### Achievement

We proposed the numerical scheme to solve elliptic PDE in deterministic domains with random apertures. The scheme is based on a boundary integral formulation of the problem using single layer potentials with Green’s kernels defined on the deterministic part of the domain only. The scheme can be applied to problems in arbitrary domains with random apertures and does not require the explicit knowledge of analytical Green’s functions to achieve the optimal complexity of the multilevel Monte Carlo estimator which was used to quantify the uncertainty.

### Significance and Impact

Problems with topological uncertainties appear in many applied fields ranging from nano-device engineering and analysis of micro electromechanical systems (MEMS) to the design of bridges. Other applications include flows over rough surfaces, surface imaging, corrosion or wear of surfaces, homogenization of random heterogeneous media and modeling of blood flow to name a few. Existing numerical methods for PDEs in random domains often rely on discretization of the whole domain making them inefficient for the problems where only part of the domain is random.

### Research Details

- developed numerical scheme based on boundary integral representation of solutions to elliptic PDEs in domains comprised of deterministic and random boundaries
- proved that when a large number of repetitive solutions is required, as is the case of Monte Carlo simulations, the proposed scheme can lead to significant reduction of computational complexity compared to standard BIE techniques

### Overview

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 effort, 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 asymptotical complexity is shown.

### Citation

Last Updated: January 14, 2021 - 11:15 am