volume | PIER Journals

Abstract

In this paper, we study a class of modified incomplete Cholesky factorization preconditioners LL^T with two control parameters including dropping rules. Before computing preconditioners, the modified incomplete Cholesky factorization algorithm allows to decide the sparsity of incomplete factorization preconditioners by two fillin control parameters: (1) p, the number of the largest number p of nonzero entries in each row; (2) dropping tolerance. With RCM reordering scheme as a crucial operation for incomplete factorization preconditioners, our numerical results show that both the number of PCOCG and PCG iterations and the total computing time are reduced evidently for appropriate fill-in control parameters. Numerical tests on harmonic analysis for 2D and 3D scattering problems show the efficiency of our method.

1. Jin, J. M., The Finite Element Method in Electromagnetics, Wiley, 1993.

2. Volakis, J. L., A. Chatterjee, and L. C. Kempel, Finite Element Method for Electromagnetics: Antennas, Microwave Circuits and Scattering Applications, IEEE Press, 1998.

3. Ahmed, S. and Q. A. Naqvi, "Electromagnetic scattering of two or more incident plane waves by a perfect, electromagnetic two or more incident plane waves by a perfect, electromagnetic," Progress In Electromagnetics Research B, Vol. 10, 75-90, 2008.
doi:10.2528/PIERB08083101

4. Fan, Z. H., D. Z. Ding, and R. S. Chen, "The efficient analysis of electromagnetic scattering from composite structures using hybrid CFIE-IEFIE," Progress In Electromagnetics Research B, Vol. 10, 131-143, 2008.
doi:10.2528/PIERB08091606

5. Botha, M. M. and D. B. Davidson, "Rigorous auxiliary variable-based implementation of a second-order ABC for the vector FEM," IEEE Trans. Antennas Propagat., Vol. 54, 3499-3504, 2006.
doi:10.1109/TAP.2006.884300

6. Harrington, R. F., Field Computation by Moment Method, 2nd edition, IEEE Press, 1993.

7. Choi, S. H., D. W. Seo, and N. H. Myung, "Scattering analysis of open-ended cavity with inner object," J. of Electromagn. Waves and Appl., Vol. 21, No. 12, 1689-1702, 2007.

8. Ruppin, R., "Scattering of electromagnetic radiation by a perfect electromagnetic conductor sphere," J. of Electromagn. Waves and Appl., Vol. 20, No. 12, 1569-1576, 2006.
doi:10.1163/156939306779292390

9. Ho, M., "Scattering of electromagnetic waves from, vibrating perfect surfaces: Simulation using relativistic boundary conditions," J. of Electromagn. Waves and Appl., Vol. 20, No. 4, 425-433, 2006.
doi:10.1163/156939306776117108

10. Lin, C. J. and J. J. More, "Incomplete cholesky factorizations with limited memory," SIAM J. Sci. Comput., Vol. 21, 24-45, 1999.
doi:10.1137/S1064827597327334

11. Fang, H. R. and P. O. Dianne, "Leary, modified Cholesky algorithms: A catalog with new approaches," Mathematical Programming, July 2007.

12. Margenov, S. and P. Popov, "MIC(0) DD preconditioning of FEM elasticity problem on non-structured meshes," Proceedings of ALGORITMY 2000 Conference on Scientific Computing, 245-253, 2000.

13. Saad, Y., "ILUT: A dual threshold incomplete LU factorization," Numer. Linear Algebra Appl., Vol. 4, 387-402, 1994.
doi:10.1002/nla.1680010405

14. Freund, R. and N. Nachtigal, "A quasi-minimal residual method for non-Hermitian linear systems," Numer. Math., Vol. 60, 315-339, 1991.
doi:10.1007/BF01385726

15. Van der Vorst, H. A. and J. B. M. Melissen, "A Petrov-Galerkin type method for solving Ax = b, where A is symmetric complex," IEEE Trans. Mag., Vol. 26, No. 2, 706-708, 1990.
doi:10.1109/20.106415

16. Barrett, R., 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, 2nd edition, SIAM, 1994.

17. Saad, Y., Iterative Methods for Sparse Linear Systems, 2nd edition, SIAM, 2003.

18. Saad, Y., "Sparskit: A basic tool kit for sparse matrix computations,", Report RIACS-90-20, Research Institute for Advanced Computer Science, NASA Ames Research Center, Moffett Field, CA, 1990.

19. Benzi, M., "Preconditioning techniques for large linear systems: A survey," J. Comp. Physics, Vol. 182, 418-477, 2002.
doi:10.1006/jcph.2002.7176

20. Meijerink, J. A. and H. A. van der Vorst, "An iterative solution method for linear equations systems of which the coefficient matrix is a symmetric M-matrix," Math. Comp., Vol. 31, 148-162, 1977.
doi:10.2307/2005786

21. Lee, I., P. Raghavan, and E. G. Ng, "Effective preconditioning through ordering interleaved with incomplete factorization," Siam J. Matrix Anal. Appl., Vol. 27, 1069-1088, 2006.
doi:10.1137/040618357

22. Ng, E. G. and P. Raghavan, "Performance of greedy ordering heuristics for sparse Cholesky factorization," Siam J. Matrix Anal. Appl., Vol. 20, No. 2, 902-914, 1999.
doi:10.1137/S0895479897319313

23. Benzi, M., D. B. Szyld, and A. van Duin, "Orderings for incomplete factorization preconditioning of nonsymmetric problems," SIAM J. Sci. Comput., Vol. 20, 1652-1670, 1999.
doi:10.1137/S1064827597326845

24. Benzi, M., W. Joubert, and G. Mateescu, "Numerical experiments with parallel orderings for ILU preconditioners," Electronic Transactions on Numerical Analysis, Vol. 8, 88-114, 1999.

25. Chan, T. C. and H. A. van der Vorst, "Approximate and incomplete factorizations,", Preprint 871, Department of Mathematics, University of Utrecht, The Netherlands, 1994.

26. Zhang, Y., T. Z. Huang, and X. P. Liu, "Modified iterative methods for nonnegative matrices and M-matrices linear systems," Computers and Mathematics with Applications, Vol. 50, 1587-1602, 2005.
doi:10.1016/j.camwa.2005.07.005

Professor Tingzhu Huang

Dr. Yong Zhang

Professor Liang Li

Prof. Wei Shao

Dr. Sheng-Jian Lai