An Adaptive Multilevel Monte Carlo Method with Stochastic Bounds for Quantities of Interest with Uncertain Data

The focus of this work is the introduction of some computable a posteriori error control to the popular multilevel Monte Carlo sampling for PDE with stochastic data. We are especially interested in applications where some quantity of interest should be estimated accurately. Based on a spatial discretization by the finite element method, a goal functional is defined which encodes the quantity of interest. The devised goal-oriented a posteriori error estimator enables one to determine guaranteed path-wise a posteriori error bounds for this quantity. An adaptive algorithm is proposed which employs the computed error estimates and adaptive meshes to control the approximation error. Moreover, the stochastic error is controlled such that the determined bounds are guaranteed in probability. The approach allows for the adaptive refinement of the mesh hierarchy used in the multilevel Monte Carlo simulation which is used for a problem-dependent construction of discretization levels. Numerical experiments illustrate the performance of the adaptive method for a posteriori error control in Monte Carlo and multilevel Monte Carlo methods with respect to localized goals. Moreover, the computational efficiency of the multilevel and classical Monte Carlo approaches are compared. It is illustrated that adaptively refine dmeshes can yield significant bene fits when combined with Monte Carlo methods. In particular, with a problem-adapted mesh hierarchy, the efficiency gains of the multilevel Monte Carlo method in terms of computational cost can also be exploited in the context of error control.