Integrability of weak mixed first-order derivatives and convergence rates of scrambled digital nets

TL;DR

The paper addresses how the variance of scrambled digital nets decays when the integrand has weak mixed first-order derivatives in $L^p$, connecting to a generalized Vitali variation with parameter $\alpha \in [\tfrac{1}{2},1]$. It proves that $V_{\alpha}(f)$ is bounded by the $L^p$ norm of the weak mixed derivative with $p = \frac{2}{3-2\alpha}$, establishing variance control for the scrambled net estimator. Consequently, for $1\le p\le 2$, the variance decays as $\mathcal{O}(N^{-4+\frac{2}{p}} \log^{s-1} N)$, tying convergence rates directly to weak derivative integrability. The results generalize prior regularity conditions and are supported by numerical experiments validating the theoretical rates.

Abstract

We consider the $L^p$ integrability of weak mixed first-order derivatives of the integrand and study convergence rates of scrambled digital nets. We show that the generalized Vitali variation with parameter $α\in [\frac{1}{2}, 1]$ from [Dick and Pillichshammer, 2010] is bounded above by the $L^p$ norm of the weak mixed first-order derivative, where $p = \frac{2}{3-2α}$. Consequently, when the weak mixed first-order derivative belongs to $L^p$ for $1 \leq p \leq 2$, the variance of the scrambled digital nets estimator convergences at a rate of $\mathcal{O}(N^{-4+\frac{2}{p}} \log^{s-1} N)$. Numerical experiments further validate the theoretical results.

Figures (2)

• Figure 1: Example 1: The boxplot characterization of squared error distributions, $(I_N - I)^2$, for scrambled Sobol' sequence estimators across various dimensions $s$ and parameters $\alpha$. Each whisker in the boxplot extends from the 1st to 99th percentile of 8,192 independent realizations of the squared errors.
• Figure 2: Example 2: The boxplot characterization of squared error distributions, $(I_N - I)^2$, for scrambled Sobol' sequence estimators across various dimensions $s$ and parameters $\alpha$. The reference value $I$ is approximated by averaging 8,192 independent realizations of scrambled Sobol' sequence estimators with quadrature size $N = 2^{25}$. Each whisker in the boxplot extends from the 1st to 99th percentile of 8,192 independent realizations.

Theorems & Definitions (5)

• Definition 1: Generalized Vitali variation of order 2
• Proposition 1: Variance of the scrambled $(t,m,s)$-net in base $b$ estimator
• Lemma 1: Alternating sum and the weak derivative
• proof
• Remark 1: On the convergence rate of the scrambled digital nets