A note on unshifted lattice rules for high-dimensional integration in weighted unanchored Sobolev spaces

2025/04/20 by Goda, Takashi
#FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.2504.14768

Abstract

This short article studies a deterministic quasi-Monte Carlo lattice rule in weighted unanchored Sobolev spaces of smoothness 1. Building on the error analysis by Kazashi and Sloan, we prove the existence of unshifted rank-1 lattice rules that achieve a worst-case error of O(n^-1/4(log n)^1/2), with the implied constant independent of the dimension, under certain summability conditions on the weights. Although this convergence rate is inferior to the one achievable for the shifted-averaged root mean squared worst-case error, the result does not rely on random shifting or transformation and holds unconditionally without any conjecture, as assumed by Kazashi and Sloan.

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