  In this paper, we consider the inverse problem of detecting a corrosion
coefficient between two layers of a conducting medium from the
Neumann-to-Dirichlet map. This inverse problem is motivated by the description
of the index of corrosion in non-destructive testing. We show a monotonicity
estimates between the Robin coefficient and the Neumann-to-Dirichlet operator.
We prove a global uniqueness result and Lipschitz stability estimate, and show
how to quantify the Lipschitz stability constant for a given setting.
  Our quantification of the Lipschitz constant does not rely on quantitative
unique continuation or analytic estimates of special functions. Instead of
deriving an analytic estimate, we show that the Lipschitz constant for a given
setting can be explicitly calculated from the a priori data by solving finitely
many well-posed PDEs. Our arguments rely on standard (non-quantitative) unique
continuation, a Runge approximation property, the monotonicity result and the
method of localized potentials.
  To solve the problem numerically, we reformulate the inverse problem into a
minimization problem using a least square functional. The reformulation of the
minimization problem as a suitable saddle point problem allows us to obtain the
optimality conditions by using differentiability properties of the min-sup
formulation. The reconstruction is then performed by means of the BFGS
algorithm. Finally, numerical results are presented to illustrate the
efficiency of the proposed alogorithm.
