{
  "id": "1304.0329",
  "version": "v1",
  "published": "2013-04-01T10:57:48.000Z",
  "updated": "2013-04-01T10:57:48.000Z",
  "title": "Explicit constructions of quasi-Monte Carlo rules for the numerical integration of high dimensional periodic functions",
  "authors": [
    "Josef Dick"
  ],
  "journal": "J. Dick, Explicit constructions of quasi-Monte Carlo rules for the numerical integration of high dimensional periodic functions. SIAM J. Numer. Anal., 45, 2141--2176, 2007",
  "doi": "10.1137/060658916",
  "categories": [
    "math.NA"
  ],
  "abstract": "In this paper we give explicit constructions of point sets in the $s$ dimensional unit cube yielding quasi-Monte Carlo algorithms which achieve the optimal rate of convergence of the worst-case error for numerically integrating high dimensional periodic functions. In the classical measure $P_{\\alpha}$ of the worst-case error introduced by Korobov the convergence is of $\\landau(N^{-\\min(\\alpha,d)} (\\log N)^{s\\alpha-2})$ for every even integer $\\alpha \\ge 1$, where $d$ is a parameter of the construction which can be chosen arbitrarily large and $N$ is the number of quadrature points. This convergence rate is known to be best possible up to some $\\log N$ factors. We prove the result for the deterministic and also a randomized setting. The construction is based on a suitable extension of digital $(t,m,s)$-nets over the finite field $\\integer_b$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-04-01T10:57:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11K38",
      "11K45",
      "65C05",
      "65D30",
      "65D32"
    ],
    "keywords": [
      "high dimensional periodic functions",
      "quasi-monte carlo rules",
      "explicit constructions",
      "unit cube yielding quasi-monte",
      "integrating high dimensional periodic"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1304.0329D"
    }
  }
}