A procedure is reported to determine accurate, invertible, block-diagonal factorizations for matrices obtained by discretizing integral equation formulations of electromagnetic interaction problems. The algorithm is based on the combination of localizing source/receiver transformations with orthogonally matched receiver/source transformations. The resulting factorization provides a single, sparse data structure for the system matrix and its inverse, and no approximation is required to convert between the two. Numerical examples illustrate the performance of the factorization for electromagnetic scattering from perfectly conducting elliptical cylinders of different electrical size.
2. Shaeffer, J., "Direct solve of electrically large integral equations for problems sizes to 1M unknowns," IEEE Transactions on Antennas and Propagation, Vol. 56, 2306-2313, 2008.
3. Jiao, D. and W. Chai, "An H2-matrix-based integral-equation solver of reduced complexity and controlled accuracy for solving electrodynamic problems," IEEE Transactions on Antennas and Propagation, Vol. 57, 3147-3159, 2009.
4. Heldring, A., J. M. Rius, J. M. Tamayo, J. Parron, and E. Ubeda, "Multiscale compressed block decomposition for fast direct solution of method of moments linear system," IEEE Transactions on Antennas and Propagation, Vol. 59, 526-536, 2011.
5. Wei, J.-G., Z. Peng, and J. F. Lee, "A fast direct matrix solver for surface integral equation methods for electromagnetic wave scattering from non-penetrable targets," Radio Science, Vol. 47, 1-9, 2012.
6. Brick, Y., V. Lomakin, and A. Boag, "Fast direct solver for essentially convex scatterers using multilevel non-uniform grids," IEEE Transactions on Antennas and Propagation, Vol. 62, 4314-4324, 2015.
7. Zhang, Y. L., X. Chen, C. Fei, Z. Li, and C. Gu, "Fast direct solution of composite conducting-dielectric arrays using Sherman-Morrison-Woodbury algorithm," Progress In Electromagnetics Research M, Vol. 49, 203-209, 2016.
8. Martinsson, P. G. and V. Rokhlin, "A fast direct solver for boundary integral equations in two dimensions," Journal of Computational Physics, Vol. 205, 1-23, 2005.
9. Adams, R. J., A. Zhu, and F. X. Canning, "Efficient solution of integral equations in a localizing basis," Journal of Electromagnetic Waves and Applications, Vol. 19, No. 12, 1583-1594, 2005.
10. Zhu, A., R. J. Adams, F. X. Canning, and S. D. Gedney, "Schur factorization of the impedance matrix in a localizing basis," Journal of Electromagnetic Waves and Applications, Vol. 20, No. 3, 351-362, 2006.
11. Adams, R. J., A. Zhu, and F. X. Canning, "Sparse factorization of the TMz impedance matrix in an overlapped localizing basis," Progress In Electromagnetics Research, Vol. 61, 291-322, 2006.
12. Adams, R. J., Y. Xu, X. Xu, J. S. Choi, S. D. Gedney, and F. X. Canning, "Modular fast direct electromagnetic analysis using local-global solution modes," IEEE Transactions on Antennas and Propagation, Vol. 56, 2427-2441, Aug. 2008.
13. Xu, X. and R. J. Adams, "Sparse matrix factorization using overlapped localizing LOGOS modes on a shifted grid," IEEE Transactions on Antennas and Propagation, Vol. 60, 1414-1424, 2012.
14. Adams, R. J. and J. C. Young, "A diagonal factorization for integral equation matrices," 2016 International Conference on Electromagnetics in Advanced Applications (ICEAA), 812-815, September 19-23, 2016.
15. Peterson, A. F., S. L. Ray, and R. Mittra, Computational Methods for Electromagnetics, IEEE Press, Piscataway, NJ, 1998.
16. Hackbusch, W. and S. Borm, "H2-matrix approximation of integral operators by interpolation," Applied Numerical Mathematics, Vol. 43, 129-143, 2002.
17. Xu, Y., X. Xu, and R. J. Adams, "A sparse factorization for fast computation of localizing modes," IEEE Transactions on Antennas and Propagation, Vol. 58, 3044-3049, 2010.
18. Qian, Z. G. and W. C. Chew, "An augmented electric field integral equation for highspeed interconnect analysis," Microwave and Optical Technology Letters, Vol. 50, 2658-2662, October 2008.
19. Qian, Z. G. and W. C. Chew, "Enhanced A-EFIE with perturbation method," IEEE Transactions on Antennas and Propagation, Vol. 58, 3256-3264, October 2010.
20. Cheng, J. and R. J. Adams, "Direct solution method using overlapped localizing LOGOS modes for AEFIE-G at low frequencies," 28th Annual Review of Progress in Applied Computational Electromagnetics, 579-584, Columbus, Ohio, 2012.