On Cauchy's problem: II. Completion, regularization and approximation

In Ben Belgacem and El Fekih (2005 On Cauchy's problem: I. A variational Steklov–Poincaré theory Inverse Problems 21 1915–36), a new variational theory is introduced for the data completion Cauchy problem, and studied in the Sobolev scales. Reformulating it, owing to the Dirichlet-to-Neumann operator, enables us to prove several mathematical results for the obtained Steklov–Poincaré problem and to establish the connection with some well-known minimization methods. In particular, when the over-specified data are incompatible and the existence fails for the Cauchy problem, it is stated that the least-squares incompatibility measure equals zero and so does the minimum value of the Kohn–Vogelius function, though all the minimizing sequences blow up. Because of the ill-posedness of the Cauchy–Steklov–Poincaré problem, an efficient numerical simulation of it can scarcely be achieved without some regularization materials. When combined with carefully chosen stopping criteria, they bring stability to the computations and dampen the noise perturbations caused by possibly erroneous measurements. This paper, part II, is the numerical counterpart of I and handles some practical issues. We are mainly involved in the Tikhonov scheme and the finite-element method applied to the unstable data completion problem. We lay down a new non-distributional space, in the Steklov–Poincaré framework, that allows for an elegant investigation of the reliability of both regularizations in: (i) approximating the exact solution, for compatible data and (ii) providing a consistent pseudo-solution, for incompatible data, that turns also to be a minimizing sequence of the Kohn–Vogelius gap function. Moreover, some convergence estimates with respect to the regularization parameters are stated for some worthy indicators such as the incompatibility measure and the minimum value of the Kohn–Vogelius and energy functions. Finally, we report and discuss some informative computing experiences to support the theoretical predictions of each regularization and to assess their reliability.