Enhanced Gaussian-mixture-model-based nonlinear probabilistic uncertainty propagation using Gaussian splitting approach

Practical engineering problems often involve stochastic uncertainty, which can cause substantial variations in the response of engineering products or even lead to failure. The coupling and propagation of uncertainty play a crucial role in this process. Hence, it is imperative to quantify, propagate and control stochastic uncertainty. Different from most traditional uncertainty propagation methods, the proposed method employs Gaussian splitting method to divide the input random variables into Gaussian mixture models. These GMMs have a limited number of components with very small variances. As a result, the input Gaussian components can be conveniently propagated to the response and remain Gaussian distributions after nonlinear uncertainty propagation, which is able to provide an effective method for high-precision nonlinear uncertainty propagation. Firstly, the probability density function of input random variable is reconstructed by Gaussian mixture models. Secondly, the K-value criterion is proposed for selecting split direction, taking into account both the nonlinearity and variance. The components of input random variables are then divided into a Gaussian mixture model with small variance along the direction determined by the K-value. Thirdly, the individual components of the Gaussian mixture model are propagated one by one to obtain the probability density function of the response. Finally, the convergence criterion based on Shannon entropy is developed to ensure the accuracy of uncertainty propagation. The efficacy of the method is verified using three numerical examples and two engineering examples.