  Many problems require to approximate an expected value by some kind of Monte
Carlo (MC) sampling, e.g. molecular dynamics (MD) or simulation of stochastic
reaction models (also termed kinetic Monte Carlo (kMC)). Often, we are
furthermore interested in some integral of the MC model's output over the input
parameters. We present a Multilevel Adaptive Sparse Grid strategy for the
numerical integration of such problems where the integrand is implicitly
defined by a Monte Carlo model. In this approach, we exploit different levels
of sampling accuracy in the Monte Carlo model to reduce the overall
computational costs compared to a single level approach. Unlike existing
approaches for Multilevel Numerical Quadrature, our approach is not based on a
telescoping sum, but we rather utilize the intrinsic multilevel structure of
the sparse grids and the employed locally supported, piecewise linear basis
functions. Besides illustrative toy models, we demonstrate the methodology on a
realistic kMC model for CO oxidation. We find significant savings compared to
the single level approach - often orders of magnitude.
