{
  "id": "1411.6510",
  "version": "v1",
  "published": "2014-11-24T16:17:22.000Z",
  "updated": "2014-11-24T16:17:22.000Z",
  "title": "Long-time Asymptotics of the Filtering Distribution for Partially Observed Chaotic Dynamical Systems",
  "authors": [
    "D. Sanz-Alonso",
    "A. M. Stuart"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "The filtering distribution is a time-evolving probability distribution on the state of a dynamical system, given noisy observations. We study the large-time asymptotics of this probability distribution for discrete-time, randomly initialized signals that evolve according to a deterministic map $\\Psi$. The observations are assumed to comprise a low-dimensional projection of the signal, given by an operator $P$, subject to additive noise. We address the question of whether these observations contain sufficient information to accurately reconstruct the signal. In a general framework, we establish conditions on $\\Psi$ and $P$ under which the filtering distributions concentrate around the signal in the small-noise, long-time asymptotic regime. Linear systems, the Lorenz '63 and '96 models, and the Navier Stokes equation on a two-dimensional torus are within the scope of the theory. Our main findings come as a by-product of computable bounds, of independent interest, for suboptimal filters based on new variants of the 3DVAR filtering algorithm.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-11-24T16:17:22.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "chaotic dynamical systems",
      "filtering distribution",
      "observations contain sufficient information",
      "long-time asymptotic regime",
      "navier stokes equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1411.6510S"
    }
  }
}