Extended Polynomial Dimensional Decomposition for Arbitrary Probability Distributions

• Published in: Journal of engineering mechanics Vol. 135; no. 12; pp. 1439 - 1451
• Main Author: Rahman, Sharif
• Format: Journal Article
• Language: English
• Published: Reston, VA American Society of Civil Engineers 01.12.2009
• Subjects: Exact sciences and technology; Fracture mechanics (crack, fatigue, damage...); Fundamental areas of phenomenology (including applications); Physics; Solid mechanics; Structural and continuum mechanics; TECHNICAL PAPERS; Uncertainty principles; Probabilistic approach; Uncertainty principle; Stochastic processes; Decomposition method; Polynomial method; Modeling; Spectral analysis; Orthogonal polynomial; Polynomial function; Quadrature; Dimension reduction; Orthogonal function; Polynomials; Reliability; Probability measure
• ISSN: 0733-9399; 1943-7889
• DOI: 10.1061/(ASCE)EM.1943-7889.0000047

This paper presents an extended polynomial dimensional decomposition method for solving stochastic problems subject to independent random input following an arbitrary probability distribution. The method involves Fourier-polynomial expansions of component functions by orthogonal polynomial bases, the Stieltjes procedure for generating the recursion coefficients of orthogonal polynomials and the Gauss quadrature rule for a specified probability measure, and dimension-reduction integration for calculating the expansion coefficients. The extension, which subsumes nonclassical orthogonal polynomials bases, generates a convergent sequence of lower-variate estimates of the probabilistic characteristics of a stochastic response. Numerical results indicate that the extended decomposition method provides accurate, convergent, and computationally efficient estimates of the tail probability of random mathematical functions or reliability of mechanical systems. The convergence of the extended method accelerates significantly when employing measure-consistent orthogonal polynomials.