Liu, Huan;
Jin, Bangti;
Lu, Xiliang;
(2022)
Imaging Anisotropic Conductivities from Current Densities.
SIAM Journal on Imaging Sciences
, 15
(2)
pp. 860-891.
10.1137/21M1437810.
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Abstract
In this paper, we propose and analyze a reconstruction algorithm for imaging an anisotropic conductivity tensor in a second-order elliptic PDE with a nonzero Dirichlet boundary condition from internal current densities. It is based on a regularized output least-squares formulation with the standard $L^2(\Omega)^{d,d}$ penalty, which is then discretized by the standard Galerkin finite element method. We establish the continuity and differentiability of the forward map with respect to the conductivity tensor in the $L^p(\Omega)^{d,d}$-norms, the existence of minimizers and optimality systems of the regularized formulation using the concept of H-convergence. Further, we provide a detailed analysis of the discretized problem, especially the convergence of the discrete approximations with respect to the mesh size, using the discrete counterpart of H-convergence. In addition, we develop a projected Newton algorithm for solving the first-order optimality system. We present extensive two-dimensional numerical examples to show the efficiency of the proposed method.
| Type: | Article |
|---|---|
| Title: | Imaging Anisotropic Conductivities from Current Densities |
| Open access status: | An open access version is available from UCL Discovery |
| DOI: | 10.1137/21M1437810 |
| Publisher version: | https://doi.org/10.1137/21M1437810 |
| Language: | English |
| Additional information: | This version is the version of record. For information on re-use, please refer to the publisher’s terms and conditions. |
| Keywords: | anisotropic conductivity, current density, Tikhonov regularization, H-convergence, Hd-convergence, projected Newton method |
| UCL classification: | UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Engineering Science UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Engineering Science > Dept of Computer Science UCL > Provost and Vice Provost Offices > UCL BEAMS UCL |
| URI: | https://discovery.ucl.ac.uk/id/eprint/10153241 |
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