An efficient strategy for information reuse in probability density evolution method considering large shift of distributions with multiple random variables

Abstract

The probability density evolution method (PDEM) is a versatile approach for analyzing stochastic dynamical systems. When combined with the change of probability measure (COM), it provides a tool to efficiently deal with the aleatory and epistemic uncertainties, where shifts of probability distributions are frequently encountered. However, when the shifts of distributions are too large, the PDEM-COM method can lead to increasing numerical errors. This paper aims to propose an extension of the PDEM-COM that can address the large shifts of distributions involving multiple random variables. In the proposed method, the concepts of the multi-dimensional augmented support and the augmented probability density function (PDF) are introduced based on the differences between the original and updated distributions. Then, an efficient numerical procedure is established for selecting a small set of additional representative points based on the augmented PDF. These additional representative points serve as a complement to the representative point set selected according to the original distributions. By incorporating the augmented representative point set and solving the generalized probability density evolution equation (GDEE), the stochastic response of the system considering the updated distributions can be evaluated. Numerical examples are presented to demonstrate the capability and effectiveness of the proposed approach. The results demonstrate that the proposed approach offers improved accuracy compared to the PDEM-COM method, particularly for large distribution shifts, while maintaining a relatively lower computational cost.