{
  "id": "2109.04106",
  "version": "v1",
  "published": "2021-09-09T08:58:34.000Z",
  "updated": "2021-09-09T08:58:34.000Z",
  "title": "Function recovery on manifolds using scattered data",
  "authors": [
    "David Krieg",
    "Mathias Sonnleitner"
  ],
  "categories": [
    "math.NA",
    "cs.NA"
  ],
  "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 $\\gamma\\in (0,\\infty]$. 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, Gr\\\"af and Oates [Stat. Comput., 29:1203-1214, 2019].",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-09-09T08:58:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "41A55",
      "65D05",
      "41A25",
      "41A63",
      "58C35",
      "65D15"
    ],
    "keywords": [
      "function recovery",
      "scattered data",
      "logarithmic gap left open",
      "finite point set",
      "optimal cubature formulas"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}