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

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 proposed by Goda and L'ecuyer (SIAM J. Sci. Comput., 2022) to the weighted unanchored Sobolev space of functions defined over $\mathbb{R}^s$ introduced by Nichols and Kuo (J. Complexity, 2014). 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 the rate of randomly shifted lattice rules obtained via a 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 an accuracy comparable to the CBC construction based method, and outperforms the Monte Carlo method.