Optimal Order Quadrature Error Bounds for Infinite-Dimensional Higher-Order Digital Sequences

Abstract

Quasi-Monte Carlo (QMC) quadrature rules using higher order digital nets and sequences have been shown to achieve the almost optimal rate of convergence of the worst-case error in Sobolev spaces of arbitrary fixed smoothness $\alpha\in \mathbb{N}$, $\alpha\geq 2$. In a recent paper by the authors, it was proved that randomly-digitally-shifted order $2\alpha$ digital nets in prime base $b$ achieve the best possible rate of convergence of the root mean square worst-case error of order $N^{-\alpha}(\log N)^{(s-1)/2}$ for $N=b^m$, where $N$ and $s$ denote the number of points and the dimension, respectively, which implies the existence of an optimal order QMC rule. More recently, the authors provided an explicit construction of such an optimal order QMC rule by using Chen-Skriganov's digital nets in conjunction with Dick's digit interlacing composition. These results were for fixed number of points. In this paper we give a more general result on an explicit construction of optimal order QMC rules for arbitrary fixed smoothness $\alpha\in \mathbb{N}$ including the endpoint case $\alpha=1$. That is, we prove that the projection of any infinite-dimensional order $2\alpha +1$ digital sequence in prime base $b$ onto the first $s$ coordinates achieves the best possible rate of convergence of the worst-case error of order $N^{-\alpha}(\log N)^{(s-1)/2}$ for $N=b^m$. The explicit construction presented in this paper is not only easy to implement but also extensible in both $N$ and $s$.