volume | PIER Journals

Abstract

An effective wavelet based multigrid preconditioned conjugate gradient method is developed to solve electromagnetic large matrix problem for millimeter wave scattering application. By using wavelet transformation we restrict the large matrix equation to a relative smaller matrix and which can be solved rapidly. The solution is prolonged as the new improvement for the conjugate gradient (CG) method. Numerical result shows that our developed wavelet based multigrid preconditioned CG method can reach large improvement of computational complexity. Due to the automaticity of wavelet transformation, this method is potential to be a block box solver without physical background.

1. Coifman, R., V. Rohklin, and S. Wandzura, "The fast multipole method for the wave equation: A pedestrian description," IEEE Antennas Propagat Mag., Vol. 35, 7-12, 1993.
doi:10.1109/74.250128 Google Scholar

2. Song, J., C. C. Lu, and W. C. Chew, "Multilevel fast multipole algorithm for electromagnetic scattering by large complex objects," IEEE Trans Antennas Propagat., Vol. 45, 1488-1493, 1997.
doi:10.1109/8.633855 Google Scholar

3. Michelsen, E. and A. Boag, "A multilevel matrix decomposition algorithm for analyzing scattering from large structures," IEEE Trans Antennas Propagat., Vol. 44, 1086-1093, 1996.
doi:10.1109/8.511816 Google Scholar

4. Canning, F. X., "Improved impedance matrix localization method," IEEE Trans Antennas Propagat., Vol. 41, 659-667, 1993.
doi:10.1109/8.222285 Google Scholar

5. Song, J. M., C. C. Lu, W. C. Chew, and S. W. Lee, "Fast Illinois solver code (FISC)," IEEE Antennas Propagat Mag., Vol. 40, 27-34, 1998.
doi:10.1109/74.706067 Google Scholar

6. Sarkar, T. K. and E. Arvas, "On a class of finite step iterative methods (conjugate directions) for the solution of an operator equation arising in electromagnetics," IEEE Trans. Antennas and Propagat., Vol. 33, No. 10, 1058-1066, Oct. 1985.
doi:10.1109/TAP.1985.1143493 Google Scholar

7. Saad, Y., "Iterative methods for sparse linear systems,", PWS Publishing Company, Boston, 1995. Google Scholar

8. Chen, R. S., E. K. N. Yung, C. H. Chan, and D. G. Fang, "Application of preconditioned CG-FFT technique to method of lines for analysis of the infinite-plane metallic grating," Microwave and Optical Technology Letters, Vol. 24, No. 3, 170-175, Feb. 5, 2000.
doi:10.1002/(SICI)1098-2760(20000205)24:3<170::AID-MOP8>3.0.CO;2-S Google Scholar

9. Axelsson, O. and L. Yu. Kolotilina, "Preconditioned conjugate gradient methods,", Proceedings 1989, in Lecture Notes in Mathematics, Vol. 1457, Edited by A. Dold, B. Eckmann, and F. Takens, Springer-Verlag, 1990. Google Scholar

10. Kershaw, D. S., "The incomplete Cholesky-conjugate gradient method for the solution of systems of linear equations," J. Comput. Phys., Vol. 26, 43-65, 1978.
doi:10.1016/0021-9991(78)90098-0 Google Scholar

11. Dupont, T., R. P. Kendall, and H. H. Rachford, "An approximate factorization procedure for solving self-adjoint elliptic difference equations," SIAM J., Numer. Anal., Vol. 5, No. 3, 559-573, 1968.
doi:10.1137/0705045 Google Scholar

12. Ahn, C. H., W. C. Chew, J. S. Zhao, and E. Michielssen, "Numerical study of approximate inverse preconditioner for two-dimensional engine inlet problems," Electromagnetics, Journal of Electromagn. Waves Appl., Vol. 19, No. 1, 131-146, 1999. Google Scholar

13. Canning, F. X., "Diagonal preconditioners for the EFIE using a wavelet basis," IEEE Trans. Antennas Propagat., Vol. 44, 1239-1246, 1996.
doi:10.1109/8.535382 Google Scholar

14. Yaghjian, A. D., "Banded matrix preconditioning for electric-field integral equations," IEEE APS Int. Symp. Dig., 1806-1809, Montreal, Canada, 1997. Google Scholar

15. Tsang, L., C. H. Chan, H. Sangani, A. Ishimaru, and P. Phu, "A banded matrix iterative approach to monte carlo simulations of large-scale random rough-surface scattering," Journal of Electromagn. Waves Appl., Vol. 7, No. 9, 1185-1200, 1993.
doi:10.1163/156939393X00200 Google Scholar

16. Wei, C., N. Inagaki, and W. Di, "Dimension-descent technique for electromagnetic problems," IEE Proc. Microw. Antennas Propag., Vol. 144, No. 5, 372-376, Oct. 1997.
doi:10.1049/ip-map:19971336 Google Scholar

17. Ooms, S. and D. De Zutter, "A new iterative diakoptic-based multilevel moments method for planar circuits," IEEE Trans. Microw. Theory and Tech., Vol. 46, No. 3, 280-291, Mar. 1998.
doi:10.1109/22.661716 Google Scholar

18. Briggs, W. L. and V. E. Henson, "Wavelets and multigrid," SIAM J. Sci. Comput., Vol. 14, No. 2, 506-510, March 1993.
doi:10.1137/0914031 Google Scholar

19. Xiang, Z. and Y. Lu, "An effective wavelet matrix transform approach for efficient solutions of electromagnetic integral equations," IEEE Trans. Antennas Propagat., Vol. 45, 1205-1213, 1997.
doi:10.1109/8.611238 Google Scholar

20. Wang, G., R. W. Dutton, and J. Hou, "A fast wavelet multigrid algorithm for solution of electromagnetic integral equations," Microwave and Optical Technology Letters, Vol. 24, No. 2, 86-91, Jan. 2000.
doi:10.1002/(SICI)1098-2760(20000120)24:2<86::AID-MOP3>3.0.CO;2-B Google Scholar

21. Mallat, S., "A theory for multi-resolution signal decomposition: The wavelet transform," IEEE Trans. Pattern and Mech. Intel., Vol. 11, No. 7, 674-693, 1989.
doi:10.1109/34.192463 Google Scholar

22. Daubechies, I., "Orthonormal bases of compactly supported wavelets," Commun. Pure and Appl. Math., Vol. 41, 909-996, 1988.
doi:10.1002/cpa.3160410705 Google Scholar

23. Beylkin, G., R. Coifman, and V. Rokhlin, "Fast wavelet transforms and numerical algorithms I," Commun. Pure Appl. Math., Vol. 44, 141-183, 1991.
doi:10.1002/cpa.3160440202 Google Scholar

24. Uchida, K., T. Noda, and T. Matsumaga, "Spectral domain analysis of electromagnetics wave scattering by infinite plane metallic grating," IEEE Trans. Antennas and Propagat., Vol. 35, No. 1, 46-52, 1987.
doi:10.1109/TAP.1987.1143973 Google Scholar

Dr. R. S. Chen

Dr. D. G. Fang

Dr. K. F. Tsang

Dr. E. K. N. Yung