  Problems with topological uncertainties appear in many fields ranging from
nano-device engineering to the design of bridges. In many of such problems, a
part of the domains boundaries is subjected to random perturbations making
inefficient conventional schemes that rely on discretization of the whole
domain. In this paper, we study elliptic PDEs in domains with boundaries
comprised of a deterministic part and random apertures, and apply the method of
modified potentials with Green's kernels defined on the deterministic part of
the domain. This approach allows to reduce the dimension of the original
differential problem by reformulating it as a boundary integral equation posed
on the random apertures only. The multilevel Monte Carlo method is then applied
to this modified integral equation and its optimal $\epsilon^{-2}$ asymptotical
complexity is shown. Finally, we provide the qualitative analysis of the
proposed technique and support it with numerical results.
