{
  "id": "1604.06008",
  "version": "v1",
  "published": "2016-04-20T15:46:29.000Z",
  "updated": "2016-04-20T15:46:29.000Z",
  "title": "A Monte Carlo method for integration of multivariate smooth functions I: Sobolev spaces",
  "authors": [
    "Mario Ullrich"
  ],
  "comment": "17 pages",
  "categories": [
    "math.NA"
  ],
  "abstract": "We study a Monte Carlo algorithm that is based on a specific (randomly shifted and dilated) lattice point set. The main result of this paper is that the mean squared error for a given compactly supported, square-integrable function is bounded by $n^{-1/2}$ times the $L_2$-norm of the Fourier transform outside a region around the origin, where $n$ is the expected number of function evaluations. As corollaries we obtain the order of convergence for the Sobolev spaces $H^s_p$ with isotropic, anisotropic or mixed smoothness for all values of the parameters. This proves, in particular, that the optimal order of convergence in the latter case is $n^{-s-1/2}$ for $p\\ge2$, which is, in contrast to the case of deterministic algorithms, independent of the dimension. This shows that Monte Carlo algorithms can improve the order by more than $n^{-1/2}$ for a whole class of practically important function classes. All results carry over to functions defined on the unit cube without boundary conditions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-04-20T15:46:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65D30",
      "65C05",
      "68Q25",
      "46E35",
      "42B10"
    ],
    "keywords": [
      "monte carlo method",
      "multivariate smooth functions",
      "sobolev spaces",
      "monte carlo algorithm",
      "integration"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160406008U"
    }
  }
}