{
  "id": "1308.3883",
  "version": "v1",
  "published": "2013-08-18T18:22:18.000Z",
  "updated": "2013-08-18T18:22:18.000Z",
  "title": "Efficient sum-of-exponentials approximations for the heat kernel and their applications",
  "authors": [
    "Shidong Jiang",
    "Leslie Greengard",
    "Shaobo Wang"
  ],
  "comment": "23 pages, 5 figures, 3 tables",
  "categories": [
    "math.NA"
  ],
  "abstract": "In this paper, we show that efficient separated sum-of-exponentials approximations can be constructed for the heat kernel in any dimension. In one space dimension, the heat kernel admits an approximation involving a number of terms that is of the order $O(\\log(\\frac{T}{\\delta}) (\\log(\\frac{1}{\\epsilon})+\\log\\log(\\frac{T}{\\delta})))$ for any $x\\in\\bbR$ and $\\delta \\leq t \\leq T$, where $\\epsilon$ is the desired precision. In all higher dimensions, the corresponding heat kernel admits an approximation involving only $O(\\log^2(\\frac{T}{\\delta}))$ terms for fixed accuracy $\\epsilon$. These approximations can be used to accelerate integral equation-based methods for boundary value problems governed by the heat equation in complex geometry. The resulting algorithms are nearly optimal. For $N_S$ points in the spatial discretization and $N_T$ time steps, the cost is $O(N_S N_T \\log^2 \\frac{T}{\\delta})$ in terms of both memory and CPU time for fixed accuracy $\\epsilon$. The algorithms can be parallelized in a straightforward manner. Several numerical examples are presented to illustrate the accuracy and stability of these approximations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-08-18T18:22:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "30E15",
      "35K05",
      "35K08",
      "45D05",
      "65E05",
      "65R10",
      "80A20"
    ],
    "keywords": [
      "efficient sum-of-exponentials approximations",
      "applications",
      "corresponding heat kernel admits",
      "fixed accuracy",
      "accelerate integral equation-based methods"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1308.3883J"
    }
  }
}