Efficient computation of global sensitivity indices using sparse polynomial chaos expansions
Introduction
The mathematical model of a physical system can be regarded as a function of a set of input parameters. In many cases, such a mapping does not have an explicit analytical expression due to the complexity of the underlying phenomena. Evaluating the model response often requires to solve a system of differential equations, which can be achieved by employing discretization schemes (e.g. finite difference and finite element methods) that are usually implemented in computer codes. The increasing availability of computational power and complexity of numerical solvers have allowed one to reduce considerably the numerical errors, and to handle more and more realistic behaviours. However, numerical simulation can provide realistic predictions only if the model input parameters are accurately selected. This is seldom the case in practice since the latter are often affected by uncertainty, which may either result from a lack of knowledge (epistemic uncertainty) or be associated to an intrinsic variability (aleatory uncertainty). In this context, quantifying the contribution of each input parameter to the output variability is of major interest. This may not be an easy task in practice though since the relationship between the input and the output variables of a complex model is not straightforward.
Sensitivity analysis allows one to address this problem. Various methods have been investigated in the literature [1], [2], which are typically separated into two groups:
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local sensitivity analysis, which studies how little variations of the input parameters in the vicinity of given values influence the model response;
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global sensitivity analysis, which is related with quantifying the output uncertainty due to changes of the input parameters (which are taken singly or in combination with others) over their entire domain of variation.
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regression-based methods, which exploit the results of the linear regression of the model response onto the input vector. These approaches are useful to measure the effects of the input variables if the model is linear or almost linear. However, they fail to produce satisfactory sensitivity measures in case of significant nonlinearity [3].
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variance-based methods, which rely upon the decomposition of the response variance as a sum of contributions of each input variables, or combinations thereof. They are known as ANOVA (for ANalysis Of VAriance) techniques in statistics [4]. The Fourier amplitude sensitivity test (FAST) indices [5], [6] and the Sobol’ indices [3], [7], [8], [9] enter this category, see also the review in [10].
Of interest are the Sobol’ sensitivity indices in this paper. These quantities are often computed using crude Monte Carlo simulation, which may make them hardly applicable for computationally demanding models. To overcome this difficulty, one may substitute the model under consideration for an analytical approximation, named metamodel, whose evaluations are inexpensive. The use of various metamodels based on non-parametric regression procedures was investigated in [15], [16], [17]. It is also worth mentioning specific developments of the so-called Gaussian process modelling approach [18], [19]. As an alternative, the metamodelling of the functions arising from the ANOVA decomposition rather than the model function was proposed in [20]. The approach belongs to the family of the high dimensional model representation (HDMR) methods, which consist in neglecting the ANOVA functionals associated with high-order interactions. The use of orthogonal polynomial bases has received much interest in this context [21], [22], [23]. This choice is relevant since the Sobol’ indices can be then computed exactly from algebraic operations on the coefficients of the polynomial expansions.
The present paper is also focused on representations onto polynomial bases, but in a more classical metamodelling framework since one approximates directly the model function. The approach devised herein is based on polynomial chaos (PC) expansions, which have been widely used in stochastic mechanics since the development of the so-called stochastic spectral methods [24]. These approaches rely upon a Fourier–Hermite expansion of the uncertain input parameters and the model output into the governing equations. The coefficients of the PC representation of the model response are eventually determined using a Galerkin minimization technique. Such methods are commonly named intrusive since the solving scheme of the deterministic problem has to be modified. As an alternative, non-intrusive schemes have recently emerged in the literature, allowing the computation of the PC coefficients by means of a set of deterministic model evaluations, i.e. without modifying the underlying computer code. Two approaches are usually distinguished:
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the projection approach: each PC coefficient is recast as a multidimensional integral [25], [26], [27], [28], [29], [30], [31] which can be computed either by simulation or quadrature methods;
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the regression approach [29], [32], [33], [34]: the PC coefficients are estimated by minimizing the mean square error of the response approximation in the mean square sense.
The aim of the present paper is to show how the sparse PC expansions may be used in global sensitivity analysis and in particular illustrate their accuracy and great computational efficiency. In this respect, various analytical models will be addressed as well as a structural mechanics problem.
The remainder of this paper is organized as follows. In the next section one presents the framework of the ANOVA decomposition of the model response. Section 3 is dedicated to the PC representation of the model response and the estimation of the PC coefficients. Two estimates of the accuracy of the PC approximations are proposed and the truncation strategy of the representation is discussed. The iterative procedure for building up a sparse PC approximation is described in Section 4. PC-based sensitivity indices are proposed in Section 5. The computational gain provided by the sparse PC expansions compared to crude Monte Carlo simulation is illustrated in Section 6 by numerical examples, i.e. the studies of three analytical functions and the structural mechanics finite element model of a frame structure.
Section snippets
Probabilistic formulation of the problem
Let us consider a physical model represented by a deterministic function . Here , is the vector of the input variables, and , is the vector of quantities of interest provided by the model, referred to as the model response in the sequel. As the input vector is assumed to be affected by uncertainty, it is represented by a random vector with prescribed joint probability density function (PDF) . This leads to introduce the probability space
Polynomial chaos representation
Let us first denote by L2i the Hilbert space of square-integrable functions with respect to the probability measure , for all i=1,…,M. Let be a complete orthonormal basis (CONB) of L2i. The orthonormality property meanswhere if j=k and 0 otherwise. In this work a focus is given to polynomial bases. In practice, most common continuous distributions can be associated to a specific family of polynomials [44], [45],
Sparse polynomial chaos approximation
Let be a non-empty finite subset of . It is possible to consider the truncated PC expansion defined byThe set is referred to as the truncation set in the sequel. Note that the common truncation scheme in Eq. (16) corresponds to the choice , where p is a fixed integer. However, it has been shown that the high cardinality of this set (say ) may lead to an unaffordable computational cost.
Of interest is the determination of truncation sets of low
ANOVA decomposition of the polynomial chaos expansion
Consider a sparse PC expansion of the model response produced by the procedure outlined in the last section:The statistical moments of the response PC expansion can be analytically derived from its coefficients. In particular, the mean and the variance, respectively, readLet us define by the set of in such that only the indices {i1,…,is} are non-zero:The set corresponds to the
Basic results
Let us consider the so-called Ishigami function which is widely used for benchmarking in global sensitivity analysis [1], [67]:where the Xi (i=1,…,3) are independent random variables that are uniformly distributed over . Note that the model under consideration is sparse in nature since:
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the maximal interaction order is 2;
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the underlying function is even with respect to the variables X2 and X3, hence the PC terms associated with odd polynomials in these variables
Conclusion
In this paper, a method is proposed to compute sensitivity indices of the response of mathematical models at a low computational cost. It relies upon the substitution of the model under consideration by its polynomial chaos approximation. In order to reduce the number of unknown PC coefficients to identify and hence the required number of computer experiments (i.e. the computational cost), an adaptive algorithm is proposed for automatically detecting the significant PC terms, leading to a
References (75)
- et al.
Latin hypercube sampling and the propagation of uncertainty in analyses of complex systems
Reliab Eng Syst Safety
(2003) - et al.
About the use of rank transformation in sensitivity of model output
Reliab Eng Syst Safety
(1995) - et al.
Survey of sampling-based methods for uncertainty and sensitivity analysis
Reliab Eng Syst Safety
(2006) - et al.
Sensitivity analysis of model output: performance of the iterated fractional factorial design method
Comput Stat Data Anal
(1995) - et al.
Plackett–Burman techniques for sensitivity analysis of many-parametered models
Ecol Model
(2001) - et al.
Multiple predictor smoothing methods for sensitivity analysis: description of techniques
Reliab Eng Syst Safety
(2008) - et al.
Multiple predictor smoothing methods for sensitivity analysis: example results
Reliab Eng Sys Safety
(2008) - et al.
Implementation and evaluation of nonparametric regression procedures for sensitivity analysis of computationally demanding models
Reliab Eng Syst Safety
(2009) - et al.
Calculations of Sobol indices for the Gaussian process metamodel
Reliab Eng Syst Safety
(2009) - et al.
Efficient input–output model representations
Comput Phys Commun
(1999)