### Chris Farmer (Mathematical Institute, University of Oxford)

#### A variational smoothing filter for sequential inverse problems

*Wednesday 7 November 2012 at 15.30, JCMB 5215 - Joint with ACM and NAIS*

##### Abstract

Uncertainty quantification can begin by specifying the initial state of a
system as a probability measure. Part of the state (the 'parameters') might
not evolve, and might not be directly observable. Many inverse problems are
generalisations of uncertainty quantification such that one modifies the
probability measure to be consistent with measurements, a forward model and
the initial measure. The main problem in the field is to devise a method for
computing the posterior probability measure of the states, including the
parameters and the variables, from a sequence of noise-corrupted observations.
Bayesian statistics provides a framework for this, but leads to very
challenging computational problems, particularly when the dimension of the
state space is very large, as with problems arising from the discretisation of
a partial differential equation theory.

In this talk we show how to motivate and implement a 'Variational Smoothing
Filter'. This is essentially a Bayesian wrapper for the sequential method
known as 'ensemble 4DVar' in which sequences of optimisation problems are
solved for an earlier state conditional on later observations. The algorithm
is outlined and the results of some numerical experiments on the often-used
'Lorenz 96' model, but with a generalisation that includes unknown parameters,
will be described. Such experiments indicate that the variational smoothing
filter can recover estimates of the posterior density even with an ensemble
consisting of as few as four members. However, the method is not without
error, as will be shown by a study of a linear problem that can be solved
exactly using a Kalman filter. Nevertheless, it will be argued that the
variational smoothing filter is worth further study from both theoretical and
computational points of view.

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