First-passage probability estimation of high-dimensional nonlinear stochastic dynamic systems by a fractional moments-based mixture distribution approach

TL;DR

This paper introduces a novel fractional moments-based mixture distribution method for efficiently estimating first-passage probabilities in high-dimensional nonlinear stochastic systems, leveraging advanced sampling and distribution modeling techniques.

Contribution

It develops a new approach combining fractional moments, mixture distributions, and parallel adaptive sampling to accurately estimate first-passage probabilities in complex systems.

Findings

Accurately estimates first-passage probabilities in high-dimensional systems.

Demonstrates efficiency and accuracy through three diverse examples.

Provides a flexible mixture distribution model for response distribution reconstruction.

Abstract

First-passage probability estimation of high-dimensional nonlinear stochastic systems is a significant task to be solved in many science and engineering fields, but remains still an open challenge. The present paper develops a novel approach, termed 'fractional moments-based mixture distribution', to address such challenge. This approach is implemented by capturing the extreme value distribution (EVD) of the system response with the concepts of fractional moments and mixture distribution. In our context, the fractional moment itself is by definition a high-dimensional integral with a complicated integrand. To efficiently compute the fractional moments, a parallel adaptive sampling scheme that allows for sample size extension is developed using the refined Latinized stratified sampling (RLSS). In this manner, both variance reduction and parallel computing are possible for evaluating the fractional moments. From the knowledge of low-order fractional moments, the EVD of interest is then expected to be reconstructed. Based on introducing an extended inverse Gaussian distribution and a log extended skew-normal distribution, one flexible mixture distribution model is proposed, where its fractional moments are derived in analytic form. By fitting a set of fractional moments, the EVD can be recovered via the proposed mixture model. Accordingly, the first-passage probabilities under different thresholds can be obtained from the recovered EVD straightforwardly. The performance of the proposed method is verified by three examples consisting of two test examples and one engineering problem.