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                Department of Mathematical Sciences                       
                   The Johns Hopkins University                           
                                                                        
                         SPECIAL SEMINAR
                                                                        
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Gregory L. Eyink                                 FRIDAY, December 8, 2000
Department of Mathematics                        1:00 p.m.
University of Arizona                            109 MARYLAND HALL

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         A VARIATIONAL APPROACH TO STOCHASTIC ESTIMATION
           FOR LARGE-SCALE NONLINEAR DYNAMICAL SYSTEMS

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                             ABSTRACT

There are many practical problems that require an estimate of a solution 
of a stochastic dynamical system in the future, the present, or the past 
based upon some partial and imperfect measurements on the system.  These 
include assimilation of model states to satellite data in numerical weather 
forecasting, recovering the release history of a groundwater contamination 
from observations of the plume in a few wells, and retrodicting past
climate based upon geological field data, such as ocean silt deposits.  Not
only is a state estimate required, but generally also needed is an
assessment of the uncertainty in that estimate.  The optimal solution is to
provide conditional statistics given the available data, such as the
conditional mean and variance.  There is a mathematical formalism for
calculating such statistics, but it is numerically intractable to apply to
the large-scale systems of interest.  The problem is formally equivalent to
the "closure problem" of fluid turbulence, a notoriously difficult one. 
We discuss a variational approach to the problem of approximating the
conditional statistics, based upon a Rayleigh-Ritz computation of a
variational cost function, the "effective action."  The conditional mean
is given as the minimizer of this convex function and the variance as its
Hessian.  This approach allows one to exploit moment-closure methods that
have become traditional in the turbulence field (and also nontraditional
closure ideas, based upon PDF models or statistical surrogates).  We give a
simple example of a toy model for climate change, which is known to be a
hard test case for the most advanced data assimilation techniques currently
employed.  Some ongoing work on more realistic hydrological applications
will also be decribed.

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