Imaging anisotropic conductivities from current densities. (English) Zbl 1495.35214
Summary: 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.
MSC:
| 35R30 | Inverse problems for PDEs |
| 35R25 | Ill-posed problems for PDEs |
| 35J25 | Boundary value problems for second-order elliptic equations |
| 47J06 | Nonlinear ill-posed problems |
Keywords:
anisotropic conductivity; current density; Tikhonov regularization; H-convergence; hd-convergence; projected Newton methodSoftware:
FreeFem++References:
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