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

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.