A robust and efficient ILU that incorporates the growth of the inverse triangular factors

SIAM Journal on Scientific Computing

Volumn 25, Issue 1, 2003, Pages 86-103

  Bollhöfer, Matthias ^a  

^a TECHNISCHE UNIVERSITÄT BERLIN   (Germany)

Author keywords

Approximate inverse; Condition estimator; ILU; Sparse direct methods; Sparse matrices

Indexed keywords

APPROXIMATION THEORY; INVERSE PROBLEMS; LOGIC PROGRAMMING; MATRIX ALGEBRA; PROBLEM SOLVING;

APPROXIMATE INVERSE; INVERSE TRIANGULAR FACTORS; SPARSE MATRIX;

NATURAL SCIENCES COMPUTING;

EID: 12144291027     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/S1064827502403411     Document Type: Article

References (36)

1. 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.
2. M. Benzi, J. C. Haws, and M. Tůma, Preconditioning highly indefinite and nonsymmetric matrices, SIAM J. Sci. Comput., 22 (2000), pp. 1333-1353.
3. M. Benzi, C. D. Meyer, and M. Tůma, A sparse approximate inverse preconditioner for the conjugate gradient method, SIAM J. Sci. Comput., 17 (1995), pp. 1135-1149.
4. M. Benzi and M. Tůma, A sparse approximate inverse preconditioner for nonsymmetric linear systems, SIAM J. Sci. Comput., 19 (1998), pp. 968-994.
5. M. Benzi and M. Tůma, A robust incomplete factorization preconditioner for positive definite matrices, Numer. Linear Algebra Appl., to appear.
6. M. Bollhöfer, A robust ILU with pivoting based on monitoring the growth of the inverse factors, Linear Algebra Appl., 338 (2001), pp. 201-218.
7. M. Bollhöfer and Y. Saad, On the relations between ILU's and factored approximate inverses, SIAM J. Matrix Anal. Appl., 24 (2002), pp. 219-237.
8. A. K. Cline, C. B. Moler, G. W. Stewart, and J. H. Wilkinson, An estimate for the condition number of a matrix, SIAM J. Numer. Anal., 16 (1979), pp. 368-375.
9. T. Davis, University of Florida Sparse Matrix Collection, http://www.cise.ufl.edu/~davis/sparse/.
10. T. A. Davis, Algorithm 8xx: Umfpack v3.2, an Unsymmetric-Pattern Multifrontal Method with a Column Pre-Ordering Strategy, Technical report TR-02-002, University of Florida, Gainesville, FL, 2002.
11. J. W. Demmel, S. C. Eisenstat, J. R. Gilbert, X. S. Li, and J. W. H. Liu, A supernodal approach to sparse partial pivoting, SIAM J. Matrix Anal. Appl., 20 (1999), pp. 720-755.
12. I. S. Duff, private communication, CERFACS, 2001.
13. I. S. Duff, A. Erisman, and J. Reid, Direct Methods for Sparse Matrices, Oxford University Press, New York, 1986.
14. I. S. Duff, R. G. Grimes, and J. G. Lewis, Sparse matrix test problems, ACM Trans. Math. Software, 15 (1989), pp. 1-14.
15. I. S. Duff, R. G. Grimes, and J. G. Lewis, Users' guide for the Harwell-Boeing sprse matrix collection (release 1), Technical report RAL-TR-92-086, Rutherford Appleton Laboratory, Oxfordshire, England, 1992.
16. 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.
17. I. S. Duff and J. Reid, The design of MA48, a code for the direct solution of sparse unsymmetric linear systems of equations, ACM Trans. Math. Software, 22 (1996), pp. 187-226.
18. S. C. Eisenstat and J. W. H. Liu, Exploiting structural symmetry in a sparse partial pivoting code, SIAM J. Sci. Comput., 14 (1993), pp. 253-257.
19. R. Freund and N. Nachtigal, QMR; A quasi-minimal residual method for non-Hermitian linear system, Numer. Math., 60 (1991), pp. 315-339.
20. R. W. Freund, G. H. Golub, and N. M. Nachtigal, Iterative solution of linear systems, in Acta Numer., Cambridge University Press, Cambridge, UK, 1992, pp. 57-100.
21. J. A. George and J. W. Liu, Computer Solution of Large Sparse Positive Definite Systems, Prentice-Hall, Englewood Cliffs, NJ, 1981.
22. J. R. Gilbert, C. Moler, and R. Schreiber, Sparse matrices in MATLAB: Design and implementation, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 333-356.
23. J. R. Gilbert and T. Peierls, Sparse partial pivoting in time proportional to arithmetic operations, SIAM J. Sci. Statist. Comput., 9 (1988), pp. 862-874.
24. J. R. Gilbert and S. Toledo, An assessment of incomplete-lu preconditioners for nonsymmetric linear systems, Informatica, 24 (2000), pp. 409-425.
25. G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed., The Johns Hopkins University Press, Baltimore, MD, 1996.
26. A. Greenbaum, Iterative Methods for Solving Linear Systems, Front. Appl. Math. 17, SIAM, Philadelphia, 1997.
27. A. Gupta, Improved symbolic and numerical factorization algorithms for unsymmetric sparse matrices, SIAM J. Matrix Anal. Appl., 24 (2002), pp. 529-552.
28. S. Kharchenko, L. Kolotilina, A. Nikishin, and A. Yeremin, A reliable AINV-type preconditioning method for constructing sparse approximate inverse preconditioners in factored form, Numer. Linear Algebra Appl., 8 (2001), pp. 165-179.
29. J. Meijerink and H. A. V. 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.
30. N. Munksgaard, Solving sparse symmetric sets of linear equations by preconditioned conjugate gradient method, ACM Trans. Math. Software, 6 (1980), pp. 206-219.
31. National Institute of Standards, Matrix market, http://math.nist.gov/MatrixMarket/ (24 September 2002).
32. Y. Saad, ILUT: A dual threshold incomplete ILU factorization, Numer. Linear Algebra Appl., 1 (1994), pp. 387-402.
33. Y. Saad, Iterative Methods for Sparse Linear Systems, PWS Publishing, Boston, 1996.
34. Y. Saad and M. Schultz, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Statist. Comput., 7 (1986), pp. 856-869.
35. MATLAB Version 6.1 Release 12, The MathWorks Inc., Natick, MA, 2001.
36. M. Tismenetsky, A new preconditioning technique for solving large sparse linear systems, Linear Algebra Appl., 154/156 (1991), pp. 331-353.