Imaging Anisotropic Conductivities from Current Densities

TL;DR

This paper introduces a new regularized reconstruction algorithm for imaging anisotropic conductivity tensors from internal current densities, with theoretical analysis and efficient numerical implementation.

Contribution

It develops a novel regularized output least-squares method, analyzes its convergence, and implements a projected Newton algorithm for anisotropic conductivity imaging.

Findings

The method accurately reconstructs anisotropic conductivities.

Theoretical convergence results are established.

Numerical examples demonstrate efficiency and robustness.

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.