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

Takashi Goda

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arXiv · 2504.14768

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

Takashi Goda

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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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Takashi Goda. 2025-04-20. A note on unshifted lattice rules for high-dimensional integration in weighted unanchored Sobolev spaces. https://arxiv.org/abs/2504.14768

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