Second order interlaced polynomial lattice rules for integration over $\mathbb{R}^s$

TL;DR

The paper develops a high-order quasi-Monte Carlo framework for integrating functions over $\mathbb{R}^s$ with product densities by mapping to $[0,1]^s$ via the inverse CDF and employing order-2 localized Walsh functions. It establishes sharp bounds on localized and classical Walsh coefficients under Assumption A, and proves that interlaced polynomial lattice rules of order two, constructed via a CBC algorithm, achieve convergence rates approaching $O(N^{-2+\delta})$ (with $\delta>0$ arbitrarily small) in a dimension-robust way under suitable weight structures. The authors apply this theory to elliptic PDEs with a finite number of log-normal coefficients, showing that, with importance sampling, the stochastic Galerkin discretization errors remain controllable and can reach near-quadratic convergence in the number of QMC points. Numerical experiments corroborate the theoretical rates across different dimensions and PDE settings, illustrating practical efficacy for uncertainty quantification in PDEs with log-normal randomness. These results advance high-dimensional, unbounded-domain integration by marrying localized basis functions with higher-order digit-interlacing lattice rules and CBC construction, enabling strong polynomial tractability in relevant applications.

Abstract

We study numerical integration of functions $f: \mathbb{R}^{s} \to \mathbb{R}$ with respect to a probability measure. By applying the corresponding inverse cumulative distribution function, the problem is transformed into integrating an induced function over the unit cube $(0,1)^{s}$. We introduce a new orthonormal system: \emph{order~2 localized Walsh functions}. These basis functions retain the approximation power of classical Walsh functions for twice-differentiable integrands while inheriting the spatial localization of Haar wavelets. Localization is crucial because the transformed integrand is typically unbounded at the boundary. We show that the worst-case quasi-Monte Carlo integration error decays like $\mathcal{O}(N^{-1/λ})$ for every $λ\in (1/2,1]$. As an application, we consider elliptic partial differential equations with a finite number of log-normal random coefficients and show that our error estimates remain valid for their stochastic Galerkin discretizations by applying a suitable importance sampling density.

Figures (5)

• Figure 1: Walsh function ${\rm wal}_{20}$
• Figure 2: Localized Walsh functions $w_{20,m,2}$ for $m \in L_{12,2}=\{0,1,2,3\}$
• Figure 3: QMC integration errors for integrating the quantity of interest $F_1(\omega)$ in the one-dimensional PDE example, shown as a function of QMC sample size for stochastic dimensions $s \in \{2^2, 2^3, \ldots, 2^6\}$. The integration errors and QMC sample sizes are reported on the base $2$ logarithmic scale. The blue and red triangles indicate reference slopes of $-1$ and $-2$, respectively.
• Figure 4: QMC integration errors for integrating the quantity of interest $F_1(\omega)$ in the one-dimensional PDE example, shown as a function of stochastic dimension for QMC sample sizes $N \in \{2^{17}, 2^{18}, \ldots, 2^{22}\}$. The integration errors and stochastic dimensions are reported on the base $2$ logarithmic scale. The blue and red triangles indicate reference slopes of $1$ and $2$, respectively.
• Figure 5: QMC integration errors for integrating the quantity of interest $F_2(\omega)$ in the two-dimensional PDE example, shown as a function of QMC sample size. The integration errors and QMC sample sizes are reported on the base $2$ logarithmic scale. The triangle indicates a reference slope of $-2$. The stochastic dimension is $s=32$ in this example.

Theorems & Definitions (35)

• Definition 1
• Definition 2: Walsh functions
• Lemma 1
• Definition 3: Localized Walsh functions
• Remark 1
• Definition 4
• Lemma 2
• Definition 5: Localized Walsh functions with respect to a PDF
• Lemma 3
• Lemma 4