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Preconditioners for large dense matrices in a low-rank format

  • Stanislav L. Stavtsev EMAIL logo
Published/Copyright: February 14, 2025

Abstract

In this paper, we apply low rank approximations to a large dense matrix in the solution of an integral equation in the problems of electromagnetic wave diffraction on a perfectly conductive object. In the case of a large wave number, the matrix of the system becomes ill-conditioned, therefore, we propose to use a preconditioner to solve the system with such a matrix by an iterative method. Based on block-sparse matrices, two types of preconditioners are constructed in this paper. Low-rank approximations are used to store the factors of the LU decomposition of the preconditioner matrix. The preconditioners make it possible to obtain a solution to the diffraction problem on a perfectly conductive object of 120 wavelengths.

MSC 2010: 65F08; 65F55; 65F50

Funding statement: The work was supported by the Moscow Center of Fundamental and Applied Mathematics at INM RAS (Agreement with the Ministry of Education and Science of the Russian Federation No. 075–15–2022–286).

References

[1] P. R. Amestoy, I. S. Duff, and J. Y. L’Excellent, A fully asynchronous multifrontal solver using distributed dynamic scheduling. SIAM J. Matrix Anal. Appl. 23 (2001), No. 1, 15–41.10.1137/S0895479899358194Search in Google Scholar

[2] P. R. Amestoy, I. S. Duff, and C. Vömel, Task scheduling in an asynchronous distributed memory multifrontal solver. SIAM J. Matrix Anal. Appl. 26 (2004), No. 2, 544–565.10.1137/S0895479802419877Search in Google Scholar

[3] A. Aparinov, A.Setukha, and S. Stavtsev, Supercomputer modelling of electromagnetic wave scattering with boundary integral equation method. In: 3rd Russian Supercomputing Days (RuSCDays), Communications in Computer and Information Science, vol. 793 (Eds. V. Voevodinand S. Sobolev). Springer Verlag, 2017, pp. 325–336.10.1007/978-3-319-71255-0_26Search in Google Scholar

[4] A. A. Aparinov, A. V. Setukha, and S. L. Stavtsev, Parallel implementation for some applications of integral equations method. Lobachevskii J. Math. 39 (2018), No. 2, 477–485.10.1134/S1995080218040029Search in Google Scholar

[5] A. A. Aparinov, A. V. Setukha, and S. L. Stavtsev, Low rank methods of approximation in an electromagnetic problem. Lobachevskii J. Math. 40 (2019), No. 11, 1771–1780.10.1134/S1995080219110064Search in Google Scholar

[6] M. Bebendorf, Hierarchical Matrices. Lecture Notes Comp. Sci. Eng., Vol. 63, 2008.Search in Google Scholar

[7] M. Bebendorf, Approximation of boundary element matrices. Numer. Math. 86 (2000), No. 4, 565–589.10.1007/PL00005410Search in Google Scholar

[8] E. Bleszynski, M. Bleszynski, and T. Jaroszewicz, AIM: Adaptive integral method for solving large-scale electromagnetic scattering and radiation problems. Radio Sci. 31, No. 5, 1225–1251.10.1029/96RS02504Search in Google Scholar

[9] S. Börm, Efficient numerical methods for non-local operators: H2-matrix compression, algorithms and analysis. European Math. Soc. 14 (2010).10.4171/091Search in Google Scholar

[10] K. Chen, Matrix Preconditioner Techniques and Applications. Cambrige, 2005.10.1017/CBO9780511543258Search in Google Scholar

[11] D. Colton and R. Kress, Integral Methods in Scattering Theory. John Willey & Sons, New York, 1983.Search in Google Scholar

[12] J. Demmel, J. Gilbert, and X. Li, An asynchronous parallel supernodal algorithm to sparse partial pivoting. SIAM J. Matrix Anal. Appl. 20 (1999), No. 4, 915–952.10.1137/S0895479897317685Search in Google Scholar

[13] Ö. Ergül and L. Gürel, The Multilevel Fast Multipole Algorithm (MLFMA) for Solving Large-Scale Computational Electromagnetics Problems. John Wiley & Sons, 2014.10.1002/9781118844977Search in Google Scholar

[14] A. Yu. Mikhalev and I. V. Oseledets, Iterative representing set selection for nested cross approximation. Numer. Linear Algebra Appl. 23 (2016), No. 2, 230–248.10.1002/nla.2021Search in Google Scholar

[15] S. Rao, D. Wilton, and A. Glisson, Electromagnetic scattering by surfaces of arbitrary shape. IEEE Transactions on Antennas and Propagation 30 (1982), No. 3, 409–418.10.1109/TAP.1982.1142818Search in Google Scholar

[16] O. Schenk and K. Gartner, PARDISO: A high performance serial and parallel sparse linear solver in semiconductor device simulation. Future Generation Comput. Syst. 789 (2001), No. 1, 1–9.10.1016/S0167-739X(00)00076-5Search in Google Scholar

[17] A. V. Setukha and A. V. Semenova, Numerical solution of a surface hypersingular integral equation by piecewise linear approximation and collocation methods. Comp. Math. Math. Phys. 59 (2019), No. 6. 942–957.10.1134/S0965542519060125Search in Google Scholar

[18] A. V. Setukha, S. L. Stavtsev, and R. M. Tret’yakova, Application of mosaic-skeleton approximations of matrices in the physical optics method for electromagnetic scattering problems. Comp. Math. Math. Phys. 62 (2022), No. 9, 1424–1437.10.1134/S0965542522090032Search in Google Scholar

[19] J. Song, C. C. Lu, and W. C. Chew, Multilevel fast multipole algorithm for electromagnetic scattering by large complex objects. IEEE Transactions on Antennas and Propagation 45 (1997), No. 10.10.1109/8.633855Search in Google Scholar

[20] S. Sukmanyuk, D. Zheltkov, and B. Valiakhmetov, Generalized minimal residual method for systems with multiple right-hand sides. arXiv preprint arXiv.2408.05513, 2024.Search in Google Scholar

[21] S. L. Stavtsev and E. E. Tyrtyshnikov, Application of mosaic-skeleton approximations for solving EFIE. In: Progress in Electromagnetics Research Symposium. Moscow, 2009, pp. 1752–1755.Search in Google Scholar

[22] S. L. Stavtsev, Low rank structures in solving electromagnetic problems. In: Large-Scale Scientific Computing: 12th Int. Conf., LSSC 2019, Sozopol, Bulgaria, June 10–14, 2019, Revised Selected Papers. 12, 2019, pp. 165–172.10.1007/978-3-030-41032-2_18Search in Google Scholar

[23] E. E. Tyrtyshnikov, Incomplete cross approximation in the mosaic skeleton method. Computing 64 (2000), No. 4, 367–380.10.1007/s006070070031Search in Google Scholar

[24] P. Zwamborn and P. M. van den Berg, The three dimensional weak form of the conjugate gradient FFT method for solving scattering problems. IEEE Trans. Antennas Propag. 40 (1992), 1757–1766.10.1109/22.156602Search in Google Scholar

Received: 2024-09-18
Accepted: 2024-11-25
Published Online: 2025-02-14
Published in Print: 2025-02-25

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