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PreprintCC BY 4.0arXiv

Random nodes with positive weights: Richter-Tchakaloff rules and optimal positive quadrature

  • Sebastian Neumayer
  • Martin Schäfer
  • Tino Ullrich
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Abstract

We show that independent random points support exact quadrature rules with positive weights. For any n-dimensional space V of integrable functions a constant multiple of nlog n points sampled from a suitable density yields, with high probability, positive weights which integrate the functions in the space exactly. The weights stay close to the natural importance sampling weights according to the Lewis density with respect to V. They are the result of a weight adaptation procedure which requires an explicit L1-norm discretization result that is proved on the way. As an application, we study numerical integration in reproducing kernel Hilbert spaces. Combining the random rules with a subsampling step, we show that positive quadrature rules with constantly oversampled nodes achieve the best possible worst-case error bound in terms of the singular numbers of the embedding into L2.

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Cite

BibTeXRIS
BibTeX
@misc{neumayer2026random,
  title = {{Random nodes with positive weights: Richter-Tchakaloff rules and optimal positive quadrature}},
  author = {Sebastian Neumayer and Martin Schäfer and Tino Ullrich},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.32699},
  eprint = {2610.32699},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.32699},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Random nodes with positive weights: Richter-Tchakaloff rules and optimal positive quadrature
AU  - Sebastian Neumayer
AU  - Martin Schäfer
AU  - Tino Ullrich
PY  - 2026
DA  - 2026/10/23
PB  - arXiv
DO  - 10.48550/arxiv.2610.32699
UR  - https://arxiv.org/abs/2610.32699
AB  - We show that independent random points support exact quadrature rules with positive weights. For any n-dimensional space V of integrable functions a constant multiple of nlog n points sampled from a suitable density yields, with high probability, positive weights which integrate the functions in the space exactly. The weights stay close to the natural importance sampling weights according to the Lewis density with respect to V. They are the result of a weight adaptation procedure which requires an explicit L_1-norm discretization result that is proved on the way. As an application, we study numerical integration in reproducing kernel Hilbert spaces. Combining the random rules with a subsampling step, we show that positive quadrature rules with constantly oversampled nodes achieve the best possible worst-case error bound in terms of the singular numbers of the embedding into L_2.
ER  -