A higher order lattice-based method for high-dimensional numerical integration using periodization
Abstract
We present a novel numerical integration method for non-periodic functions over the high-dimensional unit cube [0,1]s by a specially crafted (unequally-)weighted quadrature rule using transformed lattice points, with the additional option of subsampling. The method is designed for integrands whose mixed derivatives up to order α are square-integrable with respect to a product Chebyshev density. The ingredients are (i) periodizing the integrand, (ii) approximating the smooth periodic function by a kernel interpolant at lattice points, (iii) integrating exactly the product of the kernel interpolant and the nonsmooth density, together with subsampling alternatives for (ii) and~(iii). With |J| corresponding to the number of function evaluations, we achieve the convergence rate close to the order |J|−(α−1/2) and |J|−α (1.110721)s, where the implied constants are independent of s under favorable conditions. We can interpolate these two results to trade between the convergence rate and the growth in s. We provide numerical experiments to demonstrate our theory.
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BibTeX
@misc{bartel2026higher,
title = {{A higher order lattice-based method for high-dimensional numerical integration using periodization}},
author = {Felix Bartel and Alexander D. Gilbert and Michael Griebel and Frances Y. Kuo and Ian H. Sloan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01113},
eprint = {2610.01113},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01113},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - A higher order lattice-based method for high-dimensional numerical integration using periodization
AU - Felix Bartel
AU - Alexander D. Gilbert
AU - Michael Griebel
AU - Frances Y. Kuo
AU - Ian H. Sloan
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.01113
UR - https://arxiv.org/abs/2610.01113
AB - We present a novel numerical integration method for non-periodic functions over the high-dimensional unit cube [0,1]^s by a specially crafted (unequally-)weighted quadrature rule using transformed lattice points, with the additional option of subsampling. The method is designed for integrands whose mixed derivatives up to order α are square-integrable with respect to a product Chebyshev density. The ingredients are (i) periodizing the integrand, (ii) approximating the smooth periodic function by a kernel interpolant at lattice points, (iii) integrating exactly the product of the kernel interpolant and the nonsmooth density, together with subsampling alternatives for (ii) and~(iii). With |J| corresponding to the number of function evaluations, we achieve the convergence rate close to the order |J|^−(α−1/2) and |J|^−α (1.110721)^s, where the implied constants are independent of s under favorable conditions. We can interpolate these two results to trade between the convergence rate and the growth in s. We provide numerical experiments to demonstrate our theory.
ER -