A data-driven polynomial chaos method considering correlated random variables
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RESEARCH PAPER
A data-driven polynomial chaos method considering correlated random variables Qizhang Lin 1 & Fenfen Xiong 1 & Fenggang Wang 1 & Xin Yang 2 Received: 15 August 2019 / Revised: 26 November 2019 / Accepted: 6 April 2020 # Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Variable correlation commonly exists in practical engineering applications. However, most of the existing polynomial chaos (PC) approaches for uncertainty propagation (UP) assume that the input random variables are independent. To address variable correlation, an intrusive PC method has been developed for dynamic system, which however is not applicable to problems with black-box-type functions. Therefore, based on the existing data-driven PC method, a new non-intrusive data-driven polynomial chaos approach that can directly consider variable correlation for UP of black-box computationally expensive problems is developed in this paper. With the proposed method, the multivariate orthogonal polynomial basis corresponding to the correlated input random variables is conveniently constructed by solving the moment-matching equations based on the correlation statistical moments to consider the variable correlation. A comprehensive comparative study on several numerical examples of UP and design optimization under uncertainty with correlated input random variables is conducted to verify the effectiveness and advantage of the proposed method. The results show that the proposed method is more accurate than the existing data-driven PC method with Nataf transformation when the variable distribution is known, and it can produce accurate results with unknown variable distribution, demonstrating its effectiveness. Keywords Uncertainty propagation . Polynomial chaos . Data-driven . Variable correlation
Nomenclature bi The ith coefficient of PC model d Dimension of random inputs x Random input vector y Stochastic response value H Order of PC model P(k) The kth orthogonal polynomials for correlated variables P The orthogonal polynomials for independent variables
Q+1 μ μa, b ρ σ Ωc Γ(x) DD-PC gPC GS-PC ME-PC
Responsible Editor: Byeng D Youn Electronic supplementary material The online version of this article (https://doi.org/10.1007/s00158-020-02602-7) contains supplementary material, which is available to authorized users. * Fenfen Xiong [email protected] 1
School of Aerospace Engineering, Beijing Institute of Technology, Beijing 100081, China
2
The 6th Research Institute of China Electronics Corporation (CEC)/ Intelligence Technology of CEC Co. Ltd, Beijing 100081, China
PC UP
Number of PC coefficients Mean value Correlation statistical moment Correlation coefficient Standard deviation value Original correlated random variable space Joint cumulative distribution function The data-driven polynomial chaos method The generalized polynomial chaos method The Gram-Schmidt polynomial chaos method The multi-element generalized polynomial chaos method Polynomial chaos Uncertainty propagation
1 Introduction Uncertainty propagation (U
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