Extended Polynomial Dimensional Decomposition for Arbitrary Probability Distributions

Abstract

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.