Imaging Anisotropic Conductivities from Current Densities

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 (Ω) 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 (Ω) 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.

Imaging conductivity from current density magnitude using neural networks*

“…A linearized reconstruction technique was developed recently in [73]. The more conventional output least-squares formulation has not been employed for CDII reconstruction, but it applies more or less directly (see [1,41] for conductivity imaging from related internal data, and [29] for iterative reconstruction).…”