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Weighted stochastic response surface method considering sample weights

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Abstract

Conventional stochastic response surface methods (SRSM) based on polynomial chaos expansion (PCE) for uncertainty propagation treat every sample point equally during the regression process and may produce inaccurate estimations of PCE coefficients. To address this issue, a new weighted stochastic response surface method (WSRSM) that considers the sample probabilistic weights in regression is studied in this work. Techniques for determining sample probabilistic weights for three sampling approaches Gaussian Quadrature point (GQ), Monomial Cubature Rule (MCR), and Latin Hypercube Design (LHD) are developed. The advantage of the proposed method is demonstrated through mathematical and engineering examples. It is shown that for various sampling techniques WSRSM consistently achieves higher accuracy of uncertainty propagation without introducing extra computational cost compared to the conventional SRSM. Insights into the relative accuracy and efficiency of various sampling techniques in implementation are provided as well.

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Abbreviations

SRSM:

Stochastic response surface method;

WSRSM:

Weighted stochastic response surface method;

PCE:

Polynomial Chaos Expansion;

GQ:

Gaussian Quadrature;

MCR:

Monomial Cubature rule;

LHD:

Latin Hypercube design;

d :

Dimension of the input random variable;

N :

Number of sample points;

p :

PCE order;

b :

PCE coefficients;

P:

Number of PCE coefficients;

ξ i :

Standard normal random variable;

ϕ :

Cumulative density function (CDF);

φ :

Probability density function (PDF)

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Acknowledgments

The grant supports from National Science Foundation of China (NO. 10972034), US National Science Foundation (NO. CMMI-0928320) and China Poster-doctoral Science Foundation (20100470213) are greatly acknowledged. The views expressed are those of the authors and do not necessarily reflect the views of the sponsors.

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Authors and Affiliations

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

    Fenfen Xiong & Shuxing Yang

  2. Department of Mechanical Engineering, Northwestern University, Evanston, IL, 60208, USA

    Wei Chen

  3. School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai, 200031, China

    Wei Chen

  4. Bank of America, Charlotte, NC, 28255, USA

    Ying Xiong

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  1. Fenfen Xiong
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  2. Wei Chen
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  4. Shuxing Yang
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Correspondence to Shuxing Yang.

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Xiong, F., Chen, W., Xiong, Y. et al. Weighted stochastic response surface method considering sample weights. Struct Multidisc Optim 43, 837–849 (2011). https://doi.org/10.1007/s00158-011-0621-3

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  • DOI: https://doi.org/10.1007/s00158-011-0621-3

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