A mapping particle filter for high-dimensional applications

Particle filters have long had the promise of efficient fully nonlinear data-assimilation methods for high-dimensional systems with the recognition that proposal transition densities could be chosen much more widely than previously thought. This has led to a class of so-called equal-weight particle filters that try to change the particle positions in one time step, exploiting this proposal freedom. However, trying to move particles in one time step to the right positions can be suboptimal, and can easily break model balances.

Transport particle filters have been around for a long time. They try to convert prior particles at observation times into samples from the posterior in a continuous flow in artificial time. So for each particle a PDE is solved over an artificial-time window, and the particles must interact by construction, as together they should represent the posterior. If successful the particles are proper samples from the posterior, and hence all have equal weights.

Efficient ways to implement these filters encounter the problem that the fastest path towards the posterior depends on the flow itself, leading to an optimisation problem in an infinite dimensional function space that is hard to solve. A way forward has been to approximate the prior and posterior pdf's, e.g. to Gaussian mixtures, and efficient schemes have been developed for those problems. But, of course, many problems are not of that form, typically those with nonlinear relations between observations and model states.

A way forward has recently be advocated in which only the flow is of a restricted class, while the prior and posterior pdf's are completely general, for steady state problems, by Liu and Wang, by embedding the flow fields in a Reproducing Kernel Hilbert Space. We have extended that methodology to sequential problems, and in this talk we will discuss an applications to high-dimensional geophysical systems, the baotropic vorticity equation and a two-layer quasi-geostrophic model. We will discuss how to implement the method in an efficient way, and discuss pro's and con's.