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SUMMARY:The Bayesian Approach to Inverse Problems - Professor Andrew Stuar
 t\, University of Warwick Mathematics Institute
DTSTART:20140423T130000Z
DTEND:20140423T170000Z
UID:TALK51986@talks.cam.ac.uk
CONTACT:CCA
DESCRIPTION:Probabilistic thinking is of growing importance in many areas 
 of mathematics. This short course will demonstrate the beautiful mathemati
 cal framework\,\ncoupled with practical algorithms\, which results from th
 inking probabilistically about inverse problems arising in partial differe
 ntial equations.\n\nMany inverse problems in the physical sciences require
  the determination of an unknown field from a finite set of indirect measu
 rements. Examples include oceanography\, oil recovery\, water resource man
 agement and weather forecasting. In the Bayesian approach to these problem
 s\, the unknown and the data are modelled as a jointly varying random vari
 able\, and the solution of the inverse problem is the distribution of the 
 un- known given the data.\n\nThis approach provides a natural way to provi
 de estimates of the unknown field\, together with a quantification of the 
 uncertainty associated with\nthe estimate. It is hence a useful practical 
 modelling tool. However it also provides a very elegant mathematical frame
 work for inverse problems:\nwhilst the classical approach to inverse probl
 ems leads to ill-posedness\, the Bayesian approach leads to a natural well
 -posedness and stability theory. Furthermore this framework provides a way
  of deriving and developing algorithms which are well suited to the formid
 able computational challenges which arise from the conjunction of approxim
 ations arising from the numerical analysis of partial differential equatio
 ns\, together with approximations of central limit theorem type arising fr
 om sampling of measures.\n\nThe tools in mathematical analysis which you w
 ill be exposed to during the course lie at the intersection of probability
  and analysis\, and include Gaussian measures on Hilbert space\, metrics o
 n probability\nmeasures\, conditional probability and regularity estimates
  for elliptic PDEs and for the Navier-Stokes equation. 
LOCATION:MR5
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