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- W4286981609 abstract "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 $L_gamma(M)$-average of the geodesic distance to the point set and determine the value of $gammain (0,infty]$. This extends our findings on bounded convex domains [arXiv:2009.11275, 2020]. 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 $gamma<infty$. 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, Graf and Oates [Stat. Comput., 29:1203-1214, 2019]." @default.
- W4286981609 created "2022-07-25" @default.
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- W4286981609 date "2021-09-09" @default.
- W4286981609 modified "2023-10-16" @default.
- W4286981609 title "Function recovery on manifolds using scattered data" @default.
- W4286981609 doi "https://doi.org/10.48550/arxiv.2109.04106" @default.
- W4286981609 hasPublicationYear "2021" @default.
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