Uncertainty quantification for stationary and time-dependent PDEs subject to Gevrey regular random domain deformations
This work provides a rigorous error analysis for a more general class of random domain perturbations, benefiting researchers in computational uncertainty quantification for PDEs.
The authors develop uncertainty quantification for PDEs with random domain deformations modeled by Gevrey regular fields, achieving dimension-independent quasi-Monte Carlo cubature convergence rates for both Poisson and heat equations. Numerical experiments confirm theoretical rates.
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.