A note on unshifted lattice rules for high-dimensional integration in weighted unanchored Sobolev spaces
Year: 2025
Type: preprint
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.
Source: arXiv (Cornell University)
Author: Goda, Takashi
Language: English
Subfield: Numerical Analysis
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W6979330302
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