Towards Identification of Effective Parameters in Heterogeneous Media

Abstract: In this thesis we study a parameter identification problem for a stationary diffusion equation posed in heterogeneous media. This problem is closely related to the Calderón problem with anisotropic conductivities. The anisotropic case is particularly difficult and is ill-posed both in regards to uniqueness of solution and stability on the data. Since the present problem is posed in heterogeneous media, we can take advantage of multiscale modelling and the tools of homogenization theory in the study of the inverse problem, unlike the original Calderón problem. We investigate the possibilities of combining the theory of the Calderón problem with homogenization theory in order to obtain a well-posed parameter identification. We find that homogenization theory indeed can be used to make progress towards a well-posed identification of the diffusion coefficient. The success of the method is, however, dependent both on the precise structure of the heterogeneous media and on the modelling of the measurements in the invese problem framework. We have in mind a particular problem formulation which is motivated by an experiment to determine effective coefficients of materials used in food packaging. This experiment comes with a set of requirements on both the heterogeneous media and on the method for making measurements that, unfortunately, are in conflict with the currently available results for well-posedness. We study also an optimization approach to solving the inverse problem under these application specific requirements. Some progress towards well-posedness of the optimization problem is made by proving existence of minimizer, again with homogenization theory playing a key role in obtaining the result. In a proof-of-concept computational study this optimization approach is implemented and compared to two other optimization problems. For the two tested heterogeneous media, the only optimization method that manages to identify reasonably well the diffusion coefficient is the one which makes use of homogenization theory.