Efficient Planewise-like Preconditioners for Solving 3D Problems

Numerical Linear Algebra with Applications

Volumn 6, Issue 5, 1999, Pages 379-406

  Magolu, Monga Made ^a   Polman, Ben ^b  

^a UNIVERSITÉ LIBRE DE BRUXELLES   (Belgium)

^b RADBOUD UNIVERSITY NIJMEGEN   (Netherlands)

Author keywords

Block incomplete factorizations; Discretized partial differential equations; Large sparse linear systems; Preconditioned conjugate gradient; Red black like orderings

Indexed keywords

EID: 0042582048     PISSN: 10705325     EISSN: None     Source Type: Journal    
DOI: 10.1002/(sici)1099-1506(199907/08)6:5<379::aid-nla170>3.0.co;2-p     Document Type: Article

References (43)

1. O. Axelsson. Iterative Solution Methods. Cambridge University Press, Cambridge, 1994.
2. O. Axelsson and V. Barker. Finite Element Solution of Boundary Value Problems. Theory and Computation. Academic Press, New York, 1984.
3. O. Axelsson and V. Eijkhout. Vectorizable preconditioned for elliptic difference equations in three space dimensions. J. Comput. Appl. Math., 27, 299-321, 1989.
4. O. Axelsson and G. Lindskog. On the eigenvalue distribution of a class of preconditioning methods. Numer. Math., 48, 479-498, 1986.
5. O. Axelsson and H. Lu. On eigenvalue estimates for block incomplete factorization methods. SIAM J. Matrix Anal. Appl., 16, 1074-1085, 1995.
6. O. Axelsson and H. Lu. A survey of some estimates of eigenvalues and condition numbers for certain preconditioned matrices. Report No. 9601, Departement of Mathematics, University of Nijmegen, The Netherlands, January 1996.
7. O. Axelsson, H. Lu and B. Polman. On the numerical radius of matrices and its application to iterative methods. Lin. Multilin. Alge., 37, 225-238, 1994.
8. 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.
9. R. Beauwens. Upper eigenvalue bounds for pencils of matrices. Lin. Alg. Appl., 62, 87-104, 1984.
10. R. Beauwens. Modified incomplete factorization strategies. In O. Axelsson and L. Kolotilina, editors, Preconditioned Conjugate Gradient Methods, pages 1-16. Lectures Notes in Mathematics, Springer-Verlag, Berlin, 1990.
11. R. Beauwens and M. Ben Bouzid. On sparse block factorization iterative methods. SIAM J. Numer. Anal., 24, 1066-1076, 1987.
12. R. Beauwens and M. Ben Bouzid. Existence and conditioning properties of sparse approximate block factorizations. SIAM J. Numer. Anal., 25, 941-956, 1988.
13. T. F. Chan and H.A. van der Vorst. Approximate and incomplete factorizations. In D. E. Keyes, A. Samed and V. Venkatakrishnan, editors, Parallel Numerical Algorithms, pages 167-202. ICASE/LaRC Interdisciplinary Series in Science and Engineering, Kluwer Academic, Dordrecht. 1997.
14. P. Concus, G. H. Golub, and G. A. Meurant. Block preconditioning for the conjugate gradient method. SIAM J. Sci. Statist. Comput., 6, 220-252, 1985.
15. I. S. Duff and G. A. Meurant. The effect of ordering on preconditioned conjugate gradients. BIT, 29, 635-657, 1989.
16. L. Dujacquier. Préconditionnement par factorisations blocs approchées. Mémoire, Université Libre de Bruxelles, Service de Métrologie Nucléaire (C.P. 165), Belgium, 1994.
17. V. Eijkhout. Beware of unperturbed modified incomplete factorizations. In R. Beauwens and P. de Groen, editors, Iterative Methods in Linear Algebra, pages 583-591. North-Holland, Amsterdam, 1992.
18. V. Eijkhout and B. Polman. Decay rates of inverses of banded M-matrices that are near to Toeplitz matrices. Lin. Alg. Appl., 109, 247-277, 1988.
19. G. H. Golub and C. F. van Loan. Matrix Computations, 3rd edn. The John Hopkins University Press, Baltimore, Maryland, 1996.
20. G. H. Golub and H. A. van der Vorst. Closer to the solution: iterative linear solvers. In I. S. Duff and G. A. Watson, editors, The State of the Art in Numerical Analysis, pages 63-92. Clarendon Press, Oxford, 1997.
21. L. Kolotilina. Eigenvalues bounds for block preconditioned matrices. Research Report EM-RR, Elegant Mathematics, Inc., June 1994.
22. A. B. Kutcherov and M. M. Makarov. An approximate factorization method for solving discrete elliptic problems on stretched domains. J. Numer. Lin. Alg. Appl., 1, 1-26, 1992.
23. H. Lu. Compensative block incomplete factorizations. Report 9610, Departement of Mathematics, University of Nijmegen, The Netherlands, March 1996.
24. H. Lu and O. Axelsson. Conditioning analysis of block incomplete factorization and its application to elliptic equations. Report 9504, Department of Mathematics, University of Nijmegen, The Netherlands, 1995.
25. M. M. Magolu. Modified block-approximate factorization strategies. Numer. Math., 61, 91-110, 1992.
26. M. M. Magolu. Analytical bounds for block approximate factorization methods. Lin. Alg. Appl., 179, 33-57, 1993.
27. M. M. Magolu. Ordering strategies for modified block incomplete factorizations. SIAM J. Sci. Comput., 16, 378-399, 1995.
28. M. M. Magolu. Implementation of parallel block preconditionings on a transputer-based multiprocessor. Fut. Gen. Comp. Sys., 11, 167-173, 1995.
29. M. M. Magolu. Taking advantage of the potentialities of dynamically modified block incomplete factorizations. SIAM J. Sci. Comput., 19, 1083-1108, 1998.
30. M. M. Magolu and Y. Notay. On the conditioning analysis of block approximate factorization methods. Lin. Alg. Appl., 154-156, 583-599, 1991.
31. M. M. Magolu and Y. Notay. Dynamically relaxed block incomplete factorizations. Preprint no. 985, Department of Mathematics, Utrecht University, 1996.
32. M. M. Magolu 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, March 1996.
33. S. Nakamura. Computational Methods in Engineering and Science. John Wiley & Sons, New York, 1977.
34. Y. Notay. Solving positive (semi)definite linear systems by preconditioned iterative methods. In O. Axelsson and L. Kolotilina, editors, Preconditioned Conjugate Gradient Methods, pages 105-125. Lectures Notes in Mathematics, Springer-Verlag, Berlin, 1990.
35. Y. Notay. Conditioning analysis of modified block incomplete factorizations. Lin. Alg. Appl., 154-156, 711-722, 1991.
36. Y. Notay. On the convergence rate of the conjugate gradients in the presence of rounding errors. Numer. Math., 65, 301-317, 1993.
37. Y. Notay. DRIC : A dynamic version of the RIC method. J. Numer. Lin. Alg. Appl., 1, 511-532, 1994.
38. Y. Notay. An efficient parallel discrete PDE solver. Parallel Comput., 21, 1725-1748, 1995.
39. Y. Notay and A. Van de Velde. Coarse grid acceleration of parallel incomplete preconditioners. In S. Margenov and P. Vassilevski, editors, Iterative Methods in Linear Algebra II, pages 106-130, Vol. 3 of IMACS series in Computational and Applied Mathematics, 1996.
40. H. A. van der Vorst. The convergence behavior of some iterative methods. In R. Gruber, J. Periaux, and R. Shaw, editors, Proceedings of the fifth International Symposium on Numerical Methods in Engineering, pages 61-72. Springer-Verlag, Berlin, 1989.
41. H.A. van der Vorst. ICCG and related methods for 3D problems on vector computers. Comp. Phys. Comm., 53, 223-235, 1989.
42. H.A. van der Vorst. The convergence behaviour of preconditioned CG and CG-S. In O. Axelsson and L. Kolotilina, editors, Preconditioned Conjugate Gradient Methods, pages 126-136. Lectures Notes in Mathematics, Springer-Verlag, Berlin, 1990.
43. E. L. Wachspress. Iterative Solution of Elliptic Systems and Application to the Neutron Diffusion Equations of Reactor Physics. Prentice Hall, Englewood Cliffs, NJ, 1966.