Coupled Stochastic-Statistical Equations for Filtering Multiscale Turbulent Systems

TL;DR

This paper introduces a novel stochastic-statistical filtering framework for high-dimensional multiscale turbulent systems, enabling efficient and accurate estimation of non-Gaussian statistics from limited observations.

Contribution

It develops a coupled stochastic-statistical model and an ensemble filter that effectively captures non-Gaussian features and guarantees convergence to the optimal filter.

Findings

The finite-dimensional filter approximates the optimal filter with high accuracy.

The proposed method guarantees stable convergence to the optimal filter.

The ensemble filter accurately recovers true model statistics in turbulent systems.

Abstract

We present a new strategy for filtering high-dimensional multiscale systems characterized by high-order non-Gaussian statistics using observations from leading-order moments. A closed stochastic-statistical modeling framework suitable for systematic theoretical analysis and efficient numerical simulations is designed. Optimal filtering solutions are derived based on the explicit coupling structures of stochastic and statistical equations subject to linear operators, which satisfy an infinite-dimensional Kalman-Bucy filter with conditional Gaussian dynamics. To facilitate practical implementation, we develop a finite-dimensional stochastic filter model that approximates the optimal filter solution. We prove that this approximating filter effectively captures key non-Gaussian features, demonstrating consistent statistics with the optimal filter first in its analysis step update, then at the long-time limit guaranteeing stable convergence to the optimal filter. Finally, we build a practical ensemble filter algorithm based on the approximating filtering model, which enables accurate recovery of the true model statistics. The proposed modeling and filtering strategies are applicable to a wide range challenging problems in science and engineering, particularly for statistical prediction and uncertainty quantification of multiscale turbulent states.