{
  "id": "1809.09890",
  "version": "v1",
  "published": "2018-09-26T10:25:41.000Z",
  "updated": "2018-09-26T10:25:41.000Z",
  "title": "Optimal confidence for Monte Carlo integration of smooth functions",
  "authors": [
    "Robert J. Kunsch",
    "Daniel Rudolf"
  ],
  "comment": "23 pages",
  "categories": [
    "math.NA"
  ],
  "abstract": "We study the complexity of approximating integrals of smooth functions at absolute precision $\\varepsilon > 0$ with confidence level $1 - \\delta \\in (0,1)$. The optimal error rate for multivariate functions from classical isotropic Sobolev spaces $W_p^r(G)$ with sufficient smoothness on bounded Lipschitz domains $G \\subset \\mathbb{R}^d$ is determined. It turns out that the integrability index $p$ has an effect on the influence of the uncertainty $\\delta$ in the complexity. In the limiting case $p = 1$ we see that deterministic methods cannot be improved by randomization. In general, higher smoothness reduces the additional effort for diminishing the uncertainty. Finally, we add a discussion about this problem for function spaces with mixed smoothness.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-09-26T10:25:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65C05",
      "65D30",
      "65J05"
    ],
    "keywords": [
      "monte carlo integration",
      "smooth functions",
      "optimal confidence",
      "classical isotropic sobolev spaces",
      "optimal error rate"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}