On the numerical analysis of non-convex variational problems

For non-convex energy functionals for which minimizers fail to exist in the classical sense, it is not always easy to implement algorithms that faithfully reflect the underlying oscillations that are usually involved in these problems. We discuss an approach based on the relaxation of the non-convex functional at two different levels and their relationship. It leads to approximating the underlying parametrized measure itself by discrete families of probability measures. We analyze in what sense there might be convergence of these discrete parametrized measures to the real continuous solution and examine a few examples.