Efficient Distribution Estimation and Uncertainty Quantification for Elliptic Problems on Domains with Stochastic Boundaries

TL;DR

This paper introduces a method for uncertainty quantification in elliptic PDEs on domains with stochastic boundaries, using domain transformations to enable efficient Monte Carlo sampling and error analysis.

Contribution

It proposes simple transformations to map stochastic domains to a fixed reference, facilitating efficient uncertainty quantification and solution of elliptic boundary value problems.

Findings

Transformations enable effective error analysis.

Monte Carlo sampling becomes more efficient.

Improved solution accuracy on stochastic domains.

Abstract

We study the problem of uncertainty quantification for the numerical solution of elliptic partial differential equation boundary value problems posed on domains with stochastically varying boundaries. We also use the uncertainty quantification results to tackle the efficient solution of such problems. We introduce simple transformations that map a family of domains with stochastic boundaries to a fixed reference domain. We exploit the transformations to carry out a prior and a posteriori error analyses and to derive an efficient Monte Carlo sampling procedure.