Tent transformed order $2$ nets and quasi-Monte Carlo rules with quadratic error decay

TL;DR

This paper demonstrates that tent transformed order 2 digital nets achieve quadratic error decay in quasi-Monte Carlo integration of nonperiodic functions with bounded mixed second derivatives, improving previous bounds and suggesting potential for even better performance.

Contribution

The authors establish a quadratic decay rate for tent transformed order 2 digital nets in nonperiodic settings, surpassing previous bounds and indicating lower complexity point sets can be highly effective.

Findings

Quadratic error decay rate achieved with tent transformed order 2 nets

Improved error bounds by a factor of log N over previous results

Numerical experiments suggest potential for further bound improvements

Abstract

We investigate the use of order $2$ digital nets for quasi-Monte Carlo quadrature of nonperiodic functions with bounded mixed second derivative over the cube. By using the so-called tent transform and its mapping properties we inherit error bounds from the periodic setting. Our analysis is based on decay properties of the multivariate Faber-Schauder coefficients of functions with bounded mixed second weak derivatives. As already observed by Hinrichs, Markhasin, Oettershagen, T. Ullrich (Numerische Mathematik 2016), order $2$ nets work particularly well on tensorized (periodic) Faber splines. From this we obtain a quadratic decay rate for tent transformed order $2$ nets also in the nonperiodic setting. This improves the formerly best known bound for this class of point sets by a factor of $\log N$. We back up our findings with numerical experiments, even suggesting that the bounds for order $2$ nets can be improved even further. This particularly indicates that point sets of lower complexity (compared to previously considered constructions) may already give (near) optimal error decay rates for quadrature of functions with second order mixed smoothness.