A polynomial dimensional decomposition for stochastic computing

This article presents a new polynomial dimensional decomposition method for solving stochastic problems commonly encountered in engineering disciplines and applied sciences. The method involves a hierarchical decomposition of a multivariate response function in terms of variables with increasing dimensions, a broad range of orthonormal polynomial bases consistent with the probability measure for Fourier‐polynomial expansion of component functions, and an innovative dimension‐reduction integration for calculating the coefficients of the expansion. The new decomposition method does not require sample points as in the previous version; yet, it generates a convergent sequence of lower‐variate estimates of the probabilistic characteristics of a generic stochastic response. The results of five numerical examples indicate that the proposed decomposition method provides accurate, convergent, and computationally efficient estimates of the tail probability of random mathematical functions or the reliability of mechanical systems. Copyright © 2008 John Wiley & Sons, Ltd.

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

[2]  W. Hoeffding.  A Class of Statistics with Asymptotically Normal Distribution , 1948 .

[3]  Irene A. Stegun,et al.  Handbook of Mathematical Functions. , 1966 .

[4]  B. Efron,et al.  The Jackknife Estimate of Variance , 1981 .

[5]  J. Boyd.  Chebyshev and Fourier Spectral Methods , 1989 .

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

[7]  Niels C. Lind,et al.  Approximate reliability analysis using normal polynomial and simulation results , 1996 .

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

[9]  G. Schuëller.  A state-of-the-art report on computational stochastic mechanics , 1997 .

[10]  H. Rabitz,et al.  General foundations of high‐dimensional model representations , 1999 .

[11]  Ramana V. Grandhi,et al.  Adaptation of fast Fourier transformations to estimate the structural failure probability , 2000 .

[12]  D. Hunter.  Valuation of mortgage-backed securities using Brownian bridges to reduce effective dimension , 2000 .

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

[14]  H. Rabitz,et al.  Practical Approaches To Construct RS-HDMR Component Functions , 2002 .

[15]  M. Grigoriu.  Stochastic Calculus: Applications in Science and Engineering , 2002 .

[16]  Ilya M. Sobol,et al.  Theorems and examples on high dimensional model representation , 2003, Reliab. Eng. Syst. Saf..

[17]  R. Grandhi,et al.  Polynomial Chaos Expansion with Latin Hypercube Sampling for Estimating Response Variability , 2003 .

[18]  Ramana V. Grandhi,et al.  Adaption of fast Fourier transformations to estimate structural failure probability , 2003 .

[19]  Robert E. Melchers,et al.  A fast approximate method for parameter sensitivity estimation in Monte Carlo structural reliability , 2004 .

[20]  S. Rahman,et al.  A univariate dimension-reduction method for multi-dimensional integration in stochastic mechanics , 2004 .

[21]  Sean R Eddy,et al.  What is dynamic programming? , 2004, Nature Biotechnology.

[22]  S. Rahman,et al.  Decomposition methods for structural reliability analysis , 2005 .

[23]  Sharif Rahman,et al.  A generalized dimension‐reduction method for multi‐dimensional integration in stochastic mechanics (Int. J. Numer. Meth. Engng 2004; 61:1992–2019) , 2006 .