Quasi-Monte Carlo for partial differential equations with generalized Gaussian input uncertainty

TL;DR

This paper investigates the use of quasi-Monte Carlo methods for efficiently quantifying uncertainty in PDEs with generalized Gaussian inputs, extending previous work to more general bounds and analyzing various error sources.

Contribution

It introduces a framework for applying QMC to PDEs with generalized Gaussian input uncertainties, relaxing previous boundedness assumptions and analyzing multiple error components.

Findings

QMC methods achieve efficient integration for generalized Gaussian inputs.

Error bounds are established for QMC, dimension truncation, and finite element discretization.

The approach extends uncertainty quantification techniques to broader classes of input distributions.

Abstract

There has been a surge of interest in uncertainty quantification for parametric partial differential equations (PDEs) with Gevrey regular inputs. The Gevrey class contains functions that are infinitely smooth with a growth condition on the higher-order partial derivatives, but which are nonetheless not analytic in general. Recent studies by Chernov and Le (Comput. Math. Appl., 2024, and SIAM J. Numer. Anal., 2024) as well as Harbrecht, Schmidlin, and Schwab (Math. Models Methods Appl. Sci., 2024) analyze the setting wherein the input random field is assumed to be uniformly bounded with respect to the uncertain parameters. In this paper, we relax this assumption and allow for parameter-dependent bounds. The parametric inputs are modeled as generalized Gaussian random variables, and we analyze the application of quasi-Monte Carlo (QMC) integration to assess the PDE response statistics using randomly shifted rank-1 lattice rules. In addition to the QMC error analysis, we also consider the dimension truncation and finite element errors in this setting.