Sparse approximate inverse preconditioners for electromagnetic surface scattering simulations

Simulation of electromagnetic waves scattered by a connected three dimensional non-convex obstacle at medium frequencies (where the size of the obstacle is 10 to 100 times the incident wavelength) requires a non-asymptotic approach. Standard boundary element schemes at such frequencies require millions of unknowns. However, recently developed high-order algorithms require only tens of thousands of unknowns at medium frequencies for a class of three dimensional obstacles. At such frequencies we use a sparse approximation to the scattering matrix and so iterative solvers are required. We describe an efficient scheme to solve the associated linear systems using sparse approximate inverse preconditioners. The sparse preconditioners developed in this work facilitate efficient solutions of complex dense linear systems arising in electromagnetic scattering simulations. References B. Carpentieri, I. S. Duff, and L. Giraud. Sparse pattern selection strategies for robust frobenius-norm minimization preconditioners in electromagnetism. Numer. Linear Algebra Appl., 7:667--685, 2000. doi:10.1002/1099-1506(200010/12)7:7/8<667::AID-NLA218>3.0.CO;2-X. B. Carpentieri, I. S. Duff, L. Giraud, and G. Sylvand. Combining fast multipole techniques and an approximate inverse preconditioner for large electromagnetism calculations. SIAM J. Sci. Comput., 27:774--792, 2005. doi:10.1137/040603917. E. Chow. A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM J. Sci. Comput., 21:1804--1822, 2000. doi:10.1137/S106482759833913X. D. Colton and R. Kress. Integral Equation Methods in Scattering Theory. Wiley, 1983. M. Ganesh and S. C. Hawkins. A spectrally accurate algorithm for electromagnetic scattering in three dimensions. Numerical Algorithms, 43:25--60, 2006. doi:10.1007/s11075-006-9033-7. M. Ganesh and S. C. Hawkins. A hybrid high-order algorithm for radar cross section computations. SIAM J. Sci. Comput., 29:1217--1243, 2007. doi:10.1137/060664859. M. Ganesh and S. C. Hawkins. Sparse preconditioners for dense complex linear systems arising in some radar cross section computations. ANZIAM J., 48:C233--C248, 2007. http://anziamj.austms.org.au/ojs/index.php/ANZIAMJ/article/view/61. L. Y. Kolotolina. Explicit preconditioning of systems of linear algebraic equations. J. Sov. Math., 43:2566--2573, 1988. R. B. Melrose and M. E. Taylor. Near peak scattering and the corrected kirchhoff approximation for a convex obstacle. Adv. in Math., 55:242--315, 1985. doi:10.1016/0001-8708(85)90093-3. Y. Saad and M. H. Schultz. GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput., 7(3):856--869, July 1986. doi:10.1137/0907058. G. Alleon, M. Benzi, and L. Giraud. Sparse approximate inverse preconditioning for dense linear systems arising in computational electromagnetics. Numerical Algorithms, 16:1--15, 1997. doi:10.1023/A:1019170609950. J. M. Song, C. C. Lu, W. C. Chew, and S. W. Lee. Fast Illinois solver code (FISC). IEEE Antennas Propag. Mag., 40:27--34, 1998. doi:10.1109/74.706067. E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen. LAPACK Users' Guide. Society for Industrial and Applied Mathematics, Philadelphia, PA, third edition, 1999. B. Carpentieri. Fast iterative solution methods in electromagnetic scattering. Technical Report 17/2006, University of Graz, 2006. http://www.uni-graz.at/~carpenti/papers/IMA17-06.pdf. B. Carpentieri, I. S. Duff, and L. Giraud. Robust preconditioning of dense problems from electromagnetics. In Numer. Anal. and App. Lecture Notes in Computer Science 1988, pages 170--178. Springer, 2000. http://www.uni-graz.at/~carpenti/papers/rousse.pdf