A note on unshifted lattice rules for high-dimensional integration in weighted unanchored Sobolev spaces
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866908328919040000 |
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| author | Goda, Takashi |
| author_facet | Goda, Takashi |
| contents | This short article studies a deterministic quasi-Monte Carlo lattice rule in weighted unanchored Sobolev spaces of smoothness $1$. Building on the error analysis by Kazashi and Sloan, we prove the existence of unshifted rank-1 lattice rules that achieve a worst-case error of $O(n^{-1/4}(\log n)^{1/2})$, with the implied constant independent of the dimension, under certain summability conditions on the weights. Although this convergence rate is inferior to the one achievable for the shifted-averaged root mean squared worst-case error, the result does not rely on random shifting or transformation and holds unconditionally without any conjecture, as assumed by Kazashi and Sloan. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_14768 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A note on unshifted lattice rules for high-dimensional integration in weighted unanchored Sobolev spaces Goda, Takashi Numerical Analysis This short article studies a deterministic quasi-Monte Carlo lattice rule in weighted unanchored Sobolev spaces of smoothness $1$. Building on the error analysis by Kazashi and Sloan, we prove the existence of unshifted rank-1 lattice rules that achieve a worst-case error of $O(n^{-1/4}(\log n)^{1/2})$, with the implied constant independent of the dimension, under certain summability conditions on the weights. Although this convergence rate is inferior to the one achievable for the shifted-averaged root mean squared worst-case error, the result does not rely on random shifting or transformation and holds unconditionally without any conjecture, as assumed by Kazashi and Sloan. |
| title | A note on unshifted lattice rules for high-dimensional integration in weighted unanchored Sobolev spaces |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2504.14768 |