Inverse diffusion problems with redundant internal information

This paper concerns the reconstruction of a scalar diffusion coefficient σ(x) from redundant functionals of the form Hi(x) = σ (x)|∇ui|(x) where α ∈ R and ui is a solution of the elliptic problem ∇ · σ∇ui = 0 for 1 ≤ i ≤ I. The case α = 1 2 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 α = 1 finds applications in Magnetic Resonance Electrical Impedance Tomography (MREIT). We present two explicit reconstruction procedures of σ for appropriate choices of I and of traces of ui 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 α the results obtained in two and three dimensions in [10] and [5], respectively, in the case α = 12 . The second procedure consists of solving a system of linear elliptic equations, which we can prove admits a unique solution in specific situations.