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Developing suitable approximate models for analyzing and simulating complexnonlinear systems is practically important. This paper aims at exploring theskill of a rich class of nonlinear stochastic models, known as the conditionalGaussian nonlinear system (CGNS), as both a cheap surrogate model and a fastpreconditioner for facilitating many computationally challenging tasks. TheCGNS preserves the underlying physics to a large extent and can reproduceintermittency, extreme events and other non-Gaussian features in many complexsystems arising from practical applications. Three interrelated topics arestudied. First, the closed analytic formulae of solving the conditionalstatistics provide an efficient and accurate data assimilation scheme. It isshown that the data assimilation skill of a suitable CGNS approximate forecastmodel outweighs that by applying an ensemble method even to the perfect modelwith strong nonlinearity, where the latter suffers from filter divergence.Second, the CGNS allows the development of a fast algorithm for simultaneouslyestimating the parameters and the unobserved variables with uncertaintyquantification in the presence of only partial observations. Utilizing anappropriate CGNS as a preconditioner significantly reduces the computationalcost in accurately estimating the parameters in the original complex system.Finally, the CGNS advances rapid and statistically accurate algorithms forcomputing the probability density function and sampling the trajectories of theunobserved state variables. These fast algorithms facilitate the development ofan efficient and accurate data-driven method for predicting the linear responseof the original system with respect to parameter perturbations based on asuitable CGNS preconditioner.

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