Gradient Flows in Uncertainty Propagation and Filtering of Linear Gaussian Systems

TL;DR

This paper explains how gradient flow schemes like JKO and LMMR can be used for uncertainty propagation and filtering in linear Gaussian systems, offering insights into their mathematical foundations and potential for efficient computation.

Contribution

It provides a detailed exposition of gradient flow schemes in the context of linear Gaussian systems, connecting variational methods to classical filtering and uncertainty propagation techniques.

Findings

Clarifies the connection between gradient flows and filtering equations

Demonstrates the application of JKO and LMMR schemes to linear Gaussian systems

Recovers known results using variational and gradient flow perspectives

Abstract

The purpose of this work is mostly expository and aims to elucidate the Jordan-Kinderlehrer-Otto (JKO) scheme for uncertainty propagation, and a variant, the Laugesen-Mehta-Meyn-Raginsky (LMMR) scheme for filtering. We point out that these variational schemes can be understood as proximal operators in the space of density functions, realizing gradient flows. These schemes hold the promise of leading to efficient ways for solving the Fokker-Planck equation as well as the equations of non-linear filtering. Our aim in this paper is to develop in detail the underlying ideas in the setting of linear stochastic systems with Gaussian noise and recover known results.