An inverse problem applied to volcanology is studied. It consists in the determination of the variable pressure applied to a crack in order to fit observed ground displacements. The deformation of the volcano is assumed to be governed by linear elasticity. The direct problem is solved via a fictitious domain method, using a finite element discretization of XFEM type. The ground misfit is minimized using a combination of a domain decomposition and optimatily conditions. The gradient of the cost function is derived from a sensitivity analysis. Discretization of the problem is studied. Numerical tests (in 2D and 3D) are presented to illustrate the effectiveness of the proposed approach. In particular, we find that a quasi-Newton method is more efficient than a conjugate gradient method for solving the optimization problem.