Literature Review

Sensitivity analysis has been widely used in engineering design to understand a complex model behavior and help designers make informed decisions regarding where to spend the effort (Grier- son, 1983). In deterministic scenario, sensitivity analysis is conducted to find the rate of change in the model output by varying one input parameter at a time near a given reference point, which involves partial derivatives and thus is referred to as the local sensitivity analysis (Saltelli,2002). For design under uncertainty, the probabilistic sensitivity analysis is a study to quantify the im- pact of uncertainty in input variables on the uncertainty in the model output (Liu et al., 2004). Among existing probabilistic sensitivity analysis methods, a popular category is the so-called variance-based method for global sensitivity analysis (Saltelli et al., 2000;Sobol’, 2001). It is to study how variance in system output can be apportioned to different sources of uncertainty in model inputs (Saltelli,2002). Application of variance to quantify the uncertainty and the associ- ated sensitivity is based on the fact that the output variance is a unified summary of uncertainty regardless of the involved system model. Therefore, the global sensitivity analysis is on studying the impact of variations over the entire range of model inputs, as opposed to the local sensitivity on the variation near a reference point (Saltelli et al.,2000).

Method for variance-based sensitivity analysis is related to the concept of analysis of variance (ANOVA) in linear regression analysis, which is developed as a statistical tool to test the signif- icance of each representative factor. The corresponding regression coefficient, hence, can be

employed to measure the sensitivity of model behavior with respect to the input variable. How- ever, the standard (or classical) ANOVA is only limited to provide the effects of linear and/or second-order interaction of input variables, but it is seldom used to evaluate the highly nonlinear effects, such as the total linear effect, nonlinear main effect, and an arbitrary interaction effec- t, etc., that are critical for ranking the importance of input variables in a product development (Chen et al.,2005).

To extend the standard ANOVA for global sensitivity analysis, a number of variance-based methods have been developed, including the Fourier amplitude sensitivity test (McRae et al.,

1982; Saltelli and Bolado, 1998), various importance measures (Homma and Saltelli, 1996), and the Sobol’ total effect index (Saltelli and Sobol’, 1995;Sobol’,2003), etc. Reviews on the methods of variance-based global sensitivity analysis can be found in literature (Saltelli et al.,

2000,2008). Similar to the concept in standard ANOVA, many of these methods decompose the total output variance to the items contributed by variations of input variables, and then derive the global sensitivity index as the ratio of a partial variance contributed by an effect of interest over the total output variance. The Chapter is proposed to conduct a the global sensitivity analysis by computing the Sobol’ index, since it has been widely used in various areas of industry, such as nuclear engineering (Tarantola et al., 2006) and mechanical engineering design (Chen et al.,

2005;Liu et al.,2006), etc.

Obviously, the variance-based sensitivity analysis can be applied directly to improve the qual- ity of a product by reducing the output variance through controlling the variances of sensitive in- put random variables. Furthermore, the measurement is capable to capture the influential effect of each input variable and the interactions among a subset of inputs. In brief, the variance-based sensitivity analysis can be used in a prior-design stage to screen out variables that are proba- bilistically insignificant and to understand the interactions between design and noise variables, or applied in a post-design stage to determine where the effort should be made to reduce the variability so that the quality of a design can be improved (Chen et al.,2005).

To compute the Sobol’ index for global sensitivity analysis,Ishigami and Homma(1990) pre- sented the Monte Carlo simulation method. A mechanistic model with n input random variables, an N samples simulation needs N.2n 1/ model evaluations for Sobol’ index estimation (Saltelli and Sobol’,1995). To reduce computational cost of the crude Monte Carlo simulation,Tarantola et al.(2006) proposed random balance design (RBD) to conduct the global sensitivity analysis of a nuclear waste disposal system. Polynomial chaos expansion (PCE) developed in literature

(Ghanem and Spanos,1991;Wiener,1938) is another choice to compute the Sobol’ index. The method decomposes an original input-output relation as a summation form of a serial orthogonal polynomial chaos, in which the associated coefficient of each polynomial chaos was estimated by using Monte Carlo simulation (Li and Ghanem,1998) or the rules of Gauss quadrature (Blat- man and Sudret,2010a). Sudret(2008b) reviewed the PCE meta-model construction, in which the computation of Sobol’ index was directly related to the coefficients of the PCE meta-model. Nevertheless, most of methods for the variance-based global sensitivity analysis are without consideration that acquiring samples of model outputs is resource, i.e., demanding computation- ally cost involved in mechanistic model evaluations. The large number of functional calls related to the crude Monte Carlo simulation (Sobol’,2003) and the direct tensor Gauss quadrature (Blat- man and Sudret,2010b) is the motivation of developing an efficient computational technique for the variance-based global sensitivity analysis in this Chapter.

Rahman(2011) combined the functional polynomial decomposition with the dimensional re- duction method (DRM) for the objective. The primary benefit of using DRM is that a series low-dimensional function (based on univariate DRM or bivariate DRM) are employed to ap- proximate an original input-output relation, which reduces the high-dimensional integration as the summation of series of one- or two-dimensional integrals. Combined with the rules of Gaus- sian quadrature, the using of DRM can be trusted to remarkably reduce the number of mechanic model evaluations from N.2n 1/ (the crude Monte Carlo simulation) or Nn (the direct tensor Gaussian quadrature) to the magnitude of nN by using the univariate DRM or nN2 by bivariate DRM, respectively.

6.1.2

Objective

The proposed study is a development of computational method for the Sobol’ index computa- tion by using the dimensional reduction method (DRM) in literature (Li et al., 2006; Rahman and Xu, 2004). The conventional DRM (C-DRM) was derived as a summation of a series low- dimensional function with increasing dimensionality. A novel dimensional reduction method given as a multiplicative form of low-dimensional functions is proposed to approximate a gen- eral input-output relation. The primary benefit of M-DRM entails a high-dimensional moment integration as the product of a series of one-dimensional integrals. The separative property of M-DRM will be employed to handle with the involved high-dimensional integrations infolded in the variance-based global sensitivity analysis.

6.1.3

Organization

Organization of the Chapter is as follows. Section6.2provides background on the variance-based global sensitivity analysis, which consists of mathematical definitions on output variance de- composition and Sobol’ sensitivity index. Section6.3illustrates crucial challenge on the Sobol’ index computation, i.e., one has to evaluate a series of high-dimensional integrals in terms of the second-order moment of the conditional expectation. To overcome the involved intensive computational cost, a multiplicative dimensional reduction method (M-DRM) is proposed to ap- proximate the original high-dimensional integration as the product of a series of one-dimensional integrations. Six examples from literature are employed in Sections6.4to examine the accuracy and efficiency of M-DRM on the Sobol index computation. Section6.5summarizes the conclu- sions, and computational details are given in Appendices.

6.2

Background