A gentle introduction to approximating high-dimensional integrals with lattice rules and a maybe-not-so-gentle introduction to the relevant function spaces for QMC integration
Abstract
These lecture notes give both a practical and theoretical introduction to high-dimensional integration with lattice rules. We introduce lattice points and rules in combination with accompanying Julia code. Next we introduce the typical function spaces used in the literature. We compare classical Sobolev spaces with Sobolev spaces of dominating mixed smoothness. We derive lower bounds and show that the curse of dimensionality is unavoidable for these unweighted function spaces. Next we switch to weighted periodic spaces and derive a constructive upper bound using the component-by-component construction for the generating vector of a lattice rule. We pick periodic spaces because they are a perfect match with lattice rules. The upper bound shows that under suitable conditions on the weights we obtain dimension-independent worst-case error bounds. We then use embedding constants to transfer these bounds to some common norms for these periodic spaces.
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BibTeX
@misc{nuyens2026gentle,
title = {{A gentle introduction to approximating high-dimensional integrals with lattice rules and a maybe-not-so-gentle introduction to the relevant function spaces for QMC integration}},
author = {Dirk Nuyens},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.25331},
eprint = {2610.25331},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.25331},
}
RIS
TY - UNPB
TI - A gentle introduction to approximating high-dimensional integrals with lattice rules and a maybe-not-so-gentle introduction to the relevant function spaces for QMC integration
AU - Dirk Nuyens
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2610.25331
UR - https://arxiv.org/abs/2610.25331
AB - These lecture notes give both a practical and theoretical introduction to high-dimensional integration with lattice rules. We introduce lattice points and rules in combination with accompanying Julia code. Next we introduce the typical function spaces used in the literature. We compare classical Sobolev spaces with Sobolev spaces of dominating mixed smoothness. We derive lower bounds and show that the curse of dimensionality is unavoidable for these unweighted function spaces. Next we switch to weighted periodic spaces and derive a constructive upper bound using the component-by-component construction for the generating vector of a lattice rule. We pick periodic spaces because they are a perfect match with lattice rules. The upper bound shows that under suitable conditions on the weights we obtain dimension-independent worst-case error bounds. We then use embedding constants to transfer these bounds to some common norms for these periodic spaces.
ER -