On numerical approximation of non-convex variational problems using stochastic optimization algorithm

Abstract: "An implementation of the stochastic gradient minimization method is proposed as a viable approach for the solution of non-convex variational problems. In order to numerically characterize the highly oscillatory properties of minimizing sequences of non-convex energies, the approximation of an associated Young measure on a macroscopic mesh is used. However, the presence of a large number of undesired local minima is generic for such problems; consequently, the classical algorithms of descent are inadequate for minimization of these energies. Once a local minimizer is reached, further improvement may be sought through random restart(s). An adaptive filtering is proposed as an ad-hoc step that may be implemented in conjunction with the probabilistic search of feasible directions. Application of this method to the solution of the variational problem corresponding to the Ericksen-James energy density in two dimensions is demonstrated."