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

2025/02/04 by Junyang Liu, Liu, Yang · 2 citations
Computer Science · #65C05 #Cellular Automata and Applications #Digital Image Processing Techniques #FOS: Mathematics #Numerical Analysis (math.NA) #Petri Nets in System Modeling

paper · pdf · doi:10.48550/arxiv.2502.02266

openalex publication_date 2025/02/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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∈ [(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 = (2)/(3-2α). Consequently, when the weak mixed first-order derivative belongs to L^p for 1 ≤ p ≤ 2, the variance of the scrambled digital nets estimator convergences at a rate of O(N^-4+(2)/(p) log^s-1 N). Numerical experiments further validate the theoretical results.

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