Imaging of mass distributions from partial domain measurement

Abstract

Abstract This paper deals with an inverse source problem governed by the Poisson equation. The aim is to reconstruct multiple anomalies from internal observation data within an arbitrary sub-region. The considered model problem is motivated by various applications such as the identification of geological anomalies underneath the Earth’s surface. To overcome the ill-posedness, the inverse problem is formulated as a self-regularized topology optimization one. A least-square functional measuring the misfit between the observed quantities and the values provided by the model problem is introduced. The misfit function involves a regularization term penalizing the relative perimeter of the unknown domain. Finally, it is minimized with respect to a finite number of circular-shaped anomalies. The existence and stability of the solution to the inverse problem are demonstrated. The reconstruction process is based on the topological derivative method. The variation of the shape functional with respect to a small geometric perturbation is studied, and the leading terms of a second-order topological asymptotic expansion are derived. A non-iterative reconstruction algorithm is developed via the minimization of the obtained asymptotic formula with respect to the location and size parameters of the unknown set of anomalies. The efficiency and accuracy of the proposed approach are justified by some numerical experiments.