Median QMC method for unbounded integrands over $\mathbb{R}^s$ in unanchored weighted Sobolev spaces

TL;DR

This work extends the construction-free median QMC approach to unanchored weighted Sobolev spaces on $\mathbb{R}^s$, enabling efficient high-dimensional integration of unbounded integrands via the inverse transform to $[0,1]^s$. By using the median of $k=\mathcal{O}(\log N)$ independent randomized lattice estimators, the authors prove a probabilistic error bound that yields a mean absolute error of $\mathcal{O}(N^{-r+\varepsilon})$ for any $\varepsilon\in(0,r-\tfrac{1}{2}]$, with $r>\tfrac{1}{2}$ determined by the function space, matching the rate of CBC-based randomized lattice rules without requiring prior weight knowledge. Theoretical results are complemented by numerical experiments on option pricing and PDEs with random coefficients, showing that the median QMC method achieves comparable accuracy to CBC and outperforms Monte Carlo, while avoiding the challenging weight-tuning step. Overall, the method offers a practical, construction-free alternative for high-dimensional, unbounded-domain integration with strong convergence guarantees. The results have potential impact on risk assessment, uncertainty quantification, and PDEs with random inputs where weight structures are difficult to specify in advance.

Abstract

This paper investigates quasi-Monte Carlo (QMC) integration of Lebesgue integrable functions with respect to a density function over $\mathbb{R}^s$. We extend the construction-free median QMC rule to the unanchored weighted Sobolev space of functions defined over $\mathbb{R}^s$. By taking the median of $k=\mathcal{O}(\log N)$ independent randomized QMC estimators, we prove that for any $ε\in(0,r-\frac{1}{2}]$, our method achieves a mean absolute error bound of $\mathcal{O}(N^{-r+ε})$, where $N$ is the number of points and $r>\frac{1}{2}$ is a parameter determined by the function space. This rate matches that of the randomized lattice rules via component-by-component (CBC) construction, while our approach requires no specific CBC constructions or prior knowledge of the space's weight structure. Numerical experiments demonstrate that our method attains accuracy comparable to the CBC method and outperforms the Monte Carlo method.

Figures (3)

• Figure 1: Histograms of the $\log_2$ of the shift-averaged worst-case error $e_{s,N}^{sh}(\bm{z})$ with $\phi(x)=\frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}},\psi_j(x) = e^{-\frac{|x|}{16}},\gamma_u = \prod_{j\in u }\gamma_j,$ and $\gamma_j = \frac{1}{j^2}$ for rank-1 lattice rules with randomly chosen generating vectors with $N = 257$ (upper panels) and $N = 2053$ (lower panels). The left panels are for a single choice ($k = 1$), while for the right panels we take the median of the shift-averaged worst-case error for rank-1 lattice rules with $k = 11$ randomly chosen generating vectors. The red dashed line in each histogram represents the shift-averaged worst-case error of the vector generated by the CBC method, and the green dashed line in each histogram represents the $90$th percentile.
• Figure 2: MAE convergence for the MC method, the randomly shifted lattice rule with the CBC algorithm, and the median QMC method for approximating the option value for $K = 110$ (top left), option value for $K = 90$ (bottom left), CDF of $\Bar{S}$ at $K=110$ (top right), and CDF of $\Bar{S}$ at $K = 90$ (bottom right).
• Figure 3: MAE convergence for the MC method, the randomly shifted lattice rule with the CBC algorithm, and the median QMC method for approximating the expectations for $x_0 = \frac{1}{3}$ (left) and for $x_0 = \frac{2}{3}$ (right).

Theorems & Definitions (17)

• Lemma 3.1
• Theorem 3.2
• Proof 1
• Corollary 3.3
• Proof 2
• Theorem 3.4
• Proof 3
• Theorem 3.5
• Corollary 3.6
• Proof 4