{
  "id": "1301.0237",
  "version": "v2",
  "published": "2013-01-02T14:13:39.000Z",
  "updated": "2014-04-03T15:42:43.000Z",
  "title": "Sampling and reconstruction of solutions to the Helmholtz equation",
  "authors": [
    "Gilles Chardon",
    "Albert Cohen",
    "Laurent Daudet"
  ],
  "categories": [
    "math.NA"
  ],
  "abstract": "We consider the inverse problem of reconstructing general solutions to the Helmholtz equation on some domain $\\Omega$ from their values at scattered points $x_1,\\dots,x_n\\subset \\Omega$. This problem typically arises when sampling acoustic fields with $n$ microphones for the purpose of reconstructing this field over a region of interest $\\Omega$ contained in a larger domain $D$ in which the acoustic field propagates. In many applied settings, the shape of $D$ and the boundary conditions on its border are unknown. Our reconstruction method is based on the approximation of a general solution $u$ by linear combinations of Fourier-Bessel functions or plane waves. We analyze the convergence of the least-squares estimates to $u$ using these families of functions based on the samples $(u(x_i))_{i=1,\\dots,n}$. Our analysis describes the amount of regularization needed to guarantee the convergence of the least squares estimate towards $u$, in terms of a condition that depends on the dimension of the approximation subspace, the sample size $n$ and the distribution of the samples. It reveals the advantage of using non-uniform distributions that have more points on the boundary of $\\Omega$. Numerical illustrations show that our approach compares favorably with reconstruction methods using other basis functions, and other types of regularization.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-04-03T15:42:43.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "helmholtz equation",
      "reconstruction method",
      "acoustic field propagates",
      "linear combinations",
      "fourier-bessel functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}