Monday, March 30 2020

11:00am - 12:00pm

11:00am - 12:00pm

Computational Mathematics Colloquium

Particle Flow-based Bayesian Inference for high-dimensional geophysical problems

Bayesian Inference is the science of how to optimally combine existing information encoded in computational models with information in observations of the modeled system. Many systems in the geosciences are highly nonlinear, asking for fully nonlinear Bayesian Inference. Particle filters are one of the few fully nonlinear methods that seem to be feasible to serve that goal. The vanilla particle filters are highly sensitive to the likelihood, and the particle ensemble size needed for accurate results growths roughly exponential with the number of independent observations. Localization methods for particle filters have been developed since the early 90ties and have undergone much refinement, but fundamental problems remain. These problems are related to the fact that even with localization the local areas contain too many observations to avoid degeneracy, and creating smooth posterior particles via gluing local particles together remains troublesome (although some remarkable successes have been booked recently) Equal-weight particle filters have been developed, exploring e.g. ideas from synchronization, but up to now only their first and second moments can be made unbiased.

An older development that has recently gained new attention are particle flows. The basic idea is to iteratively move all particles from being samples from the prior to equal-weight samples from the posterior. This motion of all particles through state space is defined via iteratively decreasing the distance between the actual particle density and the posterior density. Many methods to achieve this have been proposed and will be discussed briefly. Then we will describe a solution based on minimizing the relative entropy. By restricting the transport map to a Reproducing Kernel Hilbert Space a practical solution is found.

When applying this to geophysical problems two issues arise: how to accurately represent the prior and how to choose the kernel covariance. These two problems are related, and we will demonstrate practical methods to solve this problem, based on iterative refinement of the kernel covariances. The behaviour of the new methodology will be investigated using toy problems and a highly nonlinear high-dimensional one-layer

model of the atmosphere.

An older development that has recently gained new attention are particle flows. The basic idea is to iteratively move all particles from being samples from the prior to equal-weight samples from the posterior. This motion of all particles through state space is defined via iteratively decreasing the distance between the actual particle density and the posterior density. Many methods to achieve this have been proposed and will be discussed briefly. Then we will describe a solution based on minimizing the relative entropy. By restricting the transport map to a Reproducing Kernel Hilbert Space a practical solution is found.

When applying this to geophysical problems two issues arise: how to accurately represent the prior and how to choose the kernel covariance. These two problems are related, and we will demonstrate practical methods to solve this problem, based on iterative refinement of the kernel covariances. The behaviour of the new methodology will be investigated using toy problems and a highly nonlinear high-dimensional one-layer

model of the atmosphere.

Speaker: | Peter Jan van Leeuwen |

Affiliation: | Colorado State University and University of Reading, UK |

Location: | zoom |

Done