The Mean Square Quasi-Monte Carlo Error for Digitally Shifted Digital Nets

TL;DR

This paper analyzes the mean square error in randomized quasi-Monte Carlo integration using digitally shifted digital nets, expressing it via Walsh coefficients and introducing a quality measure to predict convergence for smooth functions.

Contribution

It introduces a Walsh figure of merit for root mean square error and demonstrates its effectiveness in selecting digital nets with good convergence properties.

Findings

Walsh figure of merit correlates with convergence behavior

Quality measure effectively predicts root mean square error

Experiments confirm usefulness for smooth integrands

Abstract

In this paper, we study randomized quasi-Monte Carlo (QMC) integration using digitally shifted digital nets. We express the mean square QMC error of the $n$-th discrete approximation $f_n$ of a function $f\colon[0,1)^s\to \mathbb{R}$ for digitally shifted digital nets in terms of the Walsh coefficients of $f$. We then apply a bound on the Walsh coefficients for sufficiently smooth integrands to obtain a quality measure called Walsh figure of merit for root mean square error, which satisfies a Koksma-Hlawka type inequality on the root mean square error. Through two types of experiments, we confirm that our quality measure is of use for finding digital nets which show good convergence behaviors of the root mean square error for smooth integrands.