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

TL;DR

This work addresses high-dimensional numerical integration over $[0,1)^d$ in weighted unanchored Sobolev spaces of smoothness 1 using unshifted rank-1 lattice rules. Building on Kazashi and Sloan's framework, it proves the existence of generating vectors $oldsymbol{z}$ yielding a worst-case error $e(n,oldsymbol{z}) = O(n^{-1/4} (\, ext{log} n)^{1/2})$ with constants independent of dimension under summability of the weights, without relying on random shifting or conjectures. It additionally shows KS18's conjecture is false and provides a weaker, dimension-robust bound on the mean-square worst-case error $ar{e}^2(n) = O(n^{-1/2} \, ext{log} n)$, which implies the stated unshifted lattice-rule guarantee. For product weights, dimension independence requires $ ext{sum}_{j=1}^\infty eta_j < \,\infty$, establishing practical criteria for weight selection in high dimensions.

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.

Theorems & Definitions (10)

• Proposition 1: Kazashi and Sloan KS18
• Conjecture 1: Kazashi and Sloan KS18
• Theorem 1
• proof
• Lemma 1
• Remark 1
• proof : Proof of Lemma \ref{['lem']}
• Theorem 2
• proof
• Remark 2