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Multi-fidelity uncertainty propagation using polynomial chaos and Gaussian process modeling

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Abstract

The polynomial chaos (PC) method has been widely studied and applied for uncertainty propagation (UP) due to its high efficiency and mathematical rigor. However, the straightforward application of PC on the computationally expensive and highly complicated model for UP might be too costly. Therefore, a multi-fidelity PC approach using the Gaussian process modeling theory is developed in this work, by which the classic multi-level co-kriging multi-fidelity modeling framework in the deterministic domain is extended to the stochastic one. Meanwhile, taking advantage of the Gaussian process modeling theory, the strategies for response models with hierarchical and non-hierarchical fidelity are both addressed within the proposed multi-fidelity PC approach. The effectiveness and relative merit of the proposed method are demonstrated by comparative studies on several numerical examples for UP. It is noticed that the proposed approach can significantly improve the accuracy and robustness of UP compared to the commonly used addition correction-based multi-fidelity PC method; compared to co-kriging, the accuracy and robustness are generally also improved, especially for problems with unsymmetric distributed random input and large variation. An engineering robust aerodynamic optimization problem further verifies the applicability of the proposed multi-fidelity PC method.

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Abbreviations

GP:

Gaussian process

HF:

The high-fidelity model

LF:

The low-fidelity model

PC:

Polynomial chaos

b i :

The ith coefficient of PC model

d :

Response data

d :

Dimension of random inputs

h :

Exponential of the exponential correlation function

n i :

The number of sample points for the ith-level fidelity model

s :

The highest level fidelity

t :

The tth-level fidelity

x :

Random input vector

y :

Stochastic response value

B :

Polynomial coefficient matrix

D t :

Input sites

E :

Unknown covariance matrix between lower-fidelity models

R :

Correlation function

α :

Multi-indices for PC

δ(x):

Correction function

μ :

Mean value

θ :

Hyper-parameters vector

ρ :

Scaling factor

σ :

Standard deviation value

ξ :

Random vector in standard random space

Φ :

Orthogonal polynomial

Γ :

Input space

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Acknowledgements

The grant support from Science Challenge Project (No. TZ2018001) and Hongjian Innovation Foundation (No.BQ203-HYJJ-Q2018002) is greatly acknowledged.

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Authors and Affiliations

  1. School of Aerospace Engineering, Beijing Institute of Technology, Beijing, 100081, China

    Fenggang Wang, Fenfen Xiong & Jianmei Song

  2. Beijing Electro-Mechanical Engineering Institute, 100074, Beijing, People’s Republic of China

    Shishi Chen

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  1. Fenggang Wang
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  2. Fenfen Xiong
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Correspondence to Fenfen Xiong.

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Wang, F., Xiong, F., Chen, S. et al. Multi-fidelity uncertainty propagation using polynomial chaos and Gaussian process modeling. Struct Multidisc Optim 60, 1583–1604 (2019). https://doi.org/10.1007/s00158-019-02287-7

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