Effective Filtering on a Random Slow Manifold

TL;DR

This paper develops a method for filtering in slow-fast systems by reducing the system to a slow manifold, proving the approximation's validity, and demonstrating it with numerical examples.

Contribution

It introduces a low-dimensional filter based on an invariant slow manifold for slow-fast systems with only slow component observations, with theoretical and numerical validation.

Findings

The low-dimensional filter closely approximates the original filter.

The reduction simplifies data assimilation in complex systems.

Numerical examples confirm the effectiveness of the approach.

Abstract

This work is about a slow-fast data assimilation system when only slow components are observable. First, we obtain its low dimensional reduction via an invariant slow manifold. Second, we prove that the low dimensional filter on the slow manifold approximates the original filter in a suitable metric. Finally, we illustrate this approximate filter numerically in an example.