Inverse anisotropic diffusion from power density measurements in two dimensions

This paper concerns the reconstruction of an anisotropic diffusion tensor γ = (γij)1≤i,j≤2 from knowledge of internal functionals of the form γ∇ui ·∇uj with ui for 1 ≤ i ≤ I solutions of the elliptic equation ∇·γ∇ui = 0 on a two dimensional bounded domain with appropriate boundary conditions. We show that for I = 4 and appropriately chosen boundary conditions, γ may uniquely and stably be reconstructed from such internal functionals, which appear in coupled-physics inverse problems involving the ultrasound modulation of electrical or optical coefficients. Explicit reconstruction procedures for the diffusion tensor are presented and implemented numerically.