{
  "id": "1611.02785",
  "version": "v1",
  "published": "2016-11-09T00:48:19.000Z",
  "updated": "2016-11-09T00:48:19.000Z",
  "title": "Numerical Integration over the Unit Sphere by using spherical t-design",
  "authors": [
    "Congpei An",
    "Siyong Chen"
  ],
  "categories": [
    "math.NA"
  ],
  "abstract": "This paper studies numerical integration over the unit sphere $ \\mathbb{S}^2 \\subset \\mathbb{R}^{3} $ by using spherical $t$-design, which is an equal positive weights quadrature rule with polynomial precision $t$. We investigate two kinds of spherical $t$-designs with $t$ up to 160. One is well conditioned spherical $t$-design(WSTD), which was proposed by [1] with $ N=(t+1)^{2} $. The other is efficient spherical $t$-design(ESTD), given by Womersley [2], which is made of roughly of half cardinality of WSTD. Consequently, a series of persuasive numerical evidences indicates that WSTD is better than ESTD in the sense of worst-case error in Sobolev space $ \\mathbb{H}^{s}(\\mathbb{S}^2) $. Furthermore, WSTD is employed to approximate integrals of various of functions, especially including integrand has a point singularity over the unit sphere and a given ellipsoid. In particular, to deal with singularity of integrand, Atkinson's transformation [3] and Sidi's transformation [4] are implemented with the choices of `grading parameters' to obtain new integrand which is much smoother. Finally, the paper presents numerical results on uniform errors for approximating representive integrals over sphere with three quadrature rules: Bivariate trapezoidal rule, Equal area points and WSTD.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-11-09T00:48:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65D30"
    ],
    "keywords": [
      "unit sphere",
      "spherical t-design",
      "equal positive weights quadrature rule",
      "bivariate trapezoidal rule",
      "paper studies numerical integration"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}