Inverse diffusion problems with redundant internal information

Abstract

This paper concerns the reconstruction of a scalar diffusion coefficient $\sigma(x)$ from redundant functionals of the form $H_i(x)=\sigma^{2\alpha}(x)|\nabla u_i|^2(x)$ where $\alpha\in\Rm$ and $u_i$ is a solution of the elliptic problem $\nabla\cdot \sigma \nabla u_i=0$ for $1\leq i\leq I$. The case $\alpha=\frac12$ is used to model measurements obtained from modulating a domain of interest by ultrasound and finds applications in ultrasound modulated electrical impedance tomography (UMEIT) as well as ultrasound modulated optical tomography (UMOT). The case $\alpha=1$ finds applications in Magnetic Resonance Electrical Impedance Tomography (MREIT). We present two explicit reconstruction procedures of $\sigma$ for appropriate choices of $I$ and of traces of $u_i$ at the boundary of a domain of interest. The first procedure involves the solution of an over-determined system of ordinary differential equations and generalizes to the multi-dimensional case and to (almost) arbitrary values of $\alpha$ the results obtained in two and three dimensions in \cite{CFGK} and \cite{BBMT}, respectively, in the case $\alpha=\frac12$. The second procedure consists of solving a system of linear elliptic equations, which we can prove admits a unique solution in specific situations.