Stochastic Computing by a New Polynomial Dimensional Decomposition Method

Abstract

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 two 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.