Stochastic dynamic response analysis via dimension-reduced probability density evolution equation (DR-PDEE) with enhanced tail-accuracy

authored by: Yi Luo, Chao Dang, Matteo Broggi, Michael Beer
Abstract: The stochastic dynamic analysis of high-dimensional nonlinear systems is a critical concern in engineering fields, especially when considering the reliability analysis of low-probability events. To address this challenge, the dimension-reduced probability density evolution equation (DR-PDEE) method has recently emerged as a promising tool. The DR-PDEE is the analytical governing equation for the probability density function (PDF) evolution of any path-continuous stochastic process. For a single response quantity of interest in a multi-dimensional nonlinear dynamic system, the corresponding DR-PDEE is merely a one- or two-dimensional partial differential equation. After estimating the intrinsic drift coefficient (IDC) in the DR-PDEE from sample data, this equation can be easily solved with rather high accuracy. However, if only a limited number of deterministic analyses are affordable, there is usually no sample information for the tail estimation of the IDC, resulting in an inaccurate PDF solution in the tail. In this work, a scheme is tailored for the DR-PDEE to further enhance its tail accuracy. Specifically, to increase the occurrence probability of tail samples, an additional set of samples is obtained by simply magnifying the excitation intensity of the system. Then, at each time step, samples in the response tail from this additional set are identified. By merging these samples with samples from the original system, a better IDC estimation in the tail is achieved. Several numerical examples are investigated to validate the effectiveness of the proposed DR-PDEE method. Comparisons with MCS and the classical DR-PDEE method show that the proposed scheme improves the accuracy and robustness of the PDF results in the tail.
Organisation(s): Institute for Risk and Reliability
External Organisation(s): Rice University
TU Dortmund University
University of Liverpool
Tongji University
Type: Article
Journal: Probabilistic Engineering Mechanics
Volume: 79
ISSN: 0266-8920
Publication date: 01.2025
Publication status: Published
Peer reviewed: Yes
ASJC Scopus subject areas: Statistical and Nonlinear Physics, Civil and Structural Engineering, Nuclear Energy and Engineering, Condensed Matter Physics, Aerospace Engineering, Ocean Engineering, Mechanical Engineering
Electronic version(s): https://doi.org/10.1016/j.probengmech.2025.103735 (Access: Open)

BibTeX

@article{373566e579bc4d108fe803b2fb1af4c8,
title = "Stochastic dynamic response analysis via dimension-reduced probability density evolution equation (DR-PDEE) with enhanced tail-accuracy",
abstract = "The stochastic dynamic analysis of high-dimensional nonlinear systems is a critical concern in engineering fields, especially when considering the reliability analysis of low-probability events. To address this challenge, the dimension-reduced probability density evolution equation (DR-PDEE) method has recently emerged as a promising tool. The DR-PDEE is the analytical governing equation for the probability density function (PDF) evolution of any path-continuous stochastic process. For a single response quantity of interest in a multi-dimensional nonlinear dynamic system, the corresponding DR-PDEE is merely a one- or two-dimensional partial differential equation. After estimating the intrinsic drift coefficient (IDC) in the DR-PDEE from sample data, this equation can be easily solved with rather high accuracy. However, if only a limited number of deterministic analyses are affordable, there is usually no sample information for the tail estimation of the IDC, resulting in an inaccurate PDF solution in the tail. In this work, a scheme is tailored for the DR-PDEE to further enhance its tail accuracy. Specifically, to increase the occurrence probability of tail samples, an additional set of samples is obtained by simply magnifying the excitation intensity of the system. Then, at each time step, samples in the response tail from this additional set are identified. By merging these samples with samples from the original system, a better IDC estimation in the tail is achieved. Several numerical examples are investigated to validate the effectiveness of the proposed DR-PDEE method. Comparisons with MCS and the classical DR-PDEE method show that the proposed scheme improves the accuracy and robustness of the PDF results in the tail.",
keywords = "Dimension-reduced probability density evolution equation (DR-PDEE), Multi-dimensional nonlinear systems, Stochastic response analysis, Tail accuracy",
author = "Yi Luo and Chao Dang and Matteo Broggi and Michael Beer",
note = "Publisher Copyright: {\textcopyright} 2025 The Authors",
year = "2025",
month = jan,
doi = "10.1016/j.probengmech.2025.103735",
language = "English",
volume = "79",
journal = "Probabilistic Engineering Mechanics",
issn = "0266-8920",
publisher = "Elsevier Ltd.",
}