Digital nets with infinite digit expansions and construction of folded digital nets for quasi-Monte Carlo integration

TL;DR

This paper introduces folded digital nets with infinite digit expansions for quasi-Monte Carlo integration, analyzing their error and constructing optimal rules using higher order polynomial lattice point sets.

Contribution

It extends digital net theory to infinite digit expansions and develops a construction method for folded higher order polynomial lattice rules with optimal convergence.

Findings

Folded digital nets improve quasi-Monte Carlo integration accuracy.

Component-by-component construction finds optimal folded higher order polynomial lattice rules.

Achieves the best possible convergence rates in certain Sobolev spaces.

Abstract

In this paper we study quasi-Monte Carlo integration of smooth functions using digital nets. We fold digital nets over $\mathbb{Z}_{b}$ by means of the $b$-adic tent transformation, which has recently been introduced by the authors, and employ such \emph{folded digital nets} as quadrature points. We first analyze the worst-case error of quasi-Monte Carlo rules using folded digital nets in reproducing kernel Hilbert spaces. Here we need to permit digital nets with "infinite digit expansions," which are beyond the scope of the classical definition of digital nets. We overcome this issue by considering the infinite product of cyclic groups and the characters on it. We then give an explicit means of constructing good folded digital nets as follows: we use higher order polynomial lattice point sets for digital nets and show that the component-by-component construction can find good \emph{folded higher order polynomial lattice rules} that achieve the optimal convergence rate of the worst-case error in certain Sobolev spaces of smoothness of arbitrarily high order.