Balanced incomplete factorization

SIAM Journal on Scientific Computing

Volumn 30, Issue 5, 2007, Pages 2302-2318

  Bru, Rafael ^a   Marín, José ^a   Mas, José ^a   Tuma M ^b  

^a UNIVERSITAT POLITÈCNICA DE VALÈNCIA   (Spain)

^b INSTITUTE OF COMPUTER SCIENCE   (Czech Republic)

Author keywords

Approximate inverses; Incomplete decompositions; Preconditioned iterative methods; Sparse matrices

Indexed keywords

COMPUTATION THEORY; INVERSE PROBLEMS; ITERATIVE METHODS; MATRIX ALGEBRA;

APPROXIMATE INVERSE; INCOMPLETE DECOMPOSITION; INCOMPLETE FACTORIZATION; INVERSE FACTORS; PRECONDITIONED ITERATIVE METHODS; SHERMAN-MORRISON FORMULA; SPARSE MATRICES; SQUARE MATRIX; SYMMETRIC POSITIVE DEFINITE; TRIANGULAR FACTORS;

FACTORIZATION;

EID: 55349132334     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/070696088     Document Type: Article

References (50)

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. P. R. AMESTOY, T. A. DAVIS, AND I. S. DUFF, An approximate minimum degree ordering algorithm, SIAM J. Matrix Anal. Appl., 17 (1996), pp. 886-905.
3. O. AXELSSON AND L. KOLOTILINA, Diagonally compensated reduction and related preconditioning methods, Numer. Linear Algebra Appl., 1 (1994), pp. 155-177.
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 nonsymmetric 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, D. B. SZYLD, AND A. VAN DUIN, Orderings for incomplete factorization preconditioning of nonsymmetric problems, SIAM J. Sci. Comput., 20 (1999), pp. 1652-1670.
8. M. BENZI AND M. TŮMA, A sparse approximate inverse preconditioner for nonsymmetric linear systems, SIAM J. Sci. Comput., 19 (1998), pp. 968-994.
9. M. BENZI AND M. TŮMA, A comparative study of sparse approximate inverse preconditioners, Appl. Numer. Math., 30 (1999), pp. 305-340.
10. M. BENZI AND M. TŮMA, Orderings for factorized sparse approximate inverse preconditioners, SIAM J. Sci. Comput., 21 (2000), pp. 1851-1868.
11. M. BENZI AND M. TŮMA, A robust incomplete factorization preconditioner for positive definite matrices, Numer. Linear Algebra Appl., 10 (2003), pp. 385-400.
12. M. BENZI AND M. TŮMA, A robust preconditioner with low memory requirements for large sparse least squares problems, SIAM J. Sci. Comput., 25 (2003), pp. 499-512.
13. M. BENZI, Preconditioning techniques for large linear systems: A survey, J. Comput. Phys., 182 (2002), pp. 418-477.
14. M. BOLLHÖFER, AND Y. SAAD, On the relations between ILUs and factored approximate inverses, SIAM J. Matrix Anal. Appl., 24 (2002), pp. 219-237.
15. M. BOLLHÖFER, A robust ILU with pivoting based on monitoring the growth of the inverse factors, Linear Algebra Appl., 338 (2001), pp. 201-218.
16. M. BOLLHÖFER, A robust and efficient ILU that incorporates the growth of the inverse triangular factors, SIAM J. Sci. Comput., 25 (2003), pp. 86-103.
17. M. BOLLHÖFER, private communication, 2004.
18. R. BRIDSON AND W.-P. TANG, Ordering, anisotropy, and factored sparse approximate inverses, SIAM J. Sci. Comput., 21 (1999), pp. 867-882.
19. R. BRU, J. CERDÁN, J. MARÍN, AND J. MAS, Preconditioning sparse nonsymmetric linear systems with the Sherman-Morrison formula, SIAM J. Sci. Comput., 25 (2003), pp. 701-715.
20. N. I. BULEEV, A numerical method for solving two-dimensional diffusion equations, Atomnaja, Energija, 6 (1959), pp. 338-340.
21. N. I. BULEEV, A numerical method for solving two-dimensional and three-dimensional diffusion equations, Matematičeskij Sbornik, 51 (1960), pp. 227-238.
22. J. CERDÁN, T. FARAJ, J. MARÍN, AND J MAS, A Block Approximate Inverse Preconditioner for Sparse Nonsymmetric Linear Systems, Technical report TR-IMM2005/04, Polytechnic University of Valencia, Spain, 2005.
23. T. F. CHAN AND H. A. VAN DER VORST, Approximate and incomplete factorizations, in Parallel Numerical Algorithms, ICASE/LaRC Interdiscip. Ser. Sci. Eng. IV. Centenary Conference, D.E. Keyes, A. Sameh, and V. Venkatakrishnan, eds., Kluver Academic, Dordrecht, 1997, pp. 167-202.
24. T. A. DAVIS, The University of Florida Sparse Matrix Collection, Technical report REP-2007-298, CISE, University of Florida, Gainesville, FL, 2007; also available online from http://www.cise.ufi.edu/ ~davis.
25. P. F. DUBOIS, A. GREENBAUM, AND G. H. RODRIGUE, Approximating the inverse of a matrix for use on iterative algorithms on vector processors, Computing, 22 (1979), pp. 257-268.
26. 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., 20 (1999), pp. 889-901.
27. I. S. DUFF AND J. KOSTER, On algorithms for permuting large entries to the diagonal of a sparse matrix, SIAM J. Matrix Anal., 22 (2001), pp. 973-996.
28. I. S. DUFF AND G. A. MEURANT, The effect of ordering on preconditioned conjugate gradients, BIT, 29 (1989), pp. 635-657.
29. T. DUPONT, R. P. KENDALL, AND H. H. RACHFORD, JR., An approximate factorization procedure for the solving self-adjoint elliptic difference equations, SIAM J. Numer. Anal., 5 (1968), pp. 559-573.
30. N. I. M. GOULD, Y. HU, AND J. A. SCOTT, A numerical evaluation of sparse direct solvers for the solution of large sparse symmetric linear systems of equations, ACM Trans. Math. Software, article 10 (electronic), 33 (2007).
31. M. J. GROTE AND T. HUCKLE, Parallel preconditioning with sparse approximate inverses, SIAM J. Sci. Comput., 18 (1997), pp. 838-853.
32. I. GUSTAFSSON, A class of first order factorization methods, BIT, 18 (1978), pp. 142-156.
33. M. HAGEMANN AND O. SCHENK, Weighted matchings for preconditioning symmetric indefinite linear systems, SIAM J. Sci. Comput., 28 (2006), pp. 403-420.
34. A. JENNINGS, A compact storage scheme for the solution of symmetric linear simultaneous equations, Comput. J., 9 (1966), pp. 281-285.
35. O. G. JOHNSON, C. A. MICCHELLI, AND G. PAUL, Polynomial preconditioned for conjugate gradient calculations, SIAM J. Numer. Anal., 20 (1983), pp. 362-376.
36. M. T. JONES AND P. E. PLASSMANN, An improved incomplete Cholesky factorization, ACM Trans. Math. Software, 21 (1995), pp. 5-17.
37. TU decomposition, Numer. Linear Algebra Appl., 5 (1998), pp. 483-509.
38. D. S. KERSHAW, The incomplete Cholesky-conjugate gradient method for the iterative solution of systems of linear equations, J. Comp. Phys., 26 (1978), pp. 43-65.
39. L. YU. KOLOTILINA AND A. YU. YEREMIN, Factorized sparse approximate inverse preconditionings I. Theory, SIAM J. Matrix Anal. Appl., 14 (1993), pp. 45-58.
40. R. KOUHIA, Sparse matrices web page, http://www.hut.fi/ ~kouhia/sparse.html, 2001.
41. 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.
42. T. A. MANTEUFFEL, Shifted incomplete Cholesky factorization, in Sparse Matrix Proceedings, I. S. Duff and G. W. Stewart, eds., SIAM Publications, Philadelphia, PA, 1979.
43. T. A. MANTEUFFEL, An incomplete factorization technique for positive definite linear systems, Math. Comp., 34 (1980), pp. 473-497.
44. 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.
45. Y. SAAD, SPARSKIT: A Basic Tool Kit for Sparse Matrix Computations, Technical report 90-20, Research Institute for Advanced Computer Science, NASA Ames Research Center, Moffet Field, CA, 1990.
46. Y. SAAD, ILUT: A dual threshold incomplete LU factorization, Numer. Linear Algebra Appl., 1 (1994), pp. 387-402.
47. Y. SAAD, Iterative Methods for Sparse Linear Systems, PWS Publishing, Boston, 1996.
48. O. SCHENK. M. BoLLHOFER, AND R. A. RoMER, On large-scale diagonalization techniques for the Anderson model of localization, SIAM J. Sci. Comput., 28 (2006), pp. 963-983.
49. M. TISMENETSKY, A new preconditioning technique for solving large sparse linear systems. Linear. Algebra Appl., 154-156 (1991), pp. 331-353.
50. R. S. VARGA, Factorizations and normalized iterative methods, in Boundary problems in differential equations, University of Wisconsin Press, Madison, WI, 1960, pp. 121-142.