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INVERSE CONDUCTIVITY PROBLEM WITH INTERNAL DATA

Faouzi Triki1, Tao Yin2   

  1. 1. Laboratoire Jean Kuntzmann, UMR CNRS 5224, Université Grenoble-Alpes, 700 Avenue Centrale, 38401 Saint-Martin- d'Hères, France;
    2. Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Science, Beijing 100190, China
  • Received:2021-03-26 Revised:2021-07-31 Published:2023-03-14
  • Contact: Faouzi Triki, Email:faouzi.triki@univ-grenoble-alpes.fr
  • Supported by:
    This work was supported in part by the grant ANR-17-CE40-0029 of the French National Research Agency ANR (project MultiOnde).

Faouzi Triki, Tao Yin. INVERSE CONDUCTIVITY PROBLEM WITH INTERNAL DATA[J]. Journal of Computational Mathematics, 2023, 41(3): 483-501.

This paper concerns the reconstruction of a scalar coefficient of a second-order elliptic equation in divergence form posed on a bounded domain from internal data. This problem finds applications in multi-wave imaging, greedy methods to approximate parameterdependent elliptic problems, and image treatment with partial differential equations. We first show that the inverse problem for smooth coefficients can be rewritten as a linear transport equation. Assuming that the coefficient is known near the boundary, we study the well-posedness of associated transport equation as well as its numerical resolution using discontinuous Galerkin method. We propose a regularized transport equation that allow us to derive rigorous convergence rates of the numerical method in terms of the order of the polynomial approximation as well as the regularization parameter. We finally provide numerical examples for the inversion assuming a lower regularity of the coefficient, and using synthetic data.

CLC Number: 

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