Abstract
In this article authors present a new method to construct low-rank approximations of dense huge-size matrices. The method develops mosaic-skeleton method and belongs to kernel-independent methods. In distinction from a mosaic-skeleton method, the new one utilizes the hierarchical structure of matrix not only to define matrix block structure but also to calculate factors of low-rank matrix representation. The new method was applied to numerical calculation of boundary integral equations that appear from 3D problem of scattering monochromatic electromagnetic wave by ideal-conducting bodies. The solution of model problem is presented as an example of method evaluation.
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R. Coifman, V. Rokhlin and S. Wanzura “The fast multipole method for the wave equation: a pedestrian prescription,” IEEE Antennas Propag. Mag. 35 (3), 7–12 (1993).
J. M. Song and W. C. Chew, “Multilevel fast multipole algorithm for solving combined field integral equations of electromagnetic scattering,” Microw. Opt. Technol. Lett. 10, 14–19 (1995).
E. Tyrtyshnikov, “Mosaic-skeleton approximations,” Calcolo 33, 47–57 (1996).
S. L. Stavtsev, “Block LU preconditioner for the electric field integral equation,” in Proceedings of Progress in Electromagnetics Research Symposium, 2015, pp. 1523–1527.
A. Aparinov, A. Setukha, and S. Stavtsev, “Supercomputer modelling of electromagnetic wave scattering with boundary integral equation method,” Commun. Comput. Inform. Sci. 793, 325–336 (2017).
W. Hackbusch, Hierarchical Matrices: Algorithms and Analysis, Springer Series in Computational Mathematics (Springer, Berlin, Heidelberg, 2015).
A. Yu. Mikhalev and I. V. Oseledets, “Iterative representing set selection for nested cross approximation,” Numer. Linear Algebra Appl. 23, 230–248 (2016).
S. L. Stavtsev, “H 2 matrix and integral equation for electromagnetic scattering by a perfectly conducting object,” in Integral Methods in Science and Engineering, Ed. by by C. Constanda, Dalla M. Riva, P. Lamberti, and P. Musolino (Springer, Birkhauser, Cham, 2017), vol. 2, pp. 255–264. https://doi.org/10.1007/978-3-319-59387-6_25
E. V. Zakharov, G. V. Ryzhakov, and A. V. Setukha, “Numerical solution of 3D problems of electromagnetic wave diffraction on a system of ideally conducting surfaces by the method of hypersingular integral equations,” Difer. Equat. 50, 1240–1251 (2014).
A. Setukha and S. Fetisov, “The method of relocation of boundary condition for the problem of electromagnetic wave scattering by perfectly conducting thin objects,” J. Comput. Phys. 373, 631–647 (2018).
N. L. Zamarashkin and A. I. Osinsky, “New accuracy estimates for pseudoskeleton approximations of matrices,” Dokl. Math. 94, 643–645 (2016).
A. I. Osinsky and N. L. Zamarashkin, “Pseudo-skeleton approximations with better accuracy estimates,” Linear Algebra Appl. 537, 221–249 (2018).
S. A. Goreinov, I. V. Oseledets, D. V. Savostyanov, E. E. Tyrtyshnikov, and N. L. Zamarashkin “How to find a good submatrix,” in Matrix Methods: Theory, Algorithms and Applications, 2010, pp. 247–256.
A. Aparinov, A. Setukha, and S. Stavtsev, “Parallel implementation for some applications of integral equations method,” Lobachevskii J. Math. 39, 477–485 (2018).
F. Saez de Adana, O. Gutierrez, I. Gonzalez, M. F. Catedra, and L. Lozano, Practical Applications of Asymptotic Techniques in Electromagnetics (Artech House, Boston, London, 2011).
Acknowledgments
The research is carried out using the equipment of the shared research facilities of HPC computing resources at Lomonosov Moscow State University.
Funding
This work was supported by the Russian Science Foundation, grant no. 19-11-00338.
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Submitted by E. E. Tyrtyshnikov
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Aparinov, A.A., Setukha, A.V. & Stavtsev, S.L. Low Rank Methods of Approximation in an Electromagnetic Problem. Lobachevskii J Math 40, 1771–1780 (2019). https://doi.org/10.1134/S1995080219110064
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DOI: https://doi.org/10.1134/S1995080219110064