# Frechet-Derivative-Based Global Sensitivity Analysis and Its Physical Meanings in Structural Design
## 2 Primary Concepts and Assumptions in This Work  
Throughout this work, we consider a stochastic parametric model, or more specifically, a computational model with uncertain parameters, which is represented by  
\[X=g(\mathbf{\Theta}), \tag{1}\]  
where the linear or nonlinear mapping \(g(\cdot):\mathbb{R}^{s}\mapsto\mathbb{R}^{1}\) is assumed to be square-integrable, \(\mathbf{\Theta}=(\Theta_{1},\Theta_{2},\cdots,\Theta_{s})^{\top}\in\mathbb{R}^{s}\) is an \(s\)-dimensional random vector, and \(X\in\mathbb{R}^{1}\) is the quantity of interest (QoI) of output. It must be emphasized that the QoI may be multivariable or even time-dependent for many engineering practices. At present, global sensitivity analysis for this type of issue is still filled with challenges, see [1; 25; 26]. For simplicity, only the univariate and time-invariant QoIs are of concern in this work. Consider a more rigorous mathematical definition of Equation (1). Define the triple \((\Omega,\mathscr{F},\mathbb{P})\) as the probability space, where \(\Omega\) stands for the sample space, \(\mathscr{F}\) is the event space (\(\sigma\)-algebra), and \(\mathbb{P}\) is the probability measure. Let \(\varpi\) be the basic event, then Equation (1) can be rewritten by  
\[X(\varpi)=g(\mathbf{\Theta}(\varpi)). \tag{2}\]  
Further, denote the joint probability density function (PDF) of \(\mathbf{\Theta}\) by \(p_{\mathbf{\Theta}}(\mathbf{\theta};\mathbf{\xi})\), where \(\mathbf{\xi}\) are the distribution parameters, and denote the PDF of \(X\) by \(p_{X}(x)\); \(\mathbf{\theta}=(\theta_{1},\theta_{2},\cdots,\theta_{s})^{\top}\) and \(x\) are realizations of \(\mathbf{\Theta}\) and \(X\), respectively. For the \(i\)-th random variable \(\Theta_{i}\), the marginal PDF is defined by \(p_{\Theta_{i}}(\theta_{i};\mathbf{\xi}_{i})\), where \(\mathbf{\xi}_{i}\) are the distribution parameters of \(\Theta_{i}\). For simplicity, the independence of the \(\Theta_{i}\)'s is considered in this paper. According to the principle of preservation of probability [27] and Equation (2), for arbitrary \(\varpi\), there is  
\[\mathbb{P}\{X(\varpi)\in\Omega_{X}^{*}\}=\mathbb{P}\{\mathbf{\Theta}(\varpi)\in \Omega_{\mathbf{\Theta}}^{*}\}, \tag{3}\]  
where \(\Omega_{X}^{*}\) is the subspace of \(\Omega_{X}\) and \(\Omega_{\mathbf{\Theta}}^{*}\) is the subspace of \(\Omega_{\mathbf{\Theta}}\); \(\Omega_{X}\) and \(\Omega_{\mathbf{\Theta}}\) are sample spaces of \(X\) and \(\Omega_{\mathbf{\Theta}}\), respectively. Herein, \(\Omega_{X}^{*}\) is the output space with respect to the input space \(\Omega_{\mathbf{\Theta}}^{*}\), i.e., \(\Omega_{X}^{*}=g(\Omega_{\mathbf{\Theta}}^{*})\). Using the PDFs of \(X\) and \(\mathbf{\Theta}\), Equation (3) can be rewritten equivalently as  
\[\int_{\Omega_{X}^{*}}p_{X}(x;\mathbf{\xi})\mathrm{d}x=\int_{\Omega_{\mathbf{\Theta}}^ {*}}p_{\mathbf{\Theta}}(\mathbf{\theta};\mathbf{\xi})\mathrm{d}\mathbf{\theta}, \tag{4}\]  
and it is now much clearer that the PDF of \(X\) is also a function of \(\mathbf{\xi}\) implicitly. Now consider the following cases:  
(1) For the case that \(g(\cdot)\) is monotonic and \(s=1\), we have  
\[ \begin{split} p_{X}(x;\mathbf{\xi})&=\lim_{\lambda( \Omega_{X}^{*})\to 0}\frac{1}{\lambda(\Omega_{X}^{*})}\int_{\Omega_{\mathbf{\Theta}}^ {*}}p_{\Theta}(\theta;\mathbf{\xi})\mathrm{d}\theta\\ &=\lim_{\lambda(\Omega_{X}^{*})\to 0}\frac{\lambda(\Omega_{ \mathbf{\Theta}}^{*})}{\lambda(\Omega_{X}^{*})}p_{\Theta}\Big{(}\theta=g^{-1}(x); \mathbf{\xi}\Big{)}\\ &=|J|p_{\Theta}\Big{(}\theta=g^{-1}(x);\mathbf{\xi}\Big{)},\end{split}  \tag{5}\]  
where \(|J|=|\mathrm{d}g^{-1}(x)/\mathrm{d}x|\) is the Jacobian determinant and \(\lambda(\cdot)\) stands for the Lebesgue measure; \(g^{-1}(\cdot)\) denotes the inverse function of \(g(\cdot)\). Equation (5) is exactly the so-called change of variables formula in terms of the Lebesgue measure [28].