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Determining white noise forcing from Eulerian observations in the Navier-Stokes equation

Hoang, Viet Ha and Law, Kody J. H. and Stuart, Andrew M. (2014) Determining white noise forcing from Eulerian observations in the Navier-Stokes equation. Stochastic Partial Differential Equations: Analysis and Computations, 2 (2). pp. 233-261. ISSN 2194-041X. doi:10.1007/s40072-014-0028-4.

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The Bayesian approach to inverse problems is of paramount importance in quantifying uncertainty about the input to, and the state of, a system of interest given noisy observations. Herein we consider the forward problem of the forced 2D Navier-Stokes equation. The inverse problem is to make inference concerning the forcing, and possibly the initial condition, given noisy observations of the velocity field. We place a prior on the forcing which is in the form of a spatially-correlated and temporally-white Gaussian process, and formulate the inverse problem for the posterior distribution. Given appropriate spatial regularity conditions, we show that the solution is a continuous function of the forcing. Hence, for appropriately chosen spatial regularity in the prior, the posterior distribution on the forcing is absolutely continuous with respect to the prior and is hence well-defined. Furthermore, it may then be shown that the posterior distribution is a continuous function of the data. We complement these theoretical results with numerical simulations showing the feasibility of computing the posterior distribution, and illustrating its properties.

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Additional Information:© Springer Science+Business Media New York 2014. Received: 19 March 2013 / Published online: 29 April 2014. VHH gratefully acknowledges the financial support of the AcRF Tier 1 grant RG69/10. AMS is grateful to EPSRC, ERC, ESA and ONR for financial support for this work. KJHL is grateful to the financial support of the ESA and is currently a member of the King Abdullah University of Science and Technology (KAUST) Strategic Research Initiative (SRI) Center for Uncertainty Quantification in Computational Science.
Funding AgencyGrant Number
Ministry of Education (Singapore)RG69/10
Engineering and Physical Sciences Research Council (EPSRC)UNSPECIFIED
European Research Council (ERC)UNSPECIFIED
European Space Agency (ESA)UNSPECIFIED
Office of Naval Research (ONR)UNSPECIFIED
King Abdullah University of Science and Technology (KAUST)UNSPECIFIED
Subject Keywords:Bayesian inversion, Navier-Stokes equation, White noise forcing
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Other Numbering System NameOther Numbering System ID
Andrew StuartJ110
Issue or Number:2
Record Number:CaltechAUTHORS:20160719-150657732
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Official Citation:Hoang, V.H., Law, K.J.H. & Stuart, A.M. Stoch PDE: Anal Comp (2014) 2: 233. doi:10.1007/s40072-014-0028-4
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:69118
Deposited By: Linda Taddeo
Deposited On:20 Jul 2016 17:02
Last Modified:11 Nov 2021 04:09

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