[论文解读] Uncertainty quantification using importance-sampled quasi-Monte Carlo with dimension-independent convergence rates

Zexin Pan, Du Ouyang+1 more|arXiv (Cornell University)|Mar 1, 2026

Mathematical Approximation and Integration被引用 0

一句话总结

论文提出边界阻尼重要性采样方法，将无界积分转化为有界区域，从而随机化QMC（ scrambled nets）在高维不确定性量化（UQ）问题中实现与维度无关的收敛性，包括椭圆型PDE。

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 $O(n^{-1-α^*+\varepsilon})$ on standard UQ problems in elliptic partial differential equations (PDEs), where $\varepsilon>0$ is arbitrarily small and $α^*\in (0,1)$ 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.

研究动机与目标

• Motivate the challenge of QMC integration over unbounded domains in high dimensions for UQ tasks.
• Introduce boundary-damping importance sampling (BDIS) to enable off-the-shelf scrambled nets to be effective.
• Provide a rigorous dimension-independent MSE bound for BDIS-based scrambled nets under broad function classes.
• Extend the theory to weighted function spaces and parametric elliptic PDEs.
• Demonstrate numerical effectiveness on high-dimensional PDE problems.

提出的方法

• Formulate the integral over R^s with a product density and apply an importance sampling transformation via a transport map T and weight w to obtain f^w on the unit cube (Eq. 1.2).
• Use a transport map with independent components and a product-form weight w(u)=prod_j w_j(u_j) derived from w_theta and the inverse CDF Phi^{-1} (Eq. 1.3, 1.4).
• Dampen boundary growth of the integrand to enable faster RQMC convergence with scrambled nets (BDIS).
• Prove a dimension-independent MSE bound for scrambled net estimators with the BDIS transform for f in W^{1,q}_{mix}(R^s, varphi) with q>1 (Theorem 2.4).
• Extend the analysis to parametric PDEs and weighted Sobolev spaces (Sections 4 and 5).
• Provide corollaries showing nearly O(n^{-1- abla})-type rates under dimension-structure assumptions (Corollary 2.6).

(a) $H_{0}^{1}(D)$ Error

实验结果

研究问题

• RQ1 Can boundary-damping importance sampling transform unbounded integrands so scrambled net QMC achieves dimension-independent convergence rates?
• RQ2 What are the convergence bounds for BDIS-based scrambled nets for functions in W^{1,q}_{mix}(R^s, varphi) with q>1?
• RQ3 How does the dimension structure of parametric PDE inputs affect achievable MSE rates?
• RQ4 Can the analysis be extended to weighted function spaces and general random diffusion coefficients in elliptic PDEs?
• RQ5 What practical guidance on parameter choices (theta_j, t_omega) yields near-optimal high-dimensional performance?

主要发现

• BDIS-based scrambled nets achieve a dimension-independent MSE rate of O(n^{-(1+alpha^*+ - ε)}) for appropriate alpha^* in (0,1) and arbitrarily small ε>0.
• The method applies to f in W^{1,q}_{mix}(R^s, varphi) for any q>1, extending prior results that were limited to q=2 or q∈(1,2].
• A generalization to weighted function spaces bridges BDIS with lattice-rule approaches and shows dimension-independence under certain weight structures.
• For elliptic PDEs with high-dimensional parametric inputs, the estimator achieves the stated dimension-independent rate, reflecting the problem’s dimension structure.
• Corollaries show how t-quality parameters and ANOVA gamma norms influence the achievable rate, with practical guidance under common lattice constructions.
• Numerical experiments on elliptic PDEs with high-dimensional parameters validate the theoretical findings and demonstrate effectiveness.

(b) Pointwise Error at $\boldsymbol{x}_{c}$