Ensemble Transport Filter via Optimized Maximum Mean Discrepancy

TL;DR

This paper introduces an ensemble transport filter that uses an optimized maximum mean discrepancy to accurately approximate posterior distributions, especially in high-dimensional data assimilation tasks.

Contribution

It proposes a novel transport map approach for particle filtering that improves robustness and accuracy using MMD-based optimization with a variance penalty.

Findings

Outperforms ensemble Kalman filter in numerical examples

Enhances robustness with a variance penalty term

Effective in high-dimensional data assimilation

Abstract

In this paper, we present a new ensemble-based filter method by reconstructing the analysis step of the particle filter through a transport map, which directly transports prior particles to posterior particles. The transport map is constructed through an optimization problem described by the Maximum Mean Discrepancy loss function, which matches the expectation information of the approximated posterior and reference posterior. The proposed method inherits the accurate estimation of the posterior distribution from particle filtering while gives an extension to high dimensional assimilation problems. To improve the robustness of Maximum Mean Discrepancy, a variance penalty term is used to guide the optimization. It prioritizes minimizing the discrepancy between the expectations of highly informative statistics for the reference posteriors. The penalty term significantly enhances the robustness of the proposed method and leads to a better approximation of the posterior. A few numerical examples are presented to illustrate the advantage of the proposed method over ensemble Kalman filter.