Inverse anisotropic conductivity from internal current densities

TL;DR

This paper develops a method to uniquely reconstruct anisotropic conductivity tensors from internal current density measurements, achieving minimal data requirements and analyzing stability and derivative loss in the reconstruction process.

Contribution

It introduces a minimal set of internal current density measurements for local reconstruction of anisotropic conductivities and analyzes the stability and derivative loss involved.

Findings

Reconstruction requires at least n+2 measurements in n-dimensional space.

Unique reconstruction is possible with a loss of one derivative.

Scalar conductivities can be reconstructed without derivative loss.

Abstract

This paper concerns the reconstruction of an anisotropic conductivity tensor $\gamma$ from internal current densities of the form $J = \gamma\nabla u$, where $u$ solves a second-order elliptic equation $\nabla\cdot(\gamma\nabla u) = 0$ on a bounded domain $X$ with prescribed boundary conditions. A minimum number of such functionals equal to $n + 2$, where $n$ is the spatial dimension, is sufficient to guarantee a local reconstruction. We show that $\gamma$ can be uniquely reconstructed with a loss of one derivative compared to errors in the measurement of $J$. In the special case where $\gamma$ is scalar, it can be reconstructed with no loss of derivatives. We provide a precise statement of what components may be reconstructed with a loss of zero or one derivatives.