We will deal with the numerical approximation of some geometric inverse problems for the wave and the Lam\'e equations motivated by Elastography. We present several recent results and open questions concerning the numerical reconstruction of the unknown domain where the equations evolve. In the numerical experiments, we solve appropriate optimization problems. Two different numerical techniques will be proposed. Firstly, the finite element method for the numerical solution of the PDE's, that will be performed with \texttt{FreeFem++}. The routines on the \texttt{ff-NLopt} package, that provide an interface to a free/open-source library for nonlinear optimization, are also required. On the other hand, we will consider the numerical approximation based on the method of fundamental solutions. We present some numerical results in the 2D and 3D cases.