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Problems with topological uncertainties appear in many fields ranging fromnano-device engineering to the design of bridges. In many of such problems, apart of the domains boundaries is subjected to random perturbations makinginefficient conventional schemes that rely on discretization of the wholedomain. In this paper, we study elliptic PDEs in domains with boundariescomprised of a deterministic part and random apertures, and apply the method ofmodified potentials with Green's kernels defined on the deterministic part ofthe domain. This approach allows to reduce the dimension of the originaldifferential problem by reformulating it as a boundary integral equation posedon the random apertures only. The multilevel Monte Carlo method is then appliedto this modified integral equation and its optimal $\epsilon^{-2}$ asymptoticalcomplexity is shown. Finally, we provide the qualitative analysis of theproposed technique and support it with numerical results.

Method of Green's potentials for elliptic PDEs in domains with random apertures