{
  "id": "2007.01541",
  "version": "v1",
  "published": "2020-07-03T08:02:34.000Z",
  "updated": "2020-07-03T08:02:34.000Z",
  "title": "A fast direct solver for nonlocal operators in wavelet coordinates",
  "authors": [
    "Helmut Harbrecht",
    "Michael Multerer"
  ],
  "categories": [
    "math.NA",
    "cs.NA"
  ],
  "abstract": "In this article, we consider fast direct solvers for nonlocal operators. The pivotal idea is to combine a wavelet representation of the system matrix, yielding a quasi-sparse matrix, with the nested dissection ordering scheme. The latter drastically reduces the fill-in during the factorization of the system matrix by means of a Cholesky decomposition or an LU decomposition, respectively. This way, we end up with the exact inverse of the compressed system matrix with only a moderate increase of the number of nonzero entries in the matrix. To illustrate the efficacy of the approach, we conduct numerical experiments for different highly relevant applications of nonlocal operators: We consider (i) the direct solution of boundary integral equations in three spatial dimensions, issuing from the polarizable continuum model, (ii) a parabolic problem for the fractional Laplacian in integral form and (iii) the fast simulation of Gaussian random fields.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-07-03T08:02:34.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "fast direct solver",
      "nonlocal operators",
      "wavelet coordinates",
      "boundary integral equations",
      "gaussian random fields"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}