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

摘要

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