Uncertainty quantification of high‐dimensional complex systems by multiplicative polynomial dimensional decompositions

Abstract

SUMMARYThe 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 probabilistic solutions of a complex system. To circumvent this problem, two multiplicative versions of PDD, referred to as factorized PDD and logarithmic PDD, were developed. Both versions involve a hierarchical, multiplicative decomposition of a multivariate function, 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. Three numerical problems involving mathematical functions or 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 recycles the same component functions 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. Copyright © 2013 John Wiley & Sons, Ltd.

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

“…Tunga and Demiralp [12] originally proposed this decomposition, calling it factorized highdimensional model representation. Subsequently, Yadav and Rahman [7], and Rahman [13] derived a recursive relationship between the component functions of ANOVA and factorized dimensional decompositions, as described by Theorem 1, leading to F-PDD. 1) and (16), respectively, are…”

“…This objective was attained through foresaking the inefficient higher-variate expansions and applying only univariate hybrid PDD approximations in solving high-dimensional stochastic problems. Considering the key advantage of high efficiency of a univariate additive [1] and factorized PDD [7] approximations, only the univariate hybrid PDD method was implemented in this work. Proposition 3 formally describes the univariate hybrid PDD approximation.…”

“…However, the original PDD, referred to as the additive PDD (A-PDD) in this paper, constitutes a sum of lower-dimensional component functions and is, therefore, predicated on an additive nature of a multivariate function decomposition. In contrast, when a response function is of a multiplicative nature, suitable multiplicative-type decompositions, such as the factorized PDD (F-PDD) [7], should be explored. Nonetheless, A-PDD or F-PDD is relevant as long as the dimensional hierarchy of a stochastic response is also additive or multiplicative.…”

Post optimization for accurate and efficient reliability‐based design optimization using second‐order reliability method based on importance sampling and its stochastic sensitivity analysis

“…If the random variables are correlated, the score function can be evaluated using a copula function. In a similar fashion with Equations (14)- (16) can be rewritten using the instrumental density function and the failure region defined using the second-order Taylor expansion of the performance function in Equation (13) as…”

“…In reliability analysis, such uncertainties are usually expressed as random variables, and a reliability or probability of failure is computed in order to quantify safety of the engineering system under influence of the uncertainties. It is quite difficult to estimate the probability of failure defined as a multi-dimensional integration over a nonlinear domain in a real engineering problem especially including finite element analysis, so reliability methods based on function approximation are commonly used such as first-order reliability method (FORM) [1][2][3][4], second-order reliability method (SORM) [5][6][7][8][9][10][11], dimension reduction method [12][13][14][15][16], response surface method [17][18][19], and polynomial chaos expansion [20]. FORM and SORM approximate the performance function at the most probable point (MPP), which has the highest probability density on a limit-state surface and can be obtained by searching the minimum distance from the origin to the limit-state surface in the standard normal space (U-space).…”