{
  "id": "1906.00810",
  "version": "v1",
  "published": "2019-06-03T13:46:43.000Z",
  "updated": "2019-06-03T13:46:43.000Z",
  "title": "PBDW method for state estimation: error analysis for noisy data and nonlinear formulation",
  "authors": [
    "Helin Gong",
    "Yvon Maday",
    "Olga Mula",
    "Tommaso Taddei"
  ],
  "categories": [
    "math.NA"
  ],
  "abstract": "We present an error analysis and further numerical investigations of the Parameterized-Background Data-Weak (PBDW) formulation to variational Data Assimilation (state estimation), proposed in [Y Maday, AT Patera, JD Penn, M Yano, Int J Numer Meth Eng, 102(5), 933-965]. The PBDW algorithm is a state estimation method involving reduced models. It aims at approximating an unknown function $u^{\\rm true}$ living in a high-dimensional Hilbert space from $M$ measurement observations given in the form $y_m = \\ell_m(u^{\\rm true}),\\, m=1,\\dots,M$, where $\\ell_m$ are linear functionals. The method approximates $u^{\\rm true}$ with $\\hat{u} = \\hat{z} + \\hat{\\eta}$. The \\emph{background} $\\hat{z}$ belongs to an $N$-dimensional linear space $\\mathcal{Z}_N$ built from reduced modelling of a parameterized mathematical model, and the \\emph{update} $\\hat{\\eta}$ belongs to the space $\\mathcal{U}_M$ spanned by the Riesz representers of $(\\ell_1,\\dots, \\ell_M)$. When the measurements are noisy {--- i.e., $y_m = \\ell_m(u^{\\rm true})+\\epsilon_m$ with $\\epsilon_m$ being a noise term --- } the classical PBDW formulation is not robust in the sense that, if $N$ increases, the reconstruction accuracy degrades. In this paper, we propose to address this issue with an extension of the classical formulation, {which consists in} searching for the background $\\hat{z}$ either on the whole $\\mathcal{Z}_N$ in the noise-free case, or on a well-chosen subset $\\mathcal{K}_N \\subset \\mathcal{Z}_N$ in presence of noise. The restriction to $\\mathcal{K}_N$ makes the reconstruction be nonlinear and is the key to make the algorithm significantly more robust against noise. We {further} present an \\emph{a priori} error and stability analysis, and we illustrate the efficiency of the approach on several numerical examples.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-06-03T13:46:43.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "error analysis",
      "pbdw method",
      "nonlinear formulation",
      "noisy data",
      "reconstruction accuracy degrades"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}