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A Numerical Study on the Compressibility of Subblocks of Schur Complement Matrices Obtained from Discretized Helmholtz Equations

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Numerical Analysis and Its Applications (NAA 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10187))

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Abstract

The compressibility of Schur complement matrices is the essential ingredient for \(\mathcal{H}\)-matrix techniques, and is well understood for Laplace type problems. The Helmholtz case is more difficult: there are several theoretical results which indicate when good compression is possible with additional techniques, and in practice sometimes basic \(\mathcal{H}\)-matrix techniques work well. We investigate the compressibility here with extensive numerical experiments based on the SVD. We find that with growing wave number k, the \(\epsilon \)-rank of blocks corresponding to a fixed size in physical space of the Green’s function is always growing like \(O(k^{\alpha })\), with \(\alpha \in [\frac{3}{4},1]\) in 2d and \(\alpha \in [\frac{4}{3},2]\) in 3d.

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Acknowledgments

The research described was partially supported by RFBR grants 16-05-00800,17-01-00399 and the Russian Academy of Sciences Program “Arctic”.

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Authors and Affiliations

  1. Section of Mathematics, University of Geneva, Geneva, Switzerland

    Martin J. Gander

  2. Institute of Petroleum Geology and Geophysics SB RAS, 3 Akademika Koptyuga pr., Novosibirsk, Russia, 630090

    Sergey Solovyev

Authors
  1. Martin J. Gander
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  2. Sergey Solovyev
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Correspondence to Martin J. Gander .

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Editors and Affiliations

  1. Institute of Information and Communication Technologies, Bulgarian Academy of Sciences, Sofia, Bulgaria

    Ivan Dimov

  2. Department of Applied Analysis, Eötvös Loránd University, Budapest, Hungary

    István Faragó

  3. Department of Applied Mathematics and Statistics, University of Rousse, Ruse, Bulgaria

    Lubin Vulkov

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Gander, M.J., Solovyev, S. (2017). A Numerical Study on the Compressibility of Subblocks of Schur Complement Matrices Obtained from Discretized Helmholtz Equations. In: Dimov, I., Faragó, I., Vulkov, L. (eds) Numerical Analysis and Its Applications. NAA 2016. Lecture Notes in Computer Science(), vol 10187. Springer, Cham. https://doi.org/10.1007/978-3-319-57099-0_7

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  • DOI: https://doi.org/10.1007/978-3-319-57099-0_7

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-57098-3

  • Online ISBN: 978-3-319-57099-0

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