A parabolic multiscale inverse problem approached via homogenization: A numerical method

A method based on homogenization is studied for the solution of a multiscale inverse problem. We consider a class of parabolic problems with highly oscillatory tensors that vary on a microscopic scale. We assume that the microscopic structure is known and seek to recover a macroscopic scalar parameterization of the multiscale tensor. Classical approaches, such as finite elements methods, would require mesh resolution for the direct problem down to the finest scale, that could lead to computational difficulties when implemented. So, starting from the full fine scale model, we solve the inverse problem for a coarse model obtained by homogenization, both theoretically and numerically. The input data, which consist on measurements from the fluxes and the solutions of the direct problem in a given time, are solely based on the original fine scale model. Uniqueness and stability of the inverse problem obtained via homogenization are established under some natural conditions for the fine scale model, and a link with this latter model is established by means of G-convergence. Error estimates are proven for the method.