An explicit construction of optimal order quasi-Monte Carlo rules for smooth integrands

TL;DR

This paper presents an explicit construction of optimal order quasi-Monte Carlo rules that achieve the best convergence rates for smooth integrands, leveraging Walsh coefficient decay and sparsity.

Contribution

It provides the first explicit QMC rule construction that attains optimal convergence rates for smooth functions in a reproducing kernel Hilbert space.

Findings

Achieves the best possible convergence rate for smooth integrands.

Utilizes Walsh coefficient decay and sparsity for construction.

First explicit QMC rule with optimal convergence in this setting.

Abstract

In a recent paper by the authors, it is shown that there exists a quasi-Monte Carlo (QMC) rule which achieves the best possible rate of convergence for numerical integration in a reproducing kernel Hilbert space consisting of smooth functions. In this paper we provide an explicit construction of such an optimal order QMC rule. Our approach is to exploit both the decay and the sparsity of the Walsh coefficients of the reproducing kernel simultaneously. This can be done by applying digit interlacing composition due to Dick to digital nets with large minimum Hamming and Niederreiter-Rosenbloom-Tsfasman metrics due to Chen and Skriganov. To our best knowledge, our construction gives the first QMC rule which achieves the best possible convergence in this function space.