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    Well-posedness and accuracy of the ensemble Kalman filter in discrete and continuous time

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    Type
    Article
    Authors
    Kelly, D. T B
    Law, Kody
    Stuart, Andrew M.
    KAUST Department
    Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
    Date
    2014-09-22
    Online Publication Date
    2014-09-22
    Print Publication Date
    2014-10-01
    Permanent link to this record
    http://hdl.handle.net/10754/563763
    
    Metadata
    Show full item record
    Abstract
    The ensemble Kalman filter (EnKF) is a method for combining a dynamical model with data in a sequential fashion. Despite its widespread use, there has been little analysis of its theoretical properties. Many of the algorithmic innovations associated with the filter, which are required to make a useable algorithm in practice, are derived in an ad hoc fashion. The aim of this paper is to initiate the development of a systematic analysis of the EnKF, in particular to do so for small ensemble size. The perspective is to view the method as a state estimator, and not as an algorithm which approximates the true filtering distribution. The perturbed observation version of the algorithm is studied, without and with variance inflation. Without variance inflation well-posedness of the filter is established; with variance inflation accuracy of the filter, with respect to the true signal underlying the data, is established. The algorithm is considered in discrete time, and also for a continuous time limit arising when observations are frequent and subject to large noise. The underlying dynamical model, and assumptions about it, is sufficiently general to include the Lorenz '63 and '96 models, together with the incompressible Navier-Stokes equation on a two-dimensional torus. The analysis is limited to the case of complete observation of the signal with additive white noise. Numerical results are presented for the Navier-Stokes equation on a two-dimensional torus for both complete and partial observations of the signal with additive white noise.
    Citation
    Kelly, D. T. B., Law, K. J. H., & Stuart, A. M. (2014). Well-posedness and accuracy of the ensemble Kalman filter in discrete and continuous time. Nonlinearity, 27(10), 2579–2603. doi:10.1088/0951-7715/27/10/2579
    Sponsors
    The authors are grateful to A J Majda for helpful discussions concerning this work. DTBK is supported by ONR grant N00014-12-1-0257. The work of AMS is supported by ERC, EPSRC, ESA and ONR. KJHL was supported by the King Abdullah University of Science and Technology (KAUST) and is a member of the KAUST Strategic Research Initiative Center for Uncertainty Quantification.
    Publisher
    IOP Publishing
    Journal
    Nonlinearity
    DOI
    10.1088/0951-7715/27/10/2579
    arXiv
    1310.3167
    ae974a485f413a2113503eed53cd6c53
    10.1088/0951-7715/27/10/2579
    Scopus Count
    Collections
    Articles; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

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