Function recovery on manifolds using scattered data

09/09/2021

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by   David Krieg, et al.

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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∈ (0,∞]. 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 γ<∞. 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].