Inverse anisotropic conductivity from internal current densities

This paper concerns the reconstruction of a fully anisotropic conductivity tensor. from internal current densities of the form J = gamma del u, where u solves a second- order elliptic equation del center dot (gamma del u) = 0 on a bounded domain X with prescribed boundary conditions. A minimum number of n + 2 such functionals known on Y subset of X, where n is the spatial dimension, is sufficient to guarantee a unique and explicit reconstruction of gamma locally on Y. Moreover, we show that gamma is reconstructed with a loss of one derivative compared to errors in the measurement of J in the general case and no loss of derivatives in the special case where gamma is scalar. We also describe linear combinations of mixed partial derivatives of gamma that exhibit better stability properties and hence can be reconstructed with better resolution in practice.