In this talk, I will present some recent advances in a project I have been working on during the Ph.D., namely, the reconstruction of discontinuous coefficients in a nonlinear system of PDEs and ODEs from boundary data. After providing a general overview of the problem and recalling the most recent theoretical results, I will focus on the numerical framework that we have developed for the solution of the direct and of the inverse problems. The discretization of the nonlinear system is performed via a Newton-Galerkin Finite Element scheme, which is feasible to be applied in a very general context. Some powerful tools of numerical analysis, together with the theoretical results at our disposal, allow to prove good approximation properties for the discrete scheme, and also to device a posteriori error estimators, which pave the way for efficient adaptive algorithms. This is joint work with Marco Verani (Politecnico di Milano) and Elena Beretta (Politecnico di Milano – NYU Abu Dhabi).