We propose and analyse a fully adaptive strategy
for solving elliptic PDEs with random data in this work.
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 the computational
cost. The novel aspect of our strategy is that the hierarchy of spatial approximations
is sample dependent so that the computational effort at each collocation point can be
optimised individually. We outline a rigorous analysis for the convergence and
computational complexity of the adaptive multilevel algorithm and we provide optimal choices for error tolerances at each level. Two numerical examples demonstrate the reliability of the error control and the significant decrease in the complexity that arises when compared
to single level algorithms and multilevel algorithms that employ adaptivity solely in the spatial discretisation or in the collocation procedure.