  As a simplified model for subsurface flows elliptic equations may be
utilized. Insufficient measurements or uncertainty in those are commonly
modeled by a random coefficient, which then accounts for the uncertain
permeability of a given medium. As an extension of this methodology to flows in
heterogeneous\fractured\porous media, we incorporate jumps in the diffusion
coefficient. These discontinuities then represent transitions in the media.
More precisely, we consider a second order elliptic problem where the random
coefficient is given by the sum of a (continuous) Gaussian random field and a
(discontinuous) jump part. To estimate moments of the solution to the resulting
random partial differential equation, we use a pathwise numerical approximation
combined with multilevel Monte Carlo sampling. In order to account for the
discontinuities and improve the convergence of the pathwise approximation, the
spatial domain is decomposed with respect to the jump positions in each sample,
leading to pathdependent grids. Hence, it is not possible to create a nested
sequence of grids which is suitable for each sample path a-priori. We address
this issue by an adaptive multilevel algorithm, where the discretization on
each level is sample-dependent and fulfills given refinement conditions.
