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

TL;DR

This work studies high-dimensional quasi-Monte Carlo integration for nonperiodic functions with bounded mixed second derivatives on $[0,1]^d$, using a tent transform in conjunction with order-2 digital nets to achieve quadratic convergence. The authors establish a key decay bound for Faber-Schauder coefficients and show that the tent transform maps $H_{\text{mix}}^2([0,1]^d)$ into dyadic Korobov-type spaces, enabling a worst-case error bound of $e^{\text{QMC}}(X, H_{\text{mix}}^2([0,1]^d)) \lesssim 2^{t} N^{-2} (\log N)^{2d-1}$ for tent-transformed order-2 nets. Numerical experiments corroborate the theory and suggest that lower-complexity point sets may already achieve near-optimal rates, highlighting practical benefits in constructing effective nets. Overall, the results improve the known bounds for this class of point sets by a factor of $\log N$ and illuminate the role of the tent transform in bridging nonperiodic and periodic function spaces for QMC quadrature.

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.

Figures (8)

• Figure 1: Univariate hierarchical Faber functions on $[0,1]$ for levels $j\in \{0,1\}$ and their union.
• Figure 2: The Faber basis functions of level $j = -1$ in the periodic and nonperiodic setting.
• Figure 3: Behavior of the worst case error for tent transformed order $2$ nets in dimension $d=3$, measured for the kernels $K_j^3$. We compare this to the error for untransformed order $2$ nets, using the kernel $K_1^3$.
• Figure 4: Dimension $d=4, 5, 6, 7$, kernel $K_1^d$.
• Figure 5: The considered point sets for $d=2$. From left to right: Fibonacci ($N = 89$), Halton ($N=64$), Zaremba ($N = 64$). Top row: untransformed ($Y \subseteq \mathbb{T}^2$). Bottom row: tent transformed ($X = \mathcal{R} Y \subseteq [0, 1]^2$). Some points are colored red to show how they map under the tent transform.

Theorems & Definitions (17)

• Definition 3.1
• Definition 3.2
• Theorem 3.3
• Proposition 3.4
• Remark 3.5
• proof : Proof of Proposition \ref{['prop:Faber']}
• Remark 3.6
• Theorem 4.1
• proof : Proof of Theorem \ref{['thm:tent']}
• Definition 5.1