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Bayesian inverse problems for functions and applications to fluid mechanics
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Cotter, S. L., Dashti, M., Robinson, James C. and Stuart, A. M. (2009) Bayesian inverse problems for functions and applications to fluid mechanics. Inverse Problems, Vol.25 (No.11). Article no. 115008. doi:10.1088/02665611/25/11/115008
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Official URL: http://dx.doi.org/10.1088/02665611/25/11/115008
Abstract
In this paper we establish a mathematical framework for a range of inverse problems for functions, given a finite set of noisy observations. The problems are hence underdetermined and are often illposed. We study these problems from the viewpoint of Bayesian statistics, with the resulting posterior probability measure being defined on a space of functions. We develop an abstract framework for such problems which facilitates application of an infinitedimensional version of Bayes theorem, leads to a wellposedness result for the posterior measure (continuity in a suitable probability metric with respect to changes in data), and also leads to a theory for the existence of maximizing the posterior probability (MAP) estimators for such Bayesian inverse problems on function space. A central idea underlying these results is that continuity properties and bounds on the forward model guide the choice of the prior measure for the inverse problem, leading to the desired results on wellposedness and MAP estimators; the PDE analysis and probability theory required are thus clearly dileneated, allowing a straightforward derivation of results. We show that the abstract theory applies to some concrete applications of interest by studying problems arising from data assimilation in fluid mechanics. The objective is to make inference about the underlying velocity field, on the basis of either Eulerian or Lagrangian observations. We study problems without model error, in which case the inference is on the initial condition, and problems with model error in which case the inference is on the initial condition and on the driving noise process or, equivalently, on the entire timedependent velocity field. In order to undertake a relatively uncluttered mathematical analysis we consider the twodimensional NavierStokes equation on a torus. The case of Eulerian observationsdirect observations of the velocity field itselfis then a model for weather forecasting. The case of Lagrangian observationsobservations of passive tracers advected by the flowis then a model for data arising in oceanography. The methodology which we describe herein may be applied to many other inverse problems in which it is of interest to find, given observations, an infinitedimensional object, such as the initial condition for a PDE. A similar approach might be adopted, for example, to determine an appropriate mathematical setting for the inverse problem of determining an unknown tensor arising in a constitutive law for a PDE, given observations of the solution. The paper is structured so that the abstract theory can be read independently of the particular problems in fluid mechanics which are subsequently studied by application of the theory.
Item Type:  Journal Article  

Subjects:  Q Science > QA Mathematics Q Science > QC Physics 

Divisions:  Faculty of Science > Mathematics  
Journal or Publication Title:  Inverse Problems  
Publisher:  Institute of Physics Publishing Ltd.  
ISSN:  02665611  
Official Date:  November 2009  
Dates: 


Volume:  Vol.25  
Number:  No.11  
Number of Pages:  43  
Page Range:  Article no. 115008  
Identifier:  10.1088/02665611/25/11/115008  
Status:  Peer Reviewed  
Publication Status:  Published  
Access rights to Published version:  Restricted or Subscription Access  
Funder:  Engineering and Physical Sciences Research Council (EPSRC), ERC, ONR, Warwick Postgraduate Fellowship 
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