Recovery of region boundaries of piecewise constant coefficients of an elliptic PDE from boundary data

In this study we consider the recovery of smooth region boundaries of piecewise constant coefficients of an elliptic PDE −∇ · a∇Φ + bΦ = f from data on the exterior boundary ∂Ω. The assumption is that the values of the coefficients (a, b) are known a priori but the information about the geometry of the smooth region boundaries where a and b are discontinous is missing. For the full characterisation of (a, b) it is then sufficient to find the region boundaries separating different values of the coefficients. This results to a nonlinear ill-posed inverse problem. In this study we propose a numerical algorithm that is based on the finite element method and subdivision of the discretisation elements. We formulate the forward problem as a mapping from a set of coefficients representing boundary shapes to data on ∂Ω, and derive the Jacobian of this forward mapping. Then an iterative algorithm which seeks a boundary configuration minimising the residual norm between measured and predicted data is implemented. The method is first given for a general elliptic PDE and then applied to optical tomography where the goal is to find the diffusion and absorption coefficients of the object by transilluminating the object with visible or near infrared light. Numerical test results for this specific application are given with synthetic data.