Inverse Diffusion From Knowledge of Power Densities in Dimension 3

Francois Monard (University of Michigan) and Donsub Rim (University of Washington)

We reconstruct the diffusion coefficient in an elliptic equations from the knowledge of several power densities. The power density is the product of the diffusion coefficient with the square of the gradient of a solution. The reconstruction first involves a contour integration along a path starting from the boundary, to solve for the quaternionic parametrization of a matrix R in SO(3). Then we solve a for the log of the diffusion coefficient as a solution to a Poisson problem. This problem arises in Ultrasound Modulated Electrical Impedance Tomography (UMEIT) and Ultrasound Modulated Optical Tomography (UMOT).