Derivation and Analysis of Simplified Filters for Complex Dynamical Systems

TL;DR

This paper analyzes simplified filtering methods for complex dynamical systems, using Markov switching models to evaluate their effectiveness and propose improvements for better state estimation in turbulent regimes.

Contribution

It provides a detailed analysis of simplified filters in complex systems and introduces modifications to enhance filtering accuracy in specific time-scale regimes.

Findings

Simplified filters can effectively estimate states in turbulent systems.

Markov switching models serve as useful surrogate test-beds.

Proposed modifications improve filtering performance in certain regimes.

Abstract

Filtering is concerned with the sequential estimation of the state, and uncertainties, of a Markovian system, given noisy observations. It is particularly difficult to achieve accurate filtering in complex dynamical systems, such as those arising in turbulence, in which effective low-dimensional representation of the desired probability distribution is challenging. Nonetheless recent advances have shown considerable success in filtering based on certain carefully chosen simplifications of the underlying system, which allow closed form filters. This leads to filtering algorithms with significant, but judiciously chosen, model error. The purpose of this article is to analyze the effectiveness of these simplified filters, and to suggest modifications of them which lead to improved filtering in certain time-scale regimes. We employ a Markov switching process for the true signal underlying the data, rather than working with a fully resolved DNS PDE model. Such Markov switching models haven been demonstrated to provide an excellent surrogate test-bed for the turbulent bursting phenomena which make filtering of complex physical models, such as those arising in atmospheric sciences, so challenging.