  This work proposes a scheme for significantly reducing the computational
complexity of discretized problems involving the non-smooth forward propagation
of uncertainty by combining the adaptive hierarchical sparse grid stochastic
collocation method (ALSGC) with a hierarchy of successively finer spatial
discretizations (e.g. finite elements) of the underlying deterministic problem.
To achieve this, we build strongly upon ideas from the Multilevel Monte Carlo
method (MLMC), which represents a well-established technique for the reduction
of computational complexity in problems affected by both deterministic and
stochastic error contributions. The resulting approach is termed the Multilevel
Adaptive Sparse Grid Collocation (MLASGC) method. Preliminary results for a
low-dimensional, non-smooth parametric ODE problem are promising: the proposed
MLASGC method exhibits an error/cost-relation of $\varepsilon \sim t^{-0.95}$
and therefore significantly outperforms the single-level ALSGC ($\varepsilon
\sim t^{-0.65}$) and MLMC methods ($\varepsilon \lesssim t^{-0.5}$).
