{
  "id": "1611.07239",
  "version": "v1",
  "published": "2016-11-22T10:35:57.000Z",
  "updated": "2016-11-22T10:35:57.000Z",
  "title": "Convergence of Sparse Collocation for Functions of Countably Many Gaussian Random Variables - with Application to Lognormal Elliptic Diffusion Problems",
  "authors": [
    "Oliver G. Ernst",
    "Björn Sprungk",
    "Lorenzo Tamellini"
  ],
  "comment": "23 pages, 2 Figures",
  "categories": [
    "math.NA"
  ],
  "abstract": "We give a convergence proof of sparse collocation to approximate Hilbert space-valued functions depending on countably many Gaussian random variables. Such functions appear as solutions or quantities of interest associated with elliptic PDEs with lognormal diffusion coefficients. We outline a general $L^2$-convergence theory based on previous work by Bachmayr et al. (2016) and Chen (2016) and establish an algebraic convergence rate for sufficiently smooth functions assuming a mild growth bound for the univariate hierarchical surpluses of the interpolation scheme applied to Hermite polynomials. We verify specifically for Gauss-Hermite nodes that this assumption holds and also show algebraic convergence w.r.t. the resulting number of sparse grid points for this case.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-11-22T10:35:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65D05",
      "65D15",
      "65C30"
    ],
    "keywords": [
      "lognormal elliptic diffusion problems",
      "gaussian random variables",
      "sparse collocation",
      "application",
      "algebraic convergence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}