Weighted stochastic response surface method considering sample weights

Research Paper

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.

Keywords

Stochastic response surface method Sample probabilistic weights Gauss quadrature Monomial Cubature rule Latin hypercube design 

Nomenclature

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;

ξi

Standard normal random variable;

ϕ

Cumulative density function (CDF);

φ

Probability density function (PDF)

Notes

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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Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • Fenfen Xiong
    • 1
  • Wei Chen
    • 2
    • 3
  • Ying Xiong
    • 4
  • Shuxing Yang
    • 1
  1. 1.School of Aerospace EngineeringBeijing Institute of TechnologyBeijingChina
  2. 2.Department of Mechanical EngineeringNorthwestern UniversityEvanstonUSA
  3. 3.School of Mechanical EngineeringShanghai Jiao Tong UniversityShanghaiChina
  4. 4.Bank of AmericaCharlotteUSA

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