Uncertainty quantification for random domains using periodic random variables

TL;DR

This paper develops lattice quasi-Monte Carlo methods for efficiently quantifying uncertainty in Poisson problems with random domains modeled by periodic random variables, achieving high convergence rates.

Contribution

It introduces QMC cubature rules tailored for periodic random domain models, providing a comprehensive error analysis including truncation and discretization effects.

Findings

QMC rules achieve higher order convergence rates

Error bounds are established for truncation and finite element discretization

Numerical experiments confirm theoretical error estimates

Abstract

We consider uncertainty quantification for the Poisson problem subject to domain uncertainty. For the stochastic parameterization of the random domain, we use the model recently introduced by Kaarnioja, Kuo, and Sloan (SIAM J. Numer. Anal., 2020) in which a countably infinite number of independent random variables enter the random field as periodic functions. We develop lattice quasi-Monte Carlo (QMC) cubature rules for computing the expected value of the solution to the Poisson problem subject to domain uncertainty. These QMC rules can be shown to exhibit higher order cubature convergence rates permitted by the periodic setting independently of the stochastic dimension of the problem. In addition, we present a complete error analysis for the problem by taking into account the approximation errors incurred by truncating the input random field to a finite number of terms and discretizing the spatial domain using finite elements. The paper concludes with numerical experiments demonstrating the theoretical error estimates.