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A meshless fading regularization algorithm for solving the Cauchy problem for the three-dimensional Helmholtz equation

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Abstract

We investigate the Cauchy problem associated with the Helmholtz equation in three dimensions, namely the numerical reconstruction of the primary field (Dirichlet data) and its normal derivative (Neumann data) on a part of the boundary from the knowledge of overprescribed noisy measurements taken on the remaining boundary part. This inverse problem is solved by combining the fading regularization method with the method of fundamental solutions (MFS). A stopping regularizing/stabilizing criterion is also proposed. Two numerical examples are investigated in order to validate the proposed method in terms of its accuracy, convergence, stability and efficiency.

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Acknowledgements

L. Caillé and L. Marin have been supported by a grant of Ministry of Research and Innovation, CNCS - UEFISCDI, project number PN-III-P4-ID-PCE-2016-0083, within PNCDI III. The financial support received by L. Caillé from Région Normandie is also gratefully acknowledged.

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Authors and Affiliations

  1. Normandie Université, Caen, France

    Laëtitia Caillé & Franck Delvare

  2. UNICAEN, LMNO, 14032, Caen, France

    Laëtitia Caillé & Franck Delvare

  3. CNRS, UMR 6139, 14032, Caen, France

    Laëtitia Caillé & Franck Delvare

  4. Normandie Univ, UNICAEN, CNRS, LMNO, 14000, Caen, France

    Laëtitia Caillé & Franck Delvare

  5. Department of Mathematics, Faculty of Mathematics and Computer Science, University of Bucharest, 14 Academiei, 010014, Bucharest, Romania

    Liviu Marin

  6. Research Institute of the University of Bucharest (ICUB), University of Bucharest, 34–36 Blvd. Mihail Kogălniceanu, 050107, Bucharest, Romania

    Liviu Marin

  7. Institute of Mathematical Statistics and Applied Mathematics, Romanian Academy, 13 Calea 13 Septembrie, 050711, Bucharest, Romania

    Liviu Marin

Authors
  1. Laëtitia Caillé
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  2. Liviu Marin
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  3. Franck Delvare
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Correspondence to Franck Delvare.

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Caillé, L., Marin, L. & Delvare, F. A meshless fading regularization algorithm for solving the Cauchy problem for the three-dimensional Helmholtz equation. Numer Algor 82, 869–894 (2019). https://doi.org/10.1007/s11075-018-0631-y

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  • DOI: https://doi.org/10.1007/s11075-018-0631-y

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