Discretized Tikhonov-Phillips regularization for a naturally linearized parameter identification problem

The problem of identification of the diffusion coefficient in the partial differential equation is considered. We discuss a natural linearization of this problem and application of discretized Tikhonov-Phillips regularization to its linear version. Using recent results of regularization theory, we propose a strategy for the choice of regularization and discretization parameters which automatically adapts to unknown smoothness of the coefficient. The estimation of the accuracy will be given and various numerical test supporting theoretical results will be presented.