Uncertainty quantification for stationary and time-dependent PDEs subject to Gevrey regular random domain deformations

TL;DR

This paper develops a general framework for uncertainty quantification in PDEs with Gevrey smooth random domain deformations, using quasi-Monte Carlo methods with proven convergence rates.

Contribution

It introduces a non-parametric approach to model random domain deformations and designs QMC rules with dimension-independent convergence for PDEs.

Findings

QMC cubature rules achieve linear convergence rates

Error analysis includes dimension truncation and discretization effects

Numerical experiments confirm theoretical convergence rates

Abstract

We study uncertainty quantification for partial differential equations subject to domain uncertainty. We parameterize the random domain using the model recently considered by Chernov and Le (2024) as well as Harbrecht, Schmidlin, and Schwab (2024) in which the input random field is assumed to belong to a Gevrey smoothness class. This approach has the advantage of being substantially more general than models which assume a particular parametric representation of the input random field such as a Karhunen-Loeve series expansion. We consider both the Poisson equation as well as the heat equation and design randomly shifted lattice quasi-Monte Carlo (QMC) cubature rules for the computation of the expected solution under domain uncertainty. We show that these QMC rules exhibit dimension-independent, essentially linear cubature convergence rates in this framework. In addition, we complete the error analysis by taking into account the approximation errors incurred by dimension truncation of the random input field and finite element discretization. Numerical experiments are presented to confirm the theoretical rates.