An ensemble variational filter for sequential inverse problems - Farmer, C (IOP Publishing Ltd, 2015)
Given a model dynamical system, a model of any measuring instrument relating states to measurements, and a prior assessment of uncertainty, the probability density of subsequent system states, conditioned upon the history of the measurements, is of some practical interest. When measurements are made at discrete times, it is known that the evolving probability density is a solution of the discrete Bayesian filtering equations. This paper describes the di!culties in approximating the evolving probability density using a Gaussian mixture (i.e. a sum of Gaussian densities). In general this leads to a sequence of optimisation problems and high-dimensional integrals. Attention is given to the necessity of using a small number of densities in the mixture, the requirement to maintain sparsity of any matrices and the need to compute first and second derivatives of the misfit between predictions and measurements. Adjoint methods, Taylor expansions, Gaussian random fields and Newton’s method can be combined to, possibly, provide a solution.