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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