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.

翻译：本文旨在探讨一大批非线性非线性随机模型的技能,称为有条件的高斯非线性系统(CGNS),既是一种廉价的代用模型,也是便利许多计算上具有挑战性任务的快速先决条件。CGNS在很大程度上保存了基础物理,可以复制许多由实际应用产生的复杂系统中的中间、极端事件和其他非Gausian特性。研究了三个相互关联的专题。首先,解决有条件统计的封闭式解析公式提供了高效和准确的数据同化办法。它表明,合适的CGNS非线性预测模型的数据同化技能,既是一种廉价替代模型,又是一个便利许多计算上具有挑战性的任务的快速先决条件。CGNS在很大程度上保存了基础物理物理物理,并可以复制由实际应用产生的许多复杂系统中的中间、极端事件和其他非Gausicial特性。CGNS的封闭式解析公式提供了高效和准确的数据同前期数据同值。CGNS的适当的CGNS数据同C的精确度,最终将精确的计算方法的精确地降低了Crentral-C的精确度。