Parameter choice in the Lavrentiev regularization of the data completion problem

We use the Lavrentiev method for the regularization of the severely ill-posed data completion problem, put under a variational Steklov-Poincare form. In the choice of the regularizing parameter, we check a 'super-optimal' a priori convergence criterion. Consequently, the solutions obtained by the regularization provide a minimizing sequence of the Kohn-Vogelius function, with a 'quadratic' decaying of the minimum value of it toward zero, instead of the linear rate predicted by the general theory. We call such a result the 'super-convergence' of the approximated 'incompatibility measure' of the variational problem. Finally, we apply the a posteriori Morozov discrepancy principle to the Kohn-Vogelius functional and show how it yields a regularizing strategy. We achieve by some numerical illustrations to support our analysis.