{
  "id": "2008.08103",
  "version": "v1",
  "published": "2020-08-18T18:01:06.000Z",
  "updated": "2020-08-18T18:01:06.000Z",
  "title": "On the Numerical Solution of Nonlinear Eigenvalue Problems for the Monge-Ampère Operator",
  "authors": [
    "Roland Glowinski",
    "Shingyu Leung",
    "Hao Liu",
    "Jianliang Qian"
  ],
  "categories": [
    "math.NA",
    "cs.NA"
  ],
  "abstract": "In this article, we report the results we obtained when investigating the numerical solution of some nonlinear eigenvalue problems for the Monge-Amp\\`{e}re operator $v\\rightarrow \\det \\mathbf{D}^2 v$. The methodology we employ relies on the following ingredients: (i) A divergence formulation of the eigenvalue problems under consideration. (ii) The time discretization by operator-splitting of an initial value problem (a kind of gradient flow) associated with each eigenvalue problem. (iii) A finite element approximation relying on spaces of continuous piecewise affine functions. To validate the above methodology, we applied it to the solution of problems with known exact solutions: The results we obtained suggest convergence to the exact solution when the space discretization step $h\\rightarrow 0$. We considered also test problems with no known exact solutions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-08-18T18:01:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J60",
      "65N25",
      "65N30"
    ],
    "keywords": [
      "nonlinear eigenvalue problems",
      "numerical solution",
      "monge-ampère operator",
      "exact solution",
      "space discretization step"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}