  In this article, we develop a set-oriented numerical methodology which allows
to perform uncertainty quantification (UQ) for dynamical systems from a global
point of view. That is, for systems with uncertain parameters we approximate
the corresponding global attractors and invariant measures in the related
stochastic setting. Our methods do not rely on generalized polynomial chaos
techniques. Rather, we extend classical set-oriented methods designed for
deterministic dynamical systems to the UQ-context, and this allows us to
analyze the long-term uncertainty propagation. The algorithms have been
integrated into the software package GAIO, and we illustrate the use and
efficiency of these techniques by a couple of numerical examples.
