A hybrid polynomial dimensional decomposition for uncertainty quantification of high-dimensional complex systems

Abstract

This paper presents a novel hybrid polynomial dimensional decomposition (PDD) method for stochastic computing in high-dimensional complex systems. When a stochastic response does not possess a strongly additive or a strongly multiplicative structure alone, then the existing additive and multiplicative PDD methods may not provide a sufficiently accurate probabilistic solution of such a system. To circumvent this problem, a new hybrid PDD method was developed that is based on a linear combination of an additive and a multiplicative PDD approximation, a broad range of orthonormal polynomial bases for Fourier-polynomial expansions of component functions, and a dimension-reduction or sampling technique for estimating the expansion coefficients. Two numerical problems involving mathematical functions or uncertain dynamic systems were solved to study how and when a hybrid PDD is more accurate and efficient than the additive or the multiplicative PDD. The results show that the univariate hybrid PDD method is slightly more expensive than the univariate additive or multiplicative PDD approximations, but it yields significantly more accurate stochastic solutions than the latter two methods. Therefore, the univariate truncation of the hybrid PDD is ideally suited to solving stochastic problems that may otherwise mandate expensive bivariate or higher-variate additive or multiplicative PDD approximations. Finally, a coupled acoustic-structural analysis of a pickup truck subjected to 46 random variables was performed, demonstrating the ability of the new method to solve large-scale engineering problems.