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.

拓扑不确定性问题出现在从纳米设备工程到桥梁设计的许多领域。在许多这样的问题中，部分域边界受到随机扰动，从而导致依赖整个域离散化的低效率常规方案。在本文中，我们研究在边界由确定性部分和随机孔径组成的区域中的椭圆PDE，并应用在区域的确定性部分上定义具有格林核的修饰势的方法。该方法允许通过将原始微分问题重新构造为仅位于随机孔径上的边界积分方程来减小其尺寸。然后将多级蒙特卡罗方法应用于此修改后的积分方程及其最佳\（\ epsilon ^ {-2} \）显示了渐近复杂性。最后，我们对提出的技术进行了定性分析，并提供了数值结果。