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Licensed Unlicensed Requires Authentication Published by De Gruyter April 2, 2021

Adaptive Directional Compression of High-Frequency Helmholtz Boundary Element Matrices

  • Steffen Börm ORCID logo EMAIL logo

Abstract

Boundary element methods for the high-frequency Helmholtz equation require efficient compression techniques for the resulting matrices. Directional interpolation converges exponentially and is very robust and fast, but high accuracies lead to very large storage requirements. This problem can be solved by combining interpolation with algebraic recompression techniques that significantly reduce the storage requirements while keeping the accuracy and robustness and only moderately increasing the runtime.

MSC 2010: 35J05; 65N38; 65F25

References

[1] L. Banjai and W. Hackbusch, Hierarchical matrix techniques for low- and high-frequency Helmholtz problems, IMA J. Numer. Anal. 28 (2008), no. 1, 46–79. 10.1093/imanum/drm001Search in Google Scholar

[2] M. Bebendorf, Approximation of boundary element matrices, Numer. Math. 86 (2000), no. 4, 565–589. 10.1007/PL00005410Search in Google Scholar

[3] M. Bebendorf, C. Kuske and R. Venn, Wideband nested cross approximation for Helmholtz problems, Numer. Math. 130 (2015), no. 1, 1–34. 10.1007/s00211-014-0656-7Search in Google Scholar

[4] M. Bebendorf and S. Rjasanow, Adaptive low-rank approximation of collocation matrices, Computing 70 (2003), no. 1, 1–24. 10.1007/s00607-002-1469-6Search in Google Scholar

[5] S. Börm, Directional 2 -matrix compression for high-frequency problems, Numer. Linear Algebra Appl. 24 (2017), no. 6, Article ID e2112. 10.1002/nla.2112Search in Google Scholar

[6] S. Börm and C. Börst, Hybrid matrix compression for high-frequency problems, SIAM J. Matrix Anal. Appl. 41 (2020), no. 4, 1704–1725. 10.1137/19M124280XSearch in Google Scholar

[7] S. Börm and J. M. Melenk, Approximation of the high-frequency Helmholtz kernel by nested directional interpolation: Error analysis, Numer. Math. 137 (2017), no. 1, 1–34. 10.1007/s00211-017-0873-ySearch in Google Scholar

[8] A. Brandt, Multilevel computations of integral transforms and particle interactions with oscillatory kernels, Comput. Phys. Comm. 65 (1991), no. 1–3, 24–38. 10.1016/0010-4655(91)90151-ASearch in Google Scholar

[9] B. Engquist and L. Ying, Fast directional multilevel algorithms for oscillatory kernels, SIAM J. Sci. Comput. 29 (2007), no. 4, 1710–1737. 10.1137/07068583XSearch in Google Scholar

[10] L. Greengard, J. Huang, V. Rokhlin and S. Wandzura, Accelerating fast multipole methods for the Helmholtz equation at low frequencies, IEEE Comp. Sci. Eng. 5 (1998), no. 3, 32–38. 10.1109/99.714591Search in Google Scholar

[11] W. Hackbusch, Hierarchical Matrices: Algorithms and Analysis, Springer, Heidelberg, 2015. 10.1007/978-3-662-47324-5Search in Google Scholar

[12] M. Messner, M. Schanz and E. Darve, Fast directional multilevel summation for oscillatory kernels based on Chebyshev interpolation, J. Comput. Phys. 231 (2012), no. 4, 1175–1196. 10.1016/j.jcp.2011.09.027Search in Google Scholar

[13] V. Rokhlin, Diagonal forms of translation operators for the Helmholtz equation in three dimensions, Appl. Comput. Harmon. Anal. 1 (1993), no. 1, 82–93. 10.21236/ADA248862Search in Google Scholar

[14] E. E. Tyrtyshnikov, Incomplete cross approximation in the mosaic-skeleton method, Computing 64 (2000), 367–380. 10.1007/s006070070031Search in Google Scholar

Received: 2020-05-14
Revised: 2020-11-01
Accepted: 2021-03-03
Published Online: 2021-04-02
Published in Print: 2021-07-01

© 2021 Walter de Gruyter GmbH, Berlin/Boston

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