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

Abstract

In uncertainty quantification (UQ) problems involving random fields, the dimension of the sampling space can reach hundreds or even thousands, and numerical integration consequently faces two major difficulties: the curse of dimensionality, and the rapid growth of the integrand near the boundary of the sampling space. To restore the efficiency of quadrature rules in high dimensions, researchers have developed constructive quasi-Monte Carlo (QMC) methods such as lattice rules within the framework of weighted function spaces. Unlike these approaches that require problem-specific quadrature points, the present work transforms the underlying integrand using a newly proposed boundary-damping importance sampling, making it directly compatible with construction-free QMC methods like scrambled nets. Furthermore, by exploiting the dimensional 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, where $\varepsilon>0$ is arbitrarily small and $\alpha^*\in (0,1)$ reflects the regularity with respect to the parametric variables. Numerical experiments further validate the effectiveness of the method.