Uncertainty quantification using polynomial chaos expansion with points of monomial cubature rules

This paper proposes an efficient method for estimating uncertainty propagation and identifying influence factors contributing to uncertainty. In general, the system is dominated by some of the main effects and lower-order interactions due to the sparsity-of-effect principle. Therefore, the construction of polynomial chaos expansion with points of monomial cubature rules is particularly attractive in dealing with large computational model. This approach has advantages over many others as it needs far fewer sampling points for multivariate models and all of the points can be sampled. The proposed approach is validated via two mathematical functions and an engineering problem.

[1]  Menner A Tatang,et al.  Direct incorporation of uncertainty in chemical and environmental engineering systems , 1995 .

[2]  J. Villadsen,et al.  Solution of differential equation models by polynomial approximation , 1978 .

[3]  Gyu-Hong Kang,et al.  Prediction of torque characteristic on barrier-type SRM using stochastic response surface methodology combined with moving least square , 2004 .

[4]  Menner A. Tatang,et al.  An efficient method for parametric uncertainty analysis of numerical geophysical models , 1997 .

[5]  D. Wei,et al.  Optimization and tolerance prediction of sheet metal forming process using response surface model , 2008 .

[6]  D. Xiu,et al.  Modeling Uncertainty in Steady State Diffusion Problems via Generalized Polynomial Chaos , 2002 .

[7]  R. Ghanem.  Probabilistic characterization of transport in heterogeneous media , 1998 .

[8]  R. Ghanem,et al.  Stochastic Finite Elements: A Spectral Approach , 1990 .

[9]  P G Georgopoulos,et al.  Efficient Sensitivity/Uncertainty Analysis Using the Combined Stochastic Response Surface Method and Automated Differentiation: Application to Environmental and Biological Systems , 2000, Risk analysis : an official publication of the Society for Risk Analysis.

[10]  S. Isukapalli,et al.  Stochastic Response Surface Methods (SRSMs) for Uncertainty Propagation: Application to Environmental and Biological Systems , 1998, Risk analysis : an official publication of the Society for Risk Analysis.

[11]  A. H. Stroud,et al.  Some fifth degree integration formulas for symmetric regions II , 1967 .

[12]  A. Sarkar,et al.  Mid-frequency structural dynamics with parameter uncertainty , 2001 .

[13]  S. Finette.  A stochastic representation of environmental uncertainty and its coupling to acoustic wave propagation in ocean waveguides , 2006 .

[14]  L. Devroye.  Non-Uniform Random Variate Generation , 1986 .

[15]  Ronald Cools,et al.  An encyclopaedia of cubature formulas , 2003, J. Complex..

[16]  Wei Chen,et al.  A weighted three-point-based methodology for variance estimation , 2006 .

[17]  Douglas C. Montgomery,et al.  Response Surface Methodology: Process and Product Optimization Using Designed Experiments , 1995 .

[18]  D. Xiu,et al.  A new stochastic approach to transient heat conduction modeling with uncertainty , 2003 .

[19]  Dongbin Xiu,et al.  The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations , 2002, SIAM J. Sci. Comput..

[20]  Nestor V. Queipo,et al.  Efficient Shape Optimization Under Uncertainty Using Polynomial Chaos Expansions and Local Sensitivities , 2006 .

[21]  N. Wiener.  The Homogeneous Chaos , 1938 .

[22]  N. Zabaras,et al.  Uncertainty propagation in finite deformations––A spectral stochastic Lagrangian approach , 2006 .

[23]  S. Isukapalli.  UNCERTAINTY ANALYSIS OF TRANSPORT-TRANSFORMATION MODELS , 1999 .

[24]  R. Cools.  Monomial cubature rules since “Stroud”: a compilation—part 2 , 1999 .

[25]  Nicholas Zabaras,et al.  A non-intrusive stochastic Galerkin approach for modeling uncertainty propagation in deformation processes , 2007 .

[26]  R. Cools,et al.  Monomial cubature rules since “Stroud”: a compilation , 1993 .

[27]  A. Stroud.  Approximate calculation of multiple integrals , 1973 .