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Numerical Algorithms

, Volume 82, Issue 3, pp 869–894 | Cite as

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

  • Laëtitia Caillé
  • Liviu Marin
  • Franck DelvareEmail author
Original Paper
  • 33 Downloads

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.

Keywords

Inverse problems Cauchy problem Helmholtz equation Regularization method 

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Notes

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. 1.Normandie UniversitéCaenFrance
  2. 2.UNICAEN, LMNOCaenFrance
  3. 3.CNRS, UMR 6139CaenFrance
  4. 4.Normandie Univ, UNICAEN, CNRS, LMNOCaenFrance
  5. 5.Department of Mathematics, Faculty of Mathematics and Computer ScienceUniversity of BucharestBucharestRomania
  6. 6.Research Institute of the University of Bucharest (ICUB)University of BucharestBucharestRomania
  7. 7.Institute of Mathematical Statistics and Applied MathematicsRomanian AcademyBucharestRomania

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