Conditional Gaussian Nonlinear System: a Fast Preconditioner and a Cheap Surrogate Model For Complex Nonlinear Systems

TL;DR

This paper introduces the conditional Gaussian nonlinear system (CGNS) as an efficient surrogate model and preconditioner that preserves key physics, enabling faster data assimilation, parameter estimation, and response prediction in complex nonlinear systems.

Contribution

It develops analytic formulas for CGNS-based data assimilation, parameter estimation, and response prediction, demonstrating significant computational efficiency and accuracy improvements over existing methods.

Findings

CGNS provides more accurate data assimilation than ensemble methods in nonlinear settings.

Using CGNS as a preconditioner reduces computational cost in parameter estimation.

Fast algorithms for density computation and trajectory sampling enable efficient response prediction.

Abstract

Developing suitable approximate models for analyzing and simulating complex nonlinear systems is practically important. This paper aims at exploring the skill of a rich class of nonlinear stochastic models, known as the conditional Gaussian nonlinear system (CGNS), as both a cheap surrogate model and a fast preconditioner for facilitating many computationally challenging tasks. The CGNS preserves the underlying physics to a large extent and can reproduce intermittency, extreme events and other non-Gaussian features in many complex systems arising from practical applications. Three interrelated topics are studied. First, the closed analytic formulae of solving the conditional statistics provide an efficient and accurate data assimilation scheme. It is shown that the data assimilation skill of a suitable CGNS approximate forecast model outweighs that by applying an ensemble method even to the perfect model with strong nonlinearity, where the latter suffers from filter divergence. Second, the CGNS allows the development of a fast algorithm for simultaneously estimating the parameters and the unobserved variables with uncertainty quantification in the presence of only partial observations. Utilizing an appropriate CGNS as a preconditioner significantly reduces the computational cost in accurately estimating the parameters in the original complex system. Finally, the CGNS advances rapid and statistically accurate algorithms for computing the probability density function and sampling the trajectories of the unobserved state variables. These fast algorithms facilitate the development of an efficient and accurate data-driven method for predicting the linear response of the original system with respect to parameter perturbations based on a suitable CGNS preconditioner.