Dynamically relaxed block incomplete factorizations for solving two- and three-dimensional problems

SIAM Journal on Scientific Computing

Volumn 21, Issue 6, 2000, Pages 2008-2028

  Magolu Monga Made, Mardochée ^a   Notay, Yvan ^a  

^a UNIVERSITÉ LIBRE DE BRUXELLES   (Belgium)

Author keywords

[No Author keywords available]

Indexed keywords

DIAGONAL RELAXATION; KRYLOV SUBSPACE METHODS; LARGE SPARSE LINEAR SYSTEMS; PRECONDITIONED CONJUGATE GRADIENT METHOD; RELAXED BLOCK INCOMPLETE FACTORIZATIONS;

ALGORITHMS; APPROXIMATION THEORY; BOUNDARY CONDITIONS; FINITE DIFFERENCE METHOD; INTERPOLATION; MATHEMATICAL MODELS; MATRIX ALGEBRA; PARAMETER ESTIMATION; PROBLEM SOLVING; VECTORS;

PARTIAL DIFFERENTIAL EQUATIONS;

EID: 0034505632     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/S1064827596311591     Document Type: Article

References (40)

1. O. AXELSSON, Iterative Solution Methods, Cambridge University Press, Cambridge, UK, 1994.
2. O. AXELSSON AND V. EIJKHOUT, Vectorizable preconditioners for elliptic difference equations in three space dimensions, J. Comput. Appl. Math., 27 (1989), pp. 299-321.
3. O. AXELSSON AND G. LINDSKOG, On the eigenvalue distribution of class of preconditioning methods, Numer. Math., 48 (1986), pp. 479-498.
4. O. AXELSSON AND G. LINDSKOG, On the rate of convergence of the preconditioned conjugate gradient method, Numer. Math., 48 (1986), pp. 499-523.
5. O. AXELSSON AND H. LU, On eigenvalue estimates for block incomplete factorization methods, SIAM J. Matrix Anal. Appl., 16 (1995), pp. 1074-1085.
6. R. F. BARRET, M. BERRY, T. F. 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: Building Blocks for Iterative Methods, SIAM, Philadelphia, 1994.
7. R. BEAUWENS, Modified incomplete factorization strategies, in Preconditioned Conjugate Gradient Methods, O. Axelsson and L. Kolotilina, eds., Lectures Notes in Mathematics 1457, Springer-Verlag, Berlin, 1990, pp. 1-16.
8. R. BEAUWENS AND M. BEN BOUZID, On sparse block factorization iterative methods, SIAM J. Numer. Anal., 24 (1987), pp. 1066-1076.
9. R. BEAUWENS AND M. BEN BOUZID, Existence and conditioning properties of sparse, approximate block factorizations, SIAM J. Numer. Anal., 25 (1988), pp. 941-956.
10. A. BERMAN AND R. J. PLEMMONS, Nonnegative Matrices in the Mathematical Sciences, Academic Press, New York, 1979.
11. A. M. BRUASET, A Survey of Preconditioned Iterative Methods, Pitman Res. Notes Math. Ser., 328, Longman, Harlow, UK, 1995.
12. T. F. CHAN, Fourier analysis of relaxed incomplete factorization preconditioners, SIAM J. Sci. Statist. Comput., 12 (1991), pp. 668-680.
13. T. F. CHAN AND H. A. VAN DER VORST, Approximate and incomplete factorizations, in Parallel Numerical Algorithms, D. E. Keyes, A. Samed, and V. Venkatakrishrian, eds., ICASE/LaRC Interdiscip. Ser. Sci. Eng. 4, Kluwer Acad. Publ., Dordrecht, 1997, pp. 167-202.
14. E. CHOW AND M. A. HEROUX, BPKIT Block Preconditioning Tool Kit, Technical Report UMSI 96/183, University of Minnesota, Supercomputer Institute, Minneapolis, 1996.
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. V. EIJKHOUT, Beware of unperturbed modified incomplete factorizations, in Iterative Methods in Linear Algebra, R. Beauwens and P. de Groen, eds., North-Holland, Amsterdam, 1992, pp. 583-591.
17. H. C. ELMAN, Relaxed and stabilized incomplete factorizations for nonself-adjoint linear systems, BIT, 29 (1989), pp. 890-915.
18. G. H. GOLUB AND C. F. VAN LOAN, Matrix Computations, 3rd ed., The Johns Hopkins University Press, Baltimore, MD, 1996.
19. G. H. GOLUB AND H. A. VAN DER VORST, Closer to the solution: Iterative linear solvers, in The State of the Art in Numerical Analysis, I. S. Duff and G. A. Watson, eds., Clarendon Press, Oxford, UK, 1997, pp. 63-92.
20. H. LU, Compensative Block Incomplete Factorizations, Technical Report 9610, Department of Mathematics, University of Nijmegen, The Netherlands, 1996.
21. M. MAGOLU MONGA-MADE, Modified block-approximate factorization strategies, Numer. Math., 61 (1992), pp. 91-110.
22. M. MAGOLU MONGA-MADE, Analytical bounds for block approximate factorization methods, Linear Algebra Appl., 179 (1993), pp. 33-57.
23. M. MAGOLU MONGA-MADE, Empirically modified block incomplete factorizations, in Incomplete Decomposition (ILU) - Algorithms, Theory and Applications, W. Hackbusch and G. Wittum, eds., Notes Numer. Fluid Mech. 41, Vieweg, Braunschweig, 1993, pp. 78-87.
24. M. MAGOLU MONGA-MADE, Ordering strategies for modified block incomplete factorizations, SIAM J. Sci. Comput., 16 (1995), pp. 378-399.
25. M. MAGOLU MONGA-MADE, Taking advantage of the potentialities of dynamically modified block incomplete factorizations, SIAM J. Sci. Comput., 19 (1998), pp. 1083-1108.
26. M. MAGOLU MONGA-MADE AND Y. NOTAY, On the conditioning analysis of block approximate factorization methods, Linear Algebra Appl., 154/156 (1991), pp. 583-599.
27. M. MAGOLU MONGA-MADE AND Y. NOTAY, Performance of Block Incomplete Factorization Methods for 3D Problems, Technical Report. Service de Milieux Continus, Université Libre de Bruxelles, Brussels, Belgium, 1998.
28. M. MAGOLU MONGA-MADE AND B. POLMAN, Theoretical Comparison of Pointwise, Linewise and Planewise Preconditioners for Solving Three-Dimensional Problems, Report 9609, Department of Mathematics, University of Nijmegen, The Netherlands, 1996.
29. M. MAGOLU MONGA-MADE AND B. POLMAN, Efficient planewise like preconditioners to cope with 3D Problems, Numer. Linear Algebra Appl., 6 (1999), pp. 379-406.
30. S. NAKAMURA, Computational Methods in Engineering and Science, Wiley, New York, 1977.
31. Y. NOTAY, Solving positive (semi)definite linear systems by preconditioned iterative methods, in Preconditioned Conjugate Gradient Methods, O. Axelsson and L. Kolotilina, eds., Lecture Notes in Mathematics 1457, Springer-Verlag, Berlin, 1990, pp. 105-125.
32. Y. NOTAY, Conditioning analysis of modified block incomplete factorizations, Linear Algebra Appl., 154/156 (1991), pp. 711-722.
33. Y. NOTAY, On the convergence rate of the conjugate gradients in the presence of rounding errors, Numer. Math., 65 (1993), pp. 301-317.
34. Y. NOTAY, DRIC: A dynamic version of the RIC method, Numer. Linear Algebra Appl., 1 (1994), pp. 511-532.
35. H. L. STONE, Iterative solution of implicit approximation of multidimensional partial differential equations, SIAM J. Numer. Anal., 5 (1968). pp. 530-558.
36. A. VAN DER SLUIS, The convergence behaviour of conjugate gradients and ritz values in various circumstances, in Iterative Methods in Linear Algebra, R. Beauwens and P. de Groen, eds., North-Holland, Amsterdam, 1992, pp. 49-66.
37. A. VAN DER SLUIS AND H. A. VAN DER VORST, The rate of convergence of conjugate gradients, Numer. Math., 48 (1986), pp. 543-560.
38. H. A. VAN DER VORST, ICCG and related methods for 3D problems on vector computers, Comput. Phys. Comm., 53 (1989), pp. 223-235.
39. H. A. VAN DER VORST, The convergence behaviour of preconditioned CG and CG-S, in Preconditioned Conjugate Gradient Methods, O. Axelsson and L. Kolotilina, eds., Lecture Notes in Math. 1457, Springer-Verlag, Berlin, 1990, pp. 126-136.
40. E. L. WACHSPRESS, Iterative Solution of Elliptic Systems and Application to the Neutron Diffusion Equations of Reactor Physics, Prentice Hall, Englewood Cliffs, NJ, 1966.