The Mean Square Quasi-Monte Carlo Error for Digitally Shifted Digital Nets
For researchers in numerical integration, this work provides a new quality measure to select digital nets that improve root mean square error convergence for smooth functions.
The paper derives an expression for the mean square quasi-Monte Carlo error of digitally shifted digital nets in terms of Walsh coefficients, and introduces a Walsh figure of merit for root mean square error that satisfies a Koksma-Hlawka type inequality. Experiments confirm that this quality measure helps find digital nets with good convergence for smooth integrands.
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.