We treat the inverse problem of determining material losses, such as cavities, in a conducting body, by performing electrostatic measurements at the boundary. We develop a numerical approach, based on variational methods, to reconstruct the unknown material loss by a single boundary measurement of current and voltage type. The method is based on the use of phase-field functions to model the material losses and on a perimeter-like penalization to regularize the otherwise ill-posed problem. We justify the proposed approach by a convergence result, as the error on the measurement goes to zero. © 2011 Elsevier Inc.
Rondi, L. (2011). Reconstruction of material losses by perimeter penalization and phase-field methods. Journal of Differential Equations, 251(1), 150–175. https://doi.org/10.1016/j.jde.2011.03.014