Digital nets and sequences constructed over finite rings and their application to quasi-Monte Carlo integration
The concept of a \((t,m,s)\) net (for this notion see the excellent monograph [\textit{H. Niederreiter}, Random number generation and quasi-Monte Carlo methods, SIAM (1992; Zbl 0761.65002)]) and of \((t,s)\)-sequences is of fundamental importance for high-dimensional quasi-Monte Carlo integration, in theory as well as in practice. It allows for the construction of so-called ``low discrepancy'' point sets, which serve as the nodes at which the integrand is computed. Until now, all construction methods for \((t,m,s)\) nets of practical relevance are digital methods over certain rings \(R\). That means that digit expansions in some integer base \(b\geq 2\) are used to define the net. In this paper, the authors present a much more general version of this approach to numerical integration based on digital nets over arbitrary finite rings. They provide an error bound for integrands with rapidly converging Walsh series. Further important results of this paper concern the existence of digital nets and of digital sequences constructed over arbitrary rings. In particular, necessary and sufficient conditions for the existence of digital \((0,m,s)\)-nets and of digital \((0,s)\)-sequences are exhibited. In the final section, the authors give explicit constructions of digital \((t,m,s)\)-nets and digital \((t,s)\)-sequences in the case \(R=\mathbb{Z}_b\).
high-dimensional quasi-Monte Carlo integration
scientific article; zbMATH DE number 874082
scientific article