Optimal quasi-Monte Carlo rules on order 2 digital nets for the numerical integration of multivariate periodic functions

TL;DR

This paper demonstrates that order 2 digital nets achieve optimal convergence rates for multivariate periodic functions in Besov spaces, improving previous bounds and showing independence from integrability parameters.

Contribution

It establishes the optimal convergence rate of order 2 digital nets for Besov space integration, extending known results and providing numerical evidence for the case r=2.

Findings

Order 2 digital nets achieve the optimal rate N^{-r} with a specific logarithmic factor.

The convergence rate is independent of the integrability parameter p.

Numerical computations suggest the bound holds for r=2.

Abstract

We investigate quasi-Monte Carlo rules for the numerical integration of multivariate periodic functions from Besov spaces $S^r_{p,q}B(\mathbb{T}^d)$ with dominating mixed smoothness $1/p<r<2$. We show that order 2 digital nets achieve the optimal rate of convergence $N^{-r} (\log N)^{(d-1)(1-1/q)}$. The logarithmic term does not depend on $r$ and hence improves the known bound provided by J. Dick for the special case of Sobolev spaces $H^r_{\text{mix}}(\mathbb{T}^d)$. Secondly, the rate of convergence is independent of the integrability $p$ of the Besov space, which allows for sacrificing integrability in order to gain Besov regularity. Our method combines characterizations of periodic Besov spaces with dominating mixed smoothness via Faber bases with sharp estimates of Haar coefficients for the discrepancy function of higher order digital nets. Moreover, we provide numerical computations which indicate that this bound also holds for the case $r=2$.