  The multilevel Monte Carlo (MLMC) method has proven to be an effective
variance-reduction statistical method for Uncertainty quantification in PDE
models. It combines approximations at different levels of accuracy using a
hierarchy of meshes in a similar way as multigrid. The generation of
body-fitted mesh hierarchies is only possible for simple geometries. On top of
that, MLMC for random domains involves the generation of a mesh for every
sample. Instead, here we consider the use of embedded methods which make use of
simple background meshes of an artificial domain (a bounding-box) for which it
is easy to define a mesh hierarchy, thus eliminating the need of body-fitted
unstructured meshes, but can produce ill-conditioned discrete problems. To
avoid this complication, we consider the recent aggregated finite element
method (AgFEM). In particular, we design an embedded MLMC framework for
(geometrically and topologically) random domains implicitly defined through a
random level-set function, which makes use of a set of hierarchical background
meshes and the AgFEM. Performance predictions from existing theory are verified
statistically in three numerical experiments, namely the solution of the
Poisson equation on a circular domain of random radius, the solution of the
Poisson equation on a topologically identical but more complex domain, and the
solution of a heat-transfer problem in a domain that has geometric and
topological uncertainties. Finally, the use of AgFE is statistically
demonstrated to be crucial for complex and uncertain geometries in terms of
robustness and computational cost.
