Appl., 1 (1992), pp. 59-76.
60. B. VITAL, Etude de quelques méthodes de résolution de problèmes linéaires de grande taille sur multiprocesseur, Ph.D. thesis, Université de Rennes I, Rennes, France, 1990.
61. J. W. WATTS III, A conjugate gradient truncated direct method for the iterative solution of the reservoir simulation pressure equation, Soc. Petr. Eng. J., 21 (1981), pp. 345-353.
62. O. WIDLUND, A Lanczos method for a class of nonsymmetric systems of linear equations, SIAM J. Numer. Anal., 15 (1978), pp. 801-812.