Invited Talk. Combining variational data assimilation and particle filters: the variational mapping particle filter

PULIDO M

Vienna

Conferencia; European Geophysical Conference; 2019

European Geophysical Union

Recent works in the machine learning community have started to combine two classical statistical concepts:Monte Carlo sampling and variational inference. In the traditional variational inference, including variational dataassimilation, some parameters of a proposed posterior density are estimated through maximazing the marginallikelihood or via maximum a posteriori estimation. The idea for combining them is to use as optimizationparameters in variational inference the Monte Carlo sample points, i.e. the particles. In this way, we seek for aset of particles that best represent the posterior density by, for instance, maximazing the marginal likelihood.Following this idea, we introduce a novel particle filter that is based on local optimal transport principles. Theminimization of the Kullback-Leibler divergence between the intermediate density represented by the particles andthe posterior density is conducted by a sequence of local maps. The transformations are embedded in a reproducingkernel Hilbert space which defines the steepest descent directions. The particles are required to follow thesedirections in order to minimize the Kullback-Leibler divergence or equivalently maximize the marginal likelihood.The optimization can be interpreted as a flow in which the particles are active tracers, they are moved along thesteepest descent directions, i.e. streamlines of the flow, but at the same time they define the flow. Evaluation ofthe variational mapping particle filter will be shown with experiments using a 1000-variables Lorenz-96 systemand a 1.5 layer quasi-geostrophic model with a resolution of 256x256 in which a preconditioning step in thevariational optimization based on 3DVar schemes is introduced. We show that the deterministic mappings avoidthe resampling step, i.e. the number of effective particles remains close to the total number of particles even forlong recursive implementations. This variational particle filter framework inherits the well-known convergenceproperties and efficient implementations of optimization algorithms in high-dimensional state spaces. It shedssome light on the representation of sequential Monte Carlo methods in high-dimensional state-spaces. Thechallenges for further developments and implementation f the variational mapping particle filter will be discussed