  While multilevel Monte Carlo (MLMC) methods for the numerical approximation
of partial differential equations with random coefficients enjoy great
popularity, combinations with spatial adaptivity seem to be rare. We present an
adaptive MLMC finite element approach based on deterministic adaptive mesh
refinement for the arising "pathwise" problems and outline a convergence theory
in terms of desired accuracy and required computational cost. Our theoretical
and heuristic reasoning together with the efficiency of our new approach are
confirmed by numerical experiments.
