Numerical approximation of young measures in non-convex variational problems

摘要

In non-convex optimisation problems, in particular in non-convex variational problems, there usually does not exist any classical solution but only generalised solutions which involve Young measures. In this paper, first a suitable relaxation and approximation theory is developed together with optimality conditions, and then an adaptive scheme is proposed for the efficient numerical treatment. The Young measures solving the approximate problems are usually composed only from a few atoms. This is the main argument our effective active-set type algorithm is based on. The support of those atoms is estimated from the Weierstrass maximum principle which involves a Hamiltonian whose good guess is obtained by a multilevel technique. Numerical experiments are performed in a one-dimensional variational problem and support efficiency of the algorithm. [References: 19]