Asymptotic properties of Monte Carlo methods in elliptic PDE-constrained optimization under uncertainty

Monte Carlo approximations for random linear elliptic PDE constrained optimization problems are studied. We use empirical process theory to obtain best possible mean convergence rates \(O(n^{-\frac{1}{2}})\) $O(n^{-\frac{1}{2}})$ for optimal values and solutions, and a central limit theorem for optimal values. The latter allows to determine asymptotically consistent confidence intervals by using resampling techniques. The theoretical results are illustrated with two sets of numerical experiments. The first demonstrates the theoretical convergence rates for optimal values and optimal solutions. This is complemented by a study illustrating the usage of subsampling bootstrap methods for estimating the confidence intervals.