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

Abstract

In this study we consider the recovery of smooth region boundaries of piecewise constant coefficients of an elliptic PDE, -del.a del Phi + b Phi = f, from data on the exterior boundary partial derivative Omega. The assumption made is that the values of the coefficients (a, b) are known cc priori but the information about the geometry of the smooth region boundaries where a and b are discontinuous is missing. For the full characterization of (a, b) it is then sufficient to find the region boundaries separating different values of the coefficients. This leads 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 discretization elements. We formulate the forward problem as a mapping from a set of coefficients representing boundary shapes to data on partial derivative Omega, and derive the Jacobian of this forward mapping. Then an iterative algorithm which seeks a boundary configuration minimizing the residual norm between measured and predicted data is implemented. The method is illustrated first 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.