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arXiv · 2610.25331

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

Dirk Nuyens
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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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Dirk Nuyens. 2026-10-18. 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. https://arxiv.org/abs/2610.25331

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