Uncertainty quantification using importance-sampled quasi-Monte Carlo with dimension-independent convergence rates

TL;DR

This paper introduces a boundary-damping importance sampling approach combined with scrambled quasi-Monte Carlo methods to achieve dimension-independent convergence rates for high-dimensional uncertainty quantification problems, especially in elliptic PDEs.

Contribution

It proposes a novel integrand transformation technique that enables the use of off-the-shelf randomized QMC methods for high-dimensional unbounded integrals, with rigorous convergence analysis.

Findings

Achieves dimension-independent mean squared error rate of $O(n^{-1-eta+ ext{small}})$ for UQ problems.

Extends the applicability of scrambled nets to broader classes of unbounded functions.

Numerical results confirm the efficiency and effectiveness of the proposed method in high-dimensional elliptic PDEs.

Abstract

Quasi-Monte Carlo (QMC) integration over unbounded domains $\mathbb{R}^s$ remains challenging due to the high dimensionality of sampling space and the boundary growth of the integrand. In applications such as uncertainty quantification (UQ), the dimension $s$ can reach hundreds or even thousands. To restore the efficiency of quadrature rules in high dimensions, constructive QMC methods like lattice rules have been successfully developed within the framework of weighted function spaces. In contrast to designing problem-specific quadrature points, this paper proposes transforming the underlying integrand to accommodate the off-the-shelf scrambled nets (a construction-free randomized QMC method) via the boundary-damping importance sampling (BDIS) proposed by Pan et al. (2025). We provide a rigorous analysis of the dimension-independent convergence rate of BDIS-based scrambled nets while covering a broader class of unbounded functions than that in Pan et al. (2025). By exploiting the dimension structure of the parametric input random field, the proposed $n$-point quadrature rule achieves a dimension-independent mean squared error rate of $O(n^{-1-\alpha^*+\varepsilon})$ on standard UQ problems in elliptic partial differential equations (PDEs), where $\varepsilon>0$ is arbitrarily small and $\alpha^*\in (0,1)$ reflects the regularity with respect to the parametric variables. Numerical experiments on elliptic PDEs with high-dimensional parameters further demonstrate the effectiveness of the method.