Function recovery on manifolds using scattered data
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866917779965214720 |
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| author | Krieg, David Sonnleitner, Mathias |
| author_facet | Krieg, David Sonnleitner, Mathias |
| contents | We consider the task of recovering a Sobolev function on a connected compact Riemannian manifold $M$ when given a sample on a finite point set. We prove that the quality of the sample is given by the $L_γ(M)$-average of the geodesic distance to the point set and determine the value of $γ\in (0,\infty]$. This extends our findings on bounded convex domains [IMA J. Numer. Anal., 44:1346--1371, 2024]. As a byproduct, we prove the optimal rate of convergence of the $n$-th minimal worst case error for $L_q(M)$-approximation for all $1\le q \le \infty$.
Further, a limit theorem for moments of the average distance to a set consisting of i.i.d.\ uniform points is proven. This yields that a random sample is asymptotically as good as an optimal sample in precisely those cases with $γ<\infty$. In particular, we obtain that cubature formulas with random nodes are asymptotically as good as optimal cubature formulas if the weights are chosen correctly. This closes a logarithmic gap left open by Ehler, Gräf and Oates [Stat. Comput., 29:1203-1214, 2019]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2109_04106 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Function recovery on manifolds using scattered data Krieg, David Sonnleitner, Mathias Numerical Analysis 41A55, 65D05 (Primary) 41A25, 41A63, 58C35, 65D15 (Secondary) We consider the task of recovering a Sobolev function on a connected compact Riemannian manifold $M$ when given a sample on a finite point set. We prove that the quality of the sample is given by the $L_γ(M)$-average of the geodesic distance to the point set and determine the value of $γ\in (0,\infty]$. This extends our findings on bounded convex domains [IMA J. Numer. Anal., 44:1346--1371, 2024]. As a byproduct, we prove the optimal rate of convergence of the $n$-th minimal worst case error for $L_q(M)$-approximation for all $1\le q \le \infty$. Further, a limit theorem for moments of the average distance to a set consisting of i.i.d.\ uniform points is proven. This yields that a random sample is asymptotically as good as an optimal sample in precisely those cases with $γ<\infty$. In particular, we obtain that cubature formulas with random nodes are asymptotically as good as optimal cubature formulas if the weights are chosen correctly. This closes a logarithmic gap left open by Ehler, Gräf and Oates [Stat. Comput., 29:1203-1214, 2019]. |
| title | Function recovery on manifolds using scattered data |
| topic | Numerical Analysis 41A55, 65D05 (Primary) 41A25, 41A63, 58C35, 65D15 (Secondary) |
| url | https://arxiv.org/abs/2109.04106 |