Inverse anisotropic diffusion from power density measurements in two dimensions

This paper concerns the reconstruction of an anisotropic diffusion tensor gamma = (gamma(ij))(1 <= i, j <= 2) from knowledge of internal functionals of the form gamma del u(i) . del u(j) with ui for 1 <= i <= I solutions of the elliptic equation del . gamma del u(i) = 0 on a two-dimensional bounded domain with appropriate boundary conditions. We show that for I = 4 and appropriately chosen boundary conditions, gamma 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.