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

Handle URI:
http://hdl.handle.net/10754/563763
Title:
Well-posedness and accuracy of the ensemble Kalman filter in discrete and continuous time
Authors:
Kelly, D. T B; Law, Kody; Stuart, Andrew M.
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.
KAUST Department:
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Publisher:
IOP Publishing
Journal:
Nonlinearity
Issue Date:
22-Sep-2014
DOI:
10.1088/0951-7715/27/10/2579
Type:
Article
ISSN:
09517715
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.
Appears in Collections:
Articles; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

Full metadata record

DC FieldValue Language
dc.contributor.authorKelly, D. T Ben
dc.contributor.authorLaw, Kodyen
dc.contributor.authorStuart, Andrew M.en
dc.date.accessioned2015-08-03T12:09:19Zen
dc.date.available2015-08-03T12:09:19Zen
dc.date.issued2014-09-22en
dc.identifier.issn09517715en
dc.identifier.doi10.1088/0951-7715/27/10/2579en
dc.identifier.urihttp://hdl.handle.net/10754/563763en
dc.description.abstractThe 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.en
dc.description.sponsorshipThe 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.en
dc.publisherIOP Publishingen
dc.subjectdata assimilationen
dc.subjectensemble Kalman filteren
dc.subjectstochastic analysisen
dc.titleWell-posedness and accuracy of the ensemble Kalman filter in discrete and continuous timeen
dc.typeArticleen
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Divisionen
dc.identifier.journalNonlinearityen
dc.contributor.institutionMathematics Institute, University of WarwickCoventry, United Kingdomen
kaust.authorLaw, Kodyen
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