volume | PIER Journals

Abstract

This letter shows 50 percent memory saving for a regular Hierarchal Matrix (H-matrix) by converting it to symmetric H-matrix for large electrodynamic problems. Only the upper diagonal near-field and compressed far-field matrix blocks of the H-matrix are stored. Far-field memory saving is achieved by computing and keeping the upper diagonal far-field blocks leading to compressed column block U and row block V at a level. Due to symmetry, the lower diagonal far-field H-matrix compressed column is the transpose of V, and the compressed row block is the transpose of U. Storage and computation of lower diagonal blocks are not required. Similarly, in the case of near-field, only the upper diagonal near-field blocks are computed and stored. Numerical results show that the proposed memory reduction procedure retains the accuracy and cost of regular H-matrix.

1. Harrington, R. F., Field Computation by Moment Methods, Wiley-IEEE Press, New York, 1993.
doi:10.1109/9780470544631

2. Gibson, W. C., The Method of Moments in Electromagnetics, CRC Press, 2014.
doi:10.1007/PL00005410

3. Chew, W. C., J. M. Jin, E. Michielssen, and J. Song, Fast Efficient Algorithms in Computational Electromagnetics, Artech House, Boston, London, 2001.
doi:10.1109/20.996112

4. Bebendorf, M., "Approximation of boundary element matrices," Numerische Mathematik, Vol. 86, No. 4, 565-589, Jun. 2000.
doi:10.1007/s006070050015

5. Kurz, S., O. Rain, and S. Rjasanow, "The adaptive cross-approximation technique for the 3-D boundary element method," IEEE Transactions on Magnetics, Vol. 38, No. 2, 421-424, Mar. 2002.
doi:10.1109/20.996112

6. Hackbusch, W., "A sparse matrix arithmetic based on H-matrices. Part I: Introduction to Hmatrices," Computing, Vol. 62, No. 2, 89-108, 1999.
doi:10.1007/s006070050015

7. Hackbusch, W. and B. N. Khoromskij, "A sparse H-matrix arithmetic. Part II: Application to multi-dimensional problems," Computing, Vol. 64, 21-47, 2000.
doi:10.1049/iet-map.2009.0229

8. Borm, S., L. Grasedyck, and W. Hackbusch, "Hierarchical matrices," Lecture Notes, 21, 2003.
doi:10.1109/TAP.1982.1142818

9. Chai, W. and D. Jiao, "H and H² matrix-based fast integral-equation solvers for large-scale electromagnetic analysis," IET Microwaves, Antennas and Propagation, No. 10, 1583-1596, 2010.
doi:10.1109/TAP.1965.1138406

10. Rao, S. M., D. R. Wilton, and A. W. Glisson, "Electromagnetic scattering by surfaces of arbitrary shape," IEEE Transactions on Antennas and Propagation, Vol. 30, No. 3, 409-418, May 1982.
doi:10.1109/TAP.2013.2292079

11. Andreasen, M., "Scattering from bodies of revolution," IEEE Transactions on Antennas and Propagation, Vol. 13, No. 2, 303-310, 1965.
doi:10.1016/j.laa.2013.03.001

12. Su, T., D. Ding, Z. Fan, and R. Chen, "Efficient analysis of EM scattering from bodies of revolution via the ACA," IEEE Transactions on Antennas and Propagation, Vol. 62, No. 2, 983-985, 2013.
doi:10.1109/MCSE.1998.7102081

13. Benner, P. and T. Mach, "The LR Cholesky algorithm for symmetric hierarchical matrices," Linear Algebra and Its Applications, Vol. 439, No. 4, 1150-1166, 2013.
doi:10.1216/JIE-2009-21-3-331

14. Kapur, S. and D. E. Long, "IES3: Efficient electrostatic and electromagnetic solution," IEEE Computer Science and Engineering, Vol. 5, No. 4, 60-67, Oct.–Dec. 1998.
doi:10.1109/MCSE.1998.7102081

15. Bebendorf, M. and S. Kunis, "Recompression techniques for Adaptive Cross Approximation," Journal of Integral Equations and Applications, Vol. 21, No. 3, 331-357, 2009.
doi:10.1049/iet-map.2020.0292

16. Negi, Y. K., V. P. Padhy, and N. Balakrishnan, "Re-compressed H-matrices for fast electric field integral equation," IEEE-International Conference on Computational Electromagnetics (ICCEM 2020), Singapore, Aug. 24–26, 2020.

17. Negi, Y. K., N. Balakrishnan, and S. M. Rao, "Symmetric near-field Schur's complement preconditioner for hierarchal electric field integral equation solver," IET Microwaves, Antennas and Propagation, Vol. 14, No. 14, 1846-1856, Aug. 2020.
doi:10.1049/iet-map.2020.0292

Dr. Yoginder Kumar Negi