A sparse approximate inverse preconditioner for nonsymmetric linear systems

SIAM Journal on Scientific Computing

Volumn 19, Issue 3, 1998, Pages 968-994

  Benzi, Michele ^a   Tůma, Miroslav ^b  

^a LOS ALAMOS NATIONAL LABORATORY   (United States)

^b INSTITUTE OF COMPUTER SCIENCE   (Czech Republic)

Author keywords

Approximate inverses; Conjugate gradient type methods; Incomplete factorizations; Preconditioning; Sparse linear systems; Sparse matrices

Indexed keywords

EID: 0001135947     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/S1064827595294691     Document Type: Article

References (57)

1. E. C. ANDERSON, Parallel Implementation of Preconditioned Conjugate Gradient Methods for Solving Sparse Systems of Linear Equations, M.Sc. thesis and CSRD Report 805, Department of Computer Science, University of Illinois at Urbana-Champaign, IL, 1988.
2. O. AXELSSON, Iterative Solution Methods, Cambridge University Press, Cambridge, UK, 1994.
3. R. BARRETT, M. BERRY, T. CHAN, J. DEMMEL, J. DONATO, J. DONGARRA, V. EIJKHOUT, R. Pozo, C. ROMINE, AND H. VAN DER VORST, Templates for the Solution of Linear Systems, SIAM, Philadelphia, PA, 1994.
4. M. BENSON, J. KRETTMANN, AND M. WRIGHT, Parallel algorithms for the solution of certain large sparse linear systems, Internat. J. Comput. Math., 16 (1984), pp. 245-260.
5. M. BENZI, A Direct Row-Projection Method for Sparse Linear Systems, Ph.D. thesis, Department of Mathematics, North Carolina State University, Raleigh, NC, 1993.
6. M. BENZI AND C. D. MEYER, A direct projection method for sparse linear systems, SIAM J. Sci. Comput., 16 (1995), pp. 1159-1176.
7. M. BENZI AND C. D. MEYER, An explicit preconditioner for the conjugate gradient method, in Proc. Cornelius Lanczos International Centenary Conference, J. D. Brown, M. T. Chu, D. C. Ellison, and R. J. Plemmons, eds., SIAM, Philadelphia, PA, 1994, pp. 294-296.
8. 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.
9. M. BENZI AND M. TŮMA, A Sparse Approximate Inverse Preconditioner for Nonsymmetric Linear Systems, Research Report No. 653, Institute of Computer Science, Czech Academy of Sciences, Prague, Czech Republic, 1995.
10. H. BERRYMAN, J. SALTZ, W. GROPP, AND R. MIRCHANDANEY, Krylov methods preconditioned with incompletely factored matrices on the CM-2, J. Parallel Distrib. Comput., 8 (1990), pp. 186-190.
11. E. CHOW AND Y. SAAD, Approximate inverse preconditioners via Sparse-Sparse Iterations, SIAM J. Sci. Comput., 19 (1998), pp. 995-1023.
12. E. CHOW AND Y. SAAD, Approximate inverse techniques for block-partitioned matrices, SIAM J. Sci. Comput., 18 (1997), pp. 1657-1675.
13. E. CHU, A. GEORGE, J. W.-H. LIU, AND E. G.-Y. NG, User's Guide for SPARSPAK-A: Waterloo Sparse Linear Equations Package, Technical Report CS-84-36, University of Waterloo, Waterloo, ON, Canada, 1984.
14. M. T. CHU, R. E. FUNDERLIC, AND G. H. GOLUB, A rank-one reduction formula and its applications to matrix factorizations, SIAM Rev., 37 (1995), pp. 512-530.
15. P. CONCUS, G. H. GOLUB, AND G. A. MEURANT, Block preconditioning for the conjugate gradient method, SIAM J. Sci. Statist. Comput., 6 (1985), pp. 220-252.
16. J. D. F. COSGROVE, J. C. DIAZ, AND A. GRIEWANK, Approximate inverse preconditionings for sparse linear systems, Internat. J. Comput. Math., 44 (1992), pp. 91-110.
17. T. DAVIS, Sparse matrix collection, NA Digest, Vol. 94, Issue 42, October 1994.
18. S. DEMKO, W. F. Moss, AND P. W. SMITH, Decay rates for inverses of band matrices, Math. Comput., 43 (1984), pp. 491-499.
19. I. S. DUFF, MA28 - A Set of Fortran Subroutines for Sparse Unsymmetric Linear Equations, Harwell Report AERE R8730, HMSO, London; revised 1980.
20. I. S. DUFF, A. M. ERISMAN, AND J. K. REID, Direct Methods for Sparse Matrices, Oxford University Press, Oxford, UK, 1986.
21. I. S. DUFF, R. G. GRIMES, AND J. G. LEWIS, Users' Guide for the Harwell-Boeing Sparse Matrix Collection, Technical Report RAL-92-086, Rutherford Appleton Laboratory, Chilton, UK, 1992.
22. I. S. DUFF AND G. A. MEURANT, The effect of ordering on preconditioned conjugate gradients, BIT, 29 (1989), pp. 635-657.
23. H. ELMAN, A stability analysis of incomplete LU factorizations, Math. Comp., 47 (1986), pp. 191-217.
24. M. FIEDLER, Inversion of bigraphs and connection with the Gauss elimination, in Graphs, Hypergraphs and Block Systems, Zielona Gora, Poland, 1976, pp. 57-68.
25. L. Fox, An Introduction to Numerical Linear Algebra, Oxford University Press, Oxford, UK, 1964.
26. L. Fox, H. D. HUSKEY, AND J. H. WILKINSON, Notes on the solution of algebraic linear simultaneous equations, Quart. J. Mech. Appl. Math., 1 (1948), pp. 149-173.
27. A. GEORGE AND J. W.-H. LIU, Computer Solution of Large Sparse Positive Definite Systems, Prentice-Hall, Englewood Cliffs, NJ, 1981.
28. A. GEORGE AND J. W.-H. LIU, The evolution of the minimum degree algorithm, SIAM Rev., 31 (1989), pp. 1-19.
29. J. R. GILBERT, Predicting structure in sparse matrix computations, SIAM J. Matrix Anal. Appl., 15 (1994), pp. 62-79.
30. N. I. M. GOULD AND J. A. SCOTT, On Approximate-Inverse Preconditioners, Technical Report RAL-95-026, Rutherford Appleton Laboratory, Chilton, UK, 1995.
31. M. GROTE AND T. HUCKLE, Parallel preconditioning with sparse approximate inverses, SIAM J. Sci. Comput., 18 (1997), pp. 838-853.
32. M. GROTE AND H. SIMON, Parallel preconditioning and approximate inverses on the Connection Machine, in Proc. Sixth SIAM Conference on Parallel Processing for Scientific Computing, R. F. Sincovec, D. E. Keyes, M. R. Leuze, L. R. Petzold, and D. A. Reed, eds., SIAM, Philadelphia, PA, 1993, pp. 519-523.
33. M. A. HEROUX, P. VU, AND C. YANG, A parallel preconditioned conjugate gradient package for solving sparse linear systems on a Cray Y-MP, Appl. Numer. Math., 8 (1991), pp. 93-115.
34. M. R. HESTENES, Inversion of matrices by biorthogonalization and related results, J. Soc. Indust. Appl. Math., 6 (1958), pp. 51-90.
35. M. R. HESTENES AND E. STIEFEL, Method of conjugate gradients for solving linear systems, J. Res. Nat. Bur. Standards, 49 (1952), pp. 409-436.
36. A. S. HOUSEHOLDER, The Theory of Matrices in Numerical Analysis, Blaisdell, New York, 1964; reprinted by Dover, New York, 1975.
37. O. G. JOHNSON, C. A. MICCHELLI, AND G. PAUL, Polynomial preconditioning for conjugate gradient calculations, SIAM J. Numer. Anal., 20 (1983), pp. 362-375.
38. M. T. JONES AND P. E. PLASSMANN, The efficient parallel iterative solution of large sparse linear systems, Graph Theory and Sparse Matrix Computation, IMA Vol. Math. Appl. 56, A. George, J. R. Gilbert, and J. W.-H. Liu, eds., Springer-Verlag, New York, 1994, pp. 229-245.
39. I. E. KAPORIN, Explicitly preconditioned conjugate gradient method for the solution of unsymmetric linear systems, Internat. J. Comput. Math., 40 (1992), pp. 169-187.
40. I. E. KAPORIN, Two-level explicit preconditioning of the conjugate gradient method, Differential Equations, 28 (1992), pp. 280-289.
41. I. E. KAPORIN, New convergence results and preconditioning strategies for the conjugate gradient method, Numer. Linear Algebra Appl., 1 (1994), pp. 179-210.
42. L. YU. KOLOTILINA, A. A. NIKISHIN, AND A. YU. YEREMIN, Factorized sparse approximate inverse (FSAI) preconditionings for solving 3D FE systems on massively parallel computers II: Iterative construction of FSAI preconditioners, in Proc. IMACS International Symposium on Iterative Methods in Linear Algebra, R. Beauwens and P. de Groen, eds., North-Holland, Amsterdam, 1992, pp. 311-312.
43. L. YU. KOLOTILINA AND A. YU. YEREMIN, Factorized sparse approximate inverse preconditioning I: Theory, SIAM J. Matrix Anal. Appl., 14 (1993), pp. 45-58.
44. L. YU. KOLOTILINA AND A. YU. YEREMIN, Factorized sparse approximate inverse preconditioning II: Solution of 3D FE systems on massively parallel computers, Internat. J. High Speed Comput., 7 (1995), pp. 191-215.
45. S. L. LEE, Krylov Methods for the Numerical Solution of Initial- Value Problems in Differential-Algebraic Equations, Ph.D. thesis and Technical Report UIUCDS-R-93-1814, Department of Computer Science, University of Illinois at Urbana-Champaign, IL, 1993.
46. T. A. MANTEUFFEL, An incomplete factorization technique for positive definite linear systems, Math. Comp., 34 (1980), pp. 73-497.
47. J. A. MEIJERINK AND H. A. VAN DER VORST, An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix, Math. Comp., 31 (1977), pp. 148-162.
48. N. S. MENDELSOHN, Some properties of approximate inverses of matrices, Trans. Roy. Soc. Canada, III, 50 (1956), pp. 53-59.
49. Y. SAAD, Preconditioning techniques for nonsymmetric and indefinite linear systems, J. Comput. Appl. Math., 24 (1988), pp. 89-105.
50. Y. SAAD, SPARSKIT: A Basic Tool Kit for Sparse Matrix Computations, Technical Report CSRD TR 1029, CSRD, University of Illinois at Urbana-Champaign, IL, 1990.
51. Y. SAAD, ILUT: A dual threshold incomplete LU factorization, Numer. Linear Algebra Appl., 1 (1994), pp. 387-402.
52. G. W. STEWART, Conjugate direction methods for solving systems of linear equations, Numer. Math., 21 (1973), pp. 283-297.
53. R. E. TARJAN, Depth-first search and linear graph algorithms, SIAM J. Comput., 1 (1972), pp. 146-160.
54. H. A. VAN DER VORST, High performance preconditioning, SIAM J. Sci. Statist. Comput., 10 (1989), pp. 1174-1185.
55. R. S. VARGA, Matrix Iterative Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1962.
56. H. F. WALKER, Implementation of the GMRES method using Householder transformations, SIAM J. Sci. Statist. Comput., 9 (1988), pp. 152-163.
57. Z. ZLATEV, Computational Methods for General Sparse Matrices, Kluwer, Dordrecht, the Netherlands, 1991.