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Nested QMC designs on spheres — arXiv:2609.15767

Source: arXiv (en)
Original: https://arxiv.org/abs/2609.15767
License (as archived): CC0-1.0
Archived: 2026-10-16T01:13:56.030Z
SHA-256 of the archived text: feeb8b61ba07320bba8d2e71e4eb50efc027fc337fe10c8cf86b37899e40978d
Stable URL of this page: https://metraipolis.com/documento/dc_03cf61da2001

How to cite

Nested QMC designs on spheres — arXiv:2609.15767 — arXiv (en), https://arxiv.org/abs/2609.15767 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_03cf61da2001 (Metraipolis archive dc_03cf61da2001, sha256 feeb8b61ba07320bba8d2e71e4eb50efc027fc337fe10c8cf86b37899e40978d)

BibTeX

@misc{dc03cf61da2001,
  title = {Nested QMC designs on spheres — arXiv:2609.15767},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.15767},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_03cf61da2001; sha256 feeb8b61ba07320bba8d2e71e4eb50efc027fc337fe10c8cf86b37899e40978d},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.15767. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

Nested cubature rules, in which each refinement retains all previously used nodes and thus reuses earlier function evaluations, are natural in multilevel and adaptive integration. For every fixed $s>d/2$, we show that the equal-weight QMC integration rate on $\mathbb S^d$ is compatible with such nested point sets. In the subcritical range $d/2<s<d$, cumulative unions of geometrically growing QMC blocks yield nested QMC design sequences whose successive cardinality ratios converge to any prescribed $\rho>1$. At and above the critical index $s=d$, where the block-averaging estimate no longer yields the optimal rate, we prove an equal-weight completion theorem based on low-frequency discrepancy cancellation. A block-sensitive estimate sharpens the iteration and yields nested QMC designs for all $s\ge d$ with $N_{j+1}\lesssim(j+1)^{2s/d-1}N_j$. Every prescribed finite point set also admits an optimal-rate completion.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-09-15 in math.ST).
Categorie arXiv: Numerical Analysis (math.NA) ; Classical Analysis and ODEs (math.CA); Statistics Theory (math.ST).
Pagina del lavoro: https://arxiv.org/abs/2609.15767 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-16T00:31:08.684Z dalla pagina https://arxiv.org/catchup/math.ST/2026-09-15?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).