Inverse anisotropic diffusion from power density measurements in two dimensions

This paper concerns the reconstruction of an anisotropic diffusion tensor $\gamma =({\gamma }_{ij}{)}_{1\le i,j\le 2}$ from knowledge of internal functionals of the form $\gamma \mathrm{\nabla }{u}_i\cdot \mathrm{\nabla }{u}_j$ with $u_i$ for $1\le i\le I$ solutions of the elliptic equation $\mathrm{\nabla }\cdot \gamma \mathrm{\nabla }{u}_i=0$ 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.