Uncertainty Quantification by Alternative Decompositions of Multivariate Functions

This article advocates factorized and hybrid dimensional decompositions (FDD/HDD), as alternatives to analysis-of-variance dimensional decomposition (ADD), for second-moment statistical analysis of multivariate functions. New formulae revealing the relationships between component functions of FDD and ADD are proposed. While ADD or FDD is relevant when a function is strongly additive or strongly multiplicative, HDD, whether formed linearly or nonlinearly, requires no specific dimensional hierarchies. Furthermore, FDD and HDD lead to alternative definitions of effective dimension, reported in the current literature only for ADD. New closed-form or analytical expressions are derived for univariate truncations of all three decompositions, followed by mean-squared error analysis of univariate ADD, FDD, and HDD approximations. The analysis finds appropriate conditions when one approximation is better than the other. Numerical results affirm the theoretical finding that HDD is ideally suited to a general function approximation that may otherwise require higher-variate ADD or FDD truncations for rendering acceptable accuracy in stochastic solutions.