Long-time Asymptotics of the Filtering Distribution for Partially Observed Chaotic Dynamical Systems

TL;DR

This paper investigates the long-term behavior of filtering distributions in chaotic dynamical systems with noisy, partial observations, establishing conditions for accurate signal reconstruction and providing bounds for suboptimal filters.

Contribution

It introduces a general framework for the asymptotic analysis of filtering in chaotic systems, including new bounds for variants of the 3DVAR filtering algorithm.

Findings

Filtering distributions concentrate around the true signal over time

Conditions on system dynamics and observations ensure accurate reconstruction

Provides computable bounds for suboptimal filtering algorithms

Abstract

The filtering distribution is a time-evolving probability distribution on the state of a dynamical system, given noisy observations. We study the large-time asymptotics of this probability distribution for discrete-time, randomly initialized signals that evolve according to a deterministic map $\Psi$. The observations are assumed to comprise a low-dimensional projection of the signal, given by an operator $P$, subject to additive noise. We address the question of whether these observations contain sufficient information to accurately reconstruct the signal. In a general framework, we establish conditions on $\Psi$ and $P$ under which the filtering distributions concentrate around the signal in the small-noise, long-time asymptotic regime. Linear systems, the Lorenz '63 and '96 models, and the Navier Stokes equation on a two-dimensional torus are within the scope of the theory. Our main findings come as a by-product of computable bounds, of independent interest, for suboptimal filters based on new variants of the 3DVAR filtering algorithm.