Multiplicative polynomial dimensional decompositions for uncertainty quantification of high-dimensional complex systems

The central theme of this paper is multiplicative polynomial dimensional decomposition (PDD) methods for solving high-dimensional stochastic problems. When a stochastic response is dominantly of multiplicative nature, the standard PDD approximation, predicated on additive function decomposition, may not provide sufficiently accurate estimates of the probabilistic characteristics of a complex system. To circumvent this problem, two multiplicative versions of PDD, referred to as factorized PDD and logarithmic PDD, were developed to examine if they provide improved stochastic solutions. Both versions involve a hierarchical, multiplicative decomposition of a multivariate function in terms of variables with increasing dimensions, a broad range of orthonormal polynomial bases consistent with the input probability measure for Fourier-polynomial expansions of component functions, and a dimension-reduction integration or sampling technique for estimating the expansion coefficients. Three numerical problems involving mathematical functions or eigenvalues of uncertain dynamic systems were solved to corroborate how and when a multiplicative PDD is more efficient or accurate than the additive PDD. The results show that, indeed, both the factorized and logarithmic PDD approximations can effectively exploit the hidden multiplicative structure of a stochastic response when it exists. Since a multiplicative PDD involves component functions that are the same or similar to those of the additive PDD, no additional cost is incurred. Finally, the random eigensolutions of a sport utility vehicle comprising 40 random variables were evaluated, demonstrating the ability of the new methods to solve industrial-scale problems.