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PreprintarXiv licencearXiv

The Quality of Lattice Sequences for Arbitrary Sample Size

  • Larysa Matiukha
  • Yuhan Ding
  • Fred J. Hickernell
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Abstract

Extensible lattices are a common choice of nodes for approximating multidimensional integrals by a sample mean, typically using sample sizes of the form n = bm for a prime base b and non-negative integer m. However, a computational time budget or early termination due to hardware failure may prevent us from choosing the preferred number of samples. In this paper, we establish an upper bound on the figure of merit Pα for extensible lattice sequences \emph{with arbitrary n}, derived as a worst-case error in a Banach space setting. We show that although the upper bound on Pα can decay no faster than 𝒪(n−1) for general n, restricting n to the form n =λbp, where λ is a fixed integer, we can recover the optimal decay of nearly 𝒪(n−α) as p → ∞. We also investigate a related figure of merit, Pα,2, arising as a worst-case error for Hilbert spaces of integrands. We obtain a theoretical upper bound on Pα,2 in terms of Pα for equal sample weights and show numerically that optimal sample weights can improve the cubature error bound even further.

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Cite

BibTeXRIS
BibTeX
@misc{matiukha2026quality,
  title = {{The Quality of Lattice Sequences for Arbitrary Sample Size}},
  author = {Larysa Matiukha and Yuhan Ding and Fred J. Hickernell},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.10543},
  eprint = {2610.10543},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.10543},
}
RIS
TY  - UNPB
TI  - The Quality of Lattice Sequences for Arbitrary Sample Size
AU  - Larysa Matiukha
AU  - Yuhan Ding
AU  - Fred J. Hickernell
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2610.10543
UR  - https://arxiv.org/abs/2610.10543
AB  - Extensible lattices are a common choice of nodes for approximating multidimensional integrals by a sample mean, typically using sample sizes of the form n = b^m for a prime base b and non-negative integer m. However, a computational time budget or early termination due to hardware failure may prevent us from choosing the preferred number of samples. In this paper, we establish an upper bound on the figure of merit P_α for extensible lattice sequences \emph{with arbitrary n}, derived as a worst-case error in a Banach space setting. We show that although the upper bound on P_α can decay no faster than 𝒪(n^−1) for general n, restricting n to the form n =λb^p, where λ is a fixed integer, we can recover the optimal decay of nearly 𝒪(n^−α) as p → ∞. We also investigate a related figure of merit, P_α,2, arising as a worst-case error for Hilbert spaces of integrands. We obtain a theoretical upper bound on P_α,2 in terms of P_α for equal sample weights and show numerically that optimal sample weights can improve the cubature error bound even further.
ER  -