In this paper, we propose and analyse a new adaptive multilevel stochastic
collocation method for randomized elliptic PDEs. A hierarchical sequence of
adaptive mesh refinements for the spatial approximation is combined with
adaptive anisotropic sparse Smolyak grids in the stochastic space in such a way
as to minimize computational cost. We provide a rigorous analysis for the
convergence and computational complexity of the adaptive multilevel algorithm.
Two numerical examples demonstrate the reliability of an error control by
adaptive methods and the significant decrease in complexity versus uniform
spatial refinements and single-level stochastic sampling methods.