Evaluating the mapping particle filter in high-dimensional state spaces
The use of Monte Carlo techniques to represent sequentially the state of a system given a set of noisy observations are a challenge for high-dimensional systems with nongaussian statistics. A novel particle filter is introduced which aims to an efficient sampling of high-dimensional state spaces considering a limited number of particles. The filter is based on variational importance sampling and optimal transport. Particles are mapped from the proposal to the posterior density using the principles of optimal transport. The Kullback-Leibler divergence between the posterior density and the proposal divergence is optimized using variational principles. A key ingredient of the mapping is that the transformations are embedded in a reproducing kernel Hilbert space which constrains the dimensions of the space for the optimal transport to the number of particles. In terms of Shannon entropy, the optimization seeks to maximize the amount of information, that is present in the samples of the proposal density, from the posterior density, i.e. to minimize the relative entropy (uncertainty) between the proposal and the target densities. Evaluation of the method is conducted in a 1000-variables Lorenz-96 system and a 1.5 layer quasi-geostrophic model with a resolution of 256x256 in which a preconditioning step in the varitional optimization based on 3DVar schemes is introduced. No resampling is required even for long recursive implementations. Hence, the mapping particle filter does not suffer from sample impoverishment.
Prof. Manuel Pulido University of Reading