تجزیه و تحلیل حساسیت جهانی توسط تجزیه ابعادی چند جمله ای

This paper presents a polynomial dimensional decomposition (PDD) method for global sensitivity analysis of stochastic systems subject to independent random input following arbitrary probability distributions. The method involves Fourier-polynomial expansions of lower-variate component functions of a stochastic response by measure-consistent orthonormal polynomial bases, analytical formulae for calculating the global sensitivity indices in terms of the expansion coefficients, and dimension-reduction integration for estimating the expansion coefficients. Due to identical dimensional structures of PDD and analysis-of-variance decomposition, the proposed method facilitates simple and direct calculation of the global sensitivity indices. Numerical results of the global sensitivity indices computed for smooth systems reveal significantly higher convergence rates of the PDD approximation than those from existing methods, including polynomial chaos expansion, random balance design, state-dependent parameter, improved Sobol's method, and sampling-based methods. However, for non-smooth functions, the convergence properties of the PDD solution deteriorate to a great extent, warranting further improvements. The computational complexity of the PDD method is polynomial, as opposed to exponential, thereby alleviating the curse of dimensionality to some extent.

Mathematical modeling of complex systems often requires sensitivity analysis to determine how an output variable of interest is influenced by individual or subsets of input variables. A traditional local sensitivity analysis entails gradients or derivatives, often invoked in design optimization, describing changes in the model response due to the local variation of input. Depending on the model output, obtaining gradients or derivatives, if they exist, can be simple or difficult. In contrast, a global sensitivity analysis (GSA), increasingly becoming mainstream, characterizes how the global variation of input, due to its uncertainty, impacts the overall uncertain behavior of the model. In other words, GSA constitutes the study of how the output uncertainty from a mathematical model is divvied up, qualitatively or quantitatively, to distinct sources of input variation in the model [1]. Almost all GSA are based on the second-moment properties of random output, for which there exist a multitude of methods or techniques for calculating the global sensitivity indices. Prominent among them are a random balance design (RBD) method [2], which integrates its previous version [3] with a Fourier amplitude sensitivity test [4]; a state dependent parameter (SDP) meta-model [5] based on recursive filtering and smoothing estimation; and a variant of Sobol's method with an improved formula [6], [7] and [8]. More recent developments on GSA include application of polynomial chaos expansion (PCE) [9] as a meta-model, commonly used for uncertainty quantification of complex systems [10]. Crestaux et al. [11] examined the PCE method for calculating sensitivity indices by comparing their convergence properties with those from standard sampling-based methods, including Monte Carlo with Latin hypercube sampling (MC-LHS) [12] and quasi-Monte Carlo (QMC) simulation [13]. Their findings reveal faster convergence of the PCE solution relative to sampling-based methods for smoothly varying model responses, but the convergence rate may degrade markedly when confronted with non-smooth systems. They also found the PCE method to be cost effective for low to moderate dimensional systems, even with smooth responses, imposing a heavy computational burden when there exist a mere ten variables or more. Indeed, computational research on GSA is far from complete and, therefore, development of alternative methods for improving the accuracy or efficiency of existing methods is desirable. This paper presents an alternative method, known as the polynomial dimensional decomposition (PDD) method, for variance-based GSA of stochastic systems subject to independent random input following arbitrary probability distributions. The method is based on (1) Fourier-polynomial expansions of lower-variate component functions of a stochastic response by measure-consistent orthonormal polynomial bases; (2) analytical formulae for calculating the global sensitivity indices in terms of the expansion coefficients; and (3) dimension-reduction integration for efficiently estimating the expansion coefficients. Section 2 reviews a generic dimensional decomposition of a multivariate function, including three distinct variants. Section 3 invokes the properties of lower-variate component functions of a dimensional decomposition, leading to a formal definition of the global sensitivity index. The Fourier-polynomial expansion, calculation of sensitivity indices, dimension-reduction integration, including the computational effort, and novelties are described in Section 4. Five numerical examples illustrate the accuracy, convergence properties, and computational efficiency of the proposed method in Section 5. Finally, conclusions are drawn in Section 6.

A PDD method was developed for GSA of stochastic systems subject to independent random input following arbitrary probability distributions. The method is based on Fourier-polynomial expansions of lower-variate component functions of a stochastic response by measure-consistent orthonormal polynomial bases, analytical formulae for calculating the global sensitivity indices in terms of the expansion coefficients, and dimension-reduction integration for efficiently estimating the expansion coefficients. Compared with PCE, which contains the same orthonormal polynomials, but is arranged with respect to the order of polynomials, PDD is structured with respect to the degree of cooperativity between a finite number of random variables. As a result, PDD facilitates simple, direct, and immediate calculation of the global sensitivity indices without the need to generate the ANOVA decomposition of PCE. The PDD method employs measure-consistent orthogonal polynomials, sidestepping the need for transforming arbitrarily distributed random variables to uniform random variables, as required by the classical ANOVA decomposition. The computational complexity of the PDD method is polynomial, as opposed to exponential, consequently curbing the curse of dimensionality to some extent. The PDD method was employed to calculate the global sensitivity indices in five numerical problems, where the output functions are various mathematical constructs involving smooth or non-smooth functions and complex responses from FEA. The error analyses indicate rapid convergence of the PDD solution for smooth non-polynomials, easily outperforming MC-LHS and QMC simulations. Moreover, from the results of the smooth functions examined, the convergence rates of the PDD method are noticeably higher than those of the PCE approximation and other competing methods, including RBD, SDP, and improved Sobol's methods. However, for non-smooth functions, there is a significant loss of convergence properties of the PDD approximation, eroding its advantage over existing methods. Therefore, further improvements of PDD are necessary to effectively deal with non-differentiable functions. The final example demonstrates how the PDD method can be integrated with an external FEA code, identifying important and unimportant variables during stress analysis of a complex mechanical system.