  In this paper, we consider the inverse problem of recovering an isotropic
elastic tensor from the Neumann-to-Dirichlet map. To this end, we prove a
Lipschitz stability estimate for Lam\'e parameters with certain regularity
assumptions. In addition, we assume that the Lam\'e parameters belong to a
known finite subspace with a priori known bounds and that they fulfill a
monotonicity property. The proof relies on a monotonicity result combined with
the techniques of localized potentials. To numerically solve the inverse
problem, we propose a Kohn-Vogelius-type cost functional over a class of
admissible parameters subject to two boundary value problems. The reformulation
of the minimization problem via the Neumann-to-Dirichlet operator allows us to
obtain the optimality conditions by using the Fr\'echet differentiability of
this operator and its inverse. The reconstruction is then performed by means of
an iterative algorithm based on a quasi-Newton method. Finally, we give and
discuss several numerical examples.
