Imaging Anisotropic Conductivities from Current Densities

2022/03/04 by Liu, Huan, Jin, Bangti, Lu, Xiliang
#Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.2203.02164

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 L²(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.

Related

• Linearized internal functionals for anisotropic conductivities
• Imaging of anisotropic conductivities from current densities in two dimensions
• Inverse anisotropic conductivity from power densities in dimension n≥ 3
• Inverse conductivity problem with internal data
• Imaging Anisotropic Conductivity from Internal Measurements with Mixed Least-Squares Deep Neural Networks