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

## TL;DR

This paper introduces a boundary integral approach using Green's potentials for elliptic PDEs in domains with random apertures, reducing computational complexity and enabling efficient uncertainty quantification.

## Contribution

The paper develops a novel boundary integral method with Green's kernels for elliptic PDEs in domains with random apertures, improving efficiency over traditional discretization methods.

## Key findings

- The method reduces problem dimension by focusing on random apertures.
- Multilevel Monte Carlo achieves optimal $ ext{O}( ext{epsilon}^{-2})$ complexity.
- Numerical results validate the effectiveness of the proposed approach.

## Abstract

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.