{
  "id": "1910.05261",
  "version": "v1",
  "published": "2019-10-11T15:53:51.000Z",
  "updated": "2019-10-11T15:53:51.000Z",
  "title": "Finite element approximation of Lyapunov equations for the computation of quadratic functionals of SPDEs",
  "authors": [
    "Adam Andersson",
    "Annika Lang",
    "Andreas Petersson",
    "Leander Schroer"
  ],
  "comment": "44 pages, 1 figure",
  "categories": [
    "math.NA",
    "cs.NA",
    "math.PR"
  ],
  "abstract": "The computation of quadratic functionals of the solution to a linear stochastic partial differential equation with multiplicative noise is considered. An operator valued Lyapunov equation, whose solution admits a deterministic representation of the functional, is used for this purpose and error estimates are shown in suitable operator norms for a fully discrete approximation of this equation. Weak error rates are also derived for a fully discrete approximation of the stochastic partial differential equation, using the results obtained from the approximation of the Lyapunov equation. In the setting of finite element approximations, a computational complexity comparison reveals that approximating the Lyapunov equation allows for cheaper computation of quadratic functionals compared to applying Monte Carlo or covariance-based methods directly to the discretized stochastic partial differential equation. Numerical simulations illustrates the theoretical results.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-10-11T15:53:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65M60",
      "60H15",
      "65M12",
      "60H35",
      "65C30",
      "49J20"
    ],
    "keywords": [
      "finite element approximation",
      "lyapunov equation",
      "quadratic functionals",
      "computation",
      "fully discrete approximation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 44,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}