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 blackboxtype functions. Therefore, based on the existing datadriven PC method, a new nonintrusive datadriven polynomial chaos approach that can directly consider variable correlation for UP of blackbox 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 momentmatching 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 datadriven PC method with Nataf transformation when the variable distribution is known, and it can produce accurate results with unknown variable distribution, demonstrating its effectiveness.
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Abbreviations
 b _{ i } :

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
 \( \overline{P} \) :

The orthogonal polynomials for independent variables
 Q + 1:

Number of PC coefficients
 μ :

Mean value
 μ _{a, b} :

Correlation statistical moment
 ρ :

Correlation coefficient
 σ :

Standard deviation value
 Ω_{c} :

Original correlated random variable space
 Γ(x):

Joint cumulative distribution function
 DDPC:

The datadriven polynomial chaos method
 gPC:

The generalized polynomial chaos method
 GSPC:

The GramSchmidt polynomial chaos method
 MEPC:

The multielement generalized polynomial chaos method
 PC:

Polynomial chaos
 UP:

Uncertainty propagation
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Funding
Grant support was received from Science Challenge Project (No. TZ2018001) and Hongjian Innovation Foundation (No.BQ203HYJJQ2018002).
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Replication of results
The results shown in the manuscript can be reproduced. Considering the size limit of the uploaded supplementary material, the codes for one of the mathematical example for UP (Function 1 and Function 2 in Sect. 3.1) is uploaded as supplementary material. For the rest of the examples, it is very easy to implement by changing the response functions and sample points based on the codes provided to obtain the results shown in the manuscript
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Appendix
Appendix
The correlated statistical moments and the polynomial coefficients for Function 1 (Case 1, ρ = 0.8) in Sect. 3.1 are provided as below.
For this example, the dimension of correlated random variable is 2 and the order of model is set as H = 3. Therefore, there are ten twodimensional orthogonal polynomial bases with polynomial order on more than 3. The correlated statistical moments that are in the form of matrix in (17) are shown as below:
Based on the probabilistic distribution information of the correlated input random variables, one can obtain these correlated statistical moments that will be employed in the orthogonal polynomial basis construction. As the correlated statistical moments for k = 0, 1,..., 8 are all part of those for k = 9, only those for k = 9 is given as below:
Correspondingly, the polynomial coefficients of the ten twodimensional orthogonal polynomial bases are listed as below:
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Lin, Q., Xiong, F., Wang, F. et al. A datadriven polynomial chaos method considering correlated random variables. Struct Multidisc Optim (2020). https://doi.org/10.1007/s00158020026027
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Keywords
 Uncertainty propagation
 Polynomial chaos
 Datadriven
 Variable correlation