  We numerically implement the variational approach for reconstruction in the
inverse crack and cavity problems developed by one of the authors. The method
is based on a suitably adapted free-discontinuity problem. Its main features
are the use of phase-field functions to describe the defects to be
reconstructed and the use of perimeter-like penalizations to regularize the
ill-posed problem.
  The numerical implementation is based on the solution of the corresponding
optimality system by a gradient method. Numerical simulations are presented to
show the validity of the method.
