Global sensitivity analysis by polynomial dimensional decomposition
Introduction
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], [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.
Section snippets
Dimensional decomposition
Let be a complete probability space, where is a sample space, is a on , and is a probability measure. With representing the Borel on , consider an independent random vector , which describes statistical uncertainties in all system and input parameters of a given stochastic problem. The probability law of is completely defined by the joint probability density function , where is the
Variance decomposition
The ADD in Eq. (3) can be written more explicitly aswhere the constant y0 and component functions , , , are obtained from
Fourier-polynomial expansion
Defined on the product probability triple , , denote the space of square integrable s-variate component functions of y bywhich is a Hilbert space. Since the joint density of is separable (independence), the tensor product constitutes an orthonormal polynomial basis in . Therefore, there exists a
Numerical examples
Five numerical examples involving four well-known mathematical functions from the literature and an industrial-scale solid-mechanics problem are presented to illustrate the proposed PDD method for calculating the global sensitivity indices. The mathematical functions selected are smooth or non-smooth, permit exact solutions of the sensitivity indices, and have been studied using a number of existing methods, facilitating a critical evaluation of the PDD method developed. Whenever possible, the
Conclusions
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
Acknowledgment
The author would like to acknowledge financial support from the U.S. National Science Foundation under Grant no. CMMI-0653279.
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2018, Journal of Computational PhysicsCitation Excerpt :Compared to a projection approach [16–18], where each polynomial coefficient is obtained by computing a multi-dimensional integral, the LSR approach is more flexible (in choosing sampling points), which seems to be generally advantageous for problems involving a potentially large number of uncertain parameters. In contrast to the Blatman and Sudret [14] approach, Tang et al. [19] used ANOVA expansion and Polynomial Dimensional Decomposition (PDD) [20–24] to represent the expansion's component subspace functions. The main challenge, and motivation for improving PDD methods, is reducing the required number of samples, which corresponds to the cost of the method.