Inverse anisotropic diffusion from power density measurements in two dimensions

This paper concerns the reconstruction of an anisotropic diffusion tensor $γ=(γ_{ij})_{1\leq i,j\leq 2}$ from knowledge of internal functionals of the form $γ\nabla u_i\cdot\nabla u_j$ with $u_i$ for $1\leq i\leq I$ solutions of the elliptic equation $\nabla \cdot γ\nabla u_i=0$ on a two dimensional bounded domain with appropriate boundary conditions. We show that for I=4 and appropriately chosen boundary conditions, $γ$ 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.

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