Global sensitivity analysis by polynomial dimensional decomposition

doi:10.1016/j.ress.2011.03.002

Reliability Engineering & System Safety

Volume 96, Issue 7, July 2011, Pages 825-837

https://doi.org/10.1016/j.ress.2011.03.002 Get rights and content

Abstract

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.

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 $(Ω, F, P)$ be a complete probability space, where $Ω$ is a sample space, $F$ is a $σ ‐ field$ on $Ω$ , and $P : F \to [0, 1]$ is a probability measure. With $B^{N}$ representing the Borel $σ ‐ field$ on $R^{N}$ , consider an $R^{N} ‐ valued$ independent random vector $X = {X_{1}, \dots, X_{N}}^{T} : (Ω, F) \to (R^{N}, B^{N})$ , which describes statistical uncertainties in all system and input parameters of a given stochastic problem. The probability law of $X$ is completely defined by the joint probability density function $f_{X} (x) = \prod_{i = 1}^{i = N} f_{i} (x_{i})$ , where $f_{i} (x_{i})$ is the

Variance decomposition

The ADD in Eq. (3) can be written more explicitly as $y (X) = y_{0} + \sum_{i = 1}^{N} y_{i} (X_{i}) + \sum_{i_{1} = 1}^{N - 1} \sum_{i_{2} = i_{1} + 1}^{N} y_{i_{1} i_{2}} (X_{i_{1}}, X_{i_{2}}) + \dots + \sum_{i_{1} = 1}^{N - s + 1} \dots \sum_{i_{s} = i_{s - 1} + 1}^{N} y_{i_{1} \dots i_{s}} (X_{i_{1}}, \dots, X_{i_{s}}) + \dots + y_{12 \dots N} (X_{1}, \dots, X_{N}),$ where the constant y₀ and component functions $y_{i_{1} \dots i_{s}} (x_{i_{1}}, \dots, x_{i_{s}})$ , $1 \leq i_{1} < \dots < i_{s} \leq N$ , $s = 1, \dots, N$ , are obtained from $y_{0} ≔ \int_{R^{N}} y (x) f_{X} (x) d x,$ $y_{i} (x_{i}) ≔ \int_{R^{N - 1}} y (x) \prod_{j \neq i} f_{j} (x_{j}) {dx}_{j} - y_{0},$ $y_{i_{1} i_{2}} (x_{i_{1}}, x_{i_{2}}) ≔ \int_{R^{N - 2}} y (x) \prod_{j \neq [i_{1}, i_{2}]} f_{j} (x_{j}) {dx}_{j} - y_{i_{1}} (x_{i_{1}}) - y_{i_{2}} (x_{i_{2}}) - y_{0},$ $⋮ ⋮ ⋮$ $y_{i_{1} \dots i_{s}} (x_{i_{1}}, \dots, x_{i_{s}}) ≔ \int_{R^{N - s}} y (x) \prod_{j \neq [i_{1}, \dots, i_{s}]} f_{j} (x_{j}) {dx}_{j} - \sum_{j_{1} < \dots < j_{s - 1} \subset {i_{1}, \dots, i_{s}}} y_{j_{1} \dots j_{s - 1}} (x_{j_{1}}, \dots, x_{j_{s - 1}}) - \sum_{j_{1} < \dots < j_{s - 2} \subset {i_{1}, \dots, i_{s}}} y_{j}$

Fourier-polynomial expansion

Defined on the product probability triple $(\times_{p = 1}^{p = s} Ω_{i_{p}}, \times_{p = 1}^{p = s} F_{i_{p}}, \times_{p = 1}^{p = s} P_{i_{p}})$ , $1 \leq s \leq N$ , denote the space of square integrable s-variate component functions of y by $L_{2} (\times_{p = 1}^{p = s} Ω_{i_{p}}, \times_{p = 1}^{p = s} F_{i_{p}}, \times_{p = 1}^{p = s} P_{i_{p}}) ≔ \{y_{i_{1} \dots i_{s}} (X_{i_{1}}, \dots, X_{i_{s}}) : \int_{R^{s}} y_{i_{1} \dots i_{s}}^{2} (x_{i_{1}}, \dots, x_{i_{s}}) \times \prod_{p = 1}^{s} f_{i_{p}} (x_{i_{p}}) {dx}_{i_{p}} < \infty\},$ which is a Hilbert space. Since the joint density of ${X_{i_{1}}, \dots, X_{i_{s}}}^{T}$ is separable (independence), the tensor product $\prod_{p = 1}^{s} ψ_{i_{p} j_{p}} (X_{i_{p}}) = ψ_{i_{1} j_{1}} (X_{i_{1}}) \dots ψ_{i_{s} j_{s}} (X_{i_{s}})$ constitutes an orthonormal polynomial basis in $L_{2}$ . 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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Citation 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.
We introduce an efficient non-intrusive surrogate-based methodology for global sensitivity analysis and uncertainty quantification. Modified covariance-based sensitivity indices (mCov-SI) are defined for outputs that reflect correlated effects. The overall approach is applied to simulations of a complex plasma-coupled combustion system with disparate uncertain parameters in sub-models for chemical kinetics and a laser-induced breakdown ignition seed. The surrogate is based on an Analysis of Variance (ANOVA) expansion, such as widely used in statistics, with orthogonal polynomials representing the ANOVA subspaces and a polynomial dimensional decomposition (PDD) representing its multi-dimensional components. The coefficients of the PDD expansion are obtained using a least-squares regression, which both avoids the direct computation of high-dimensional integrals and affords an attractive flexibility in choosing sampling points. This facilitates importance sampling using a Bayesian calibrated posterior distribution, which is fast and thus particularly advantageous in common practical cases, such as our large-scale demonstration, for which the asymptotic convergence properties of polynomial expansions cannot be realized due to computation expense. Effort, instead, is focused on efficient finite-resolution sampling. Standard covariance-based sensitivity indices (Cov-SI) are employed to account for correlation of the uncertain parameters. Magnitude of Cov-SI is unfortunately unbounded, which can produce extremely large indices that limit their utility. Alternatively, mCov-SI are then proposed in order to bound this magnitude $\in [0, 1]$ . The polynomial expansion is coupled with an adaptive ANOVA strategy to provide an accurate surrogate as the union of several low-dimensional spaces, avoiding the typical computational cost of a high-dimensional expansion. It is also adaptively simplified according to the relative contribution of the different polynomials to the total variance. The approach is demonstrated for a laser-induced turbulent combustion simulation model, which includes parameters with correlated effects.

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Global sensitivity analysis by polynomial dimensional decomposition

Abstract

Introduction

Section snippets

Dimensional decomposition

Variance decomposition

Fourier-polynomial expansion

Numerical examples

Conclusions

Acknowledgment

Reliability Engineering and System Safety

Journal of Computational Physics

Computer Physics Communications

Computer Physics Communications

Computer Physics Communications

Reliability Engineering and System Safety

Reliability Engineering and System Safety

USSR Computational Mathematics and Mathematical Physics

Probabilistic Engineering Mechanics

Probabilistic Engineering Mechanics

Reliability Engineering and System Safety

Reliability Engineering and System Safety

Reliability Engineering and System Safety

Global sensitivity analysis: the primer

Random balance experimentation

Technometrics

Sensitivity analysis for nonlinear mathematical models

Mathematical Modelling & Computational Experiment

Stochastic finite elements: a spectral approach