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Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2020+1 pp 31–66Cite as

Broadband Recursive Skeletonization

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Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE,volume 137)

Abstract

The dense linear systems arising from the discretization of integral equations have in last few decades been rendered tractable through the development of techniques such as the Fast Multipole Method, Fast Direct Solvers, \(\mathcal {H}\)-matrix methods, etc. These algorithms depend crucially on the low-rank approximation of dense interactions between disjoint subsets of the computational domain. The key result of the present work is the discovery that for time-harmonic wave scattering problems, it is possible to build “universal bases” that enable efficient low rank approximation across a whole range of wavenumbers. In many cases, the number of basis functions is almost exactly the same as the number required to resolve only the highest wavenumber in the band, which is known to be determined by the “two points per wavelength” heuristic of Shannon. As an application of the new observation, the manuscript describes a numerical technique for solving time-harmonic scattering problems in the regime where a large set of wavenumbers within a specified range are considered. The technique relies on a boundary integral equation formulation that is coupled with a fast direct solver that relies on the universal basis idea to greatly accelerate the “compression stage” where a rank-structured approximation to a large dense matrix is constructed. The accuracy and effectiveness of the procedure is illustrated with several numerical experiments.

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Acknowledgements

The authors would like to thank Alex Barnett and Vladimir Rokhlin for valuable discussions about the topics under consideration. The work reported was supported by the Office of Naval Research (N00014-18-1-2354), by the National Science Foundation (DMS-1952735 and DMS-2012606), and by the Department of Energy ASCR (DE-SC0022251).

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Authors and Affiliations

  1. Yale University, New Haven, CT, USA

    Abinand Gopal

  2. University of Texas at Austin, Austin, TX, USA

    Per-Gunnar Martinsson

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  1. Abinand Gopal
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  2. Per-Gunnar Martinsson
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Correspondence to Per-Gunnar Martinsson .

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Editors and Affiliations

  1. TU Wien, Wien, Austria

    Jens M. Melenk

  2. Universität Wien, Wien, Austria

    Ilaria Perugia

  3. TU Wien, Wien, Austria

    Joachim Schöberl

  4. ETH Zürich, Zürich, Switzerland

    Christoph Schwab

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Gopal, A., Martinsson, PG. (2023). Broadband Recursive Skeletonization. In: Melenk, J.M., Perugia, I., Schöberl, J., Schwab, C. (eds) Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2020+1. Lecture Notes in Computational Science and Engineering, vol 137. Springer, Cham. https://doi.org/10.1007/978-3-031-20432-6_2

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