Abstract

The problem of modelling processes with partial differential equations posed on random domains arises in various applications like biology or engineering. We study uncertainty quantification for partial differential equations subject to domain uncertainty, where we parameterize the random domain using the model recently considered by Chernov and Lê (Comput. Math. Appl.,2024, and SIAM J. Numer. Anal., 2024) as well as Harbrecht, Schmidlin, and Schwab (Math. Models Methods Appl. Sci., 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–Loève series expansion. As model problems 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 response statistics subject to domain uncertainty. We show that these QMC rules exhibit dimension-independent, faster-than-Monte Carlo cubature convergence rates in this framework. Our theoretical results will be illustrated by numerical examples. This is a joint work with Ana Djurdjevac, Vesa Kaarnioja and Claudia Schillings.

The talk is based on the recently published paper Uncertainty quantification for stationary and time-dependent PDEs subject to Gevrey regular random domain deformations.