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Volume 41, Issue 3
Inverse conductivity Problem with Internal Data

Faouzi Triki & Tao Yin

DOI: 10.4208/jcm.2111-m2021-0093

J. Comp. Math., 41 (2023), pp. 482-500.

Published online: 2023-02

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  • Abstract

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 parameter-dependent 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.

  • Keywords

Inverse problems, Multi-wave imaging, Static transport equation, Internal data, Diffusion coeffcient, Stability estimates, Regularization.

  • AMS Subject Headings

35R30, 65N21

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

Faouzi.Triki@univ-grenoble-alpes.fr (Faouzi Triki)

yintao@lsec.cc.ac.cn (Tao Yin)

  • BibTex
  • RIS
  • TXT
@Article{JCM-41-482, author = {Triki , Faouzi and Yin , Tao}, title = {Inverse conductivity Problem with Internal Data}, journal = {Journal of Computational Mathematics}, year = {2023}, volume = {41}, number = {3}, pages = {482--500}, abstract = {

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 parameter-dependent 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.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2111-m2021-0093}, url = {http://global-sci.org/intro/article_detail/jcm/21394.html} }
TY - JOUR T1 - Inverse conductivity Problem with Internal Data AU - Triki , Faouzi AU - Yin , Tao JO - Journal of Computational Mathematics VL - 3 SP - 482 EP - 500 PY - 2023 DA - 2023/02 SN - 41 DO - http://doi.org/10.4208/jcm.2111-m2021-0093 UR - https://global-sci.org/intro/article_detail/jcm/21394.html KW - Inverse problems, Multi-wave imaging, Static transport equation, Internal data, Diffusion coeffcient, Stability estimates, Regularization. AB -

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 parameter-dependent 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.

Triki , Faouzi and Yin , Tao. (2023). Inverse conductivity Problem with Internal Data. Journal of Computational Mathematics. 41 (3). 482-500. doi:10.4208/jcm.2111-m2021-0093
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