# Frechet-Derivative-Based Global Sensitivity Analysis and Its Physical Meanings in Structural Design
## 2 Primary Concepts and Assumptions in This Work
(2) For the case that \(g(\cdot)\) is non-monotonic and \(s=1\), let \(\{\Omega_{\Theta,k}^{*}\}_{k=1}^{r}\) be a partition of \(\Omega_{\Theta}^{*}\) such that the inverse \(x\mapsto\theta\) is monotonic in each subspace \(\Omega_{\Theta,k}^{*}\). Let the corresponding inverse function be \(g_{k}^{-1}(\cdot)\) for \(k=1,2,\cdots,r\), then there is  
\[ \begin{split} p_{X}(x;\mathbf{\xi})&=\lim_{\lambda( \Omega_{X}^{*})\to 0}\frac{1}{\lambda(\Omega_{X}^{*})}\sum_{k=1}^{r}\int_{ \Omega_{\Theta,k}^{*}}p_{\Theta}(\theta;\mathbf{\xi})\mathrm{d}\theta\\ &=\lim_{\lambda(\Omega_{X}^{*})\to 0}\sum_{k=1}^{r}\frac{\lambda( \Omega_{\Theta,k}^{*})}{\lambda(\Omega_{X}^{*})}\,p_{\Theta}\Big{(}\theta=g_{ k}^{-1}(x);\mathbf{\xi}\Big{)}\\ &=\sum_{k=1}^{r}|I_{k}|p_{\Theta}\Big{(}\theta=g_{k}^{-1}(x);\mathbf{ \xi}\Big{)},\end{split}  \tag{6}\]  
where \(|J_{k}|=|\mathrm{d}g_{k}^{-1}(x)/\mathrm{d}x|\) is the \(k\)-th Jacobian determinant with respect to \(g_{k}^{-1}(\cdot)\). The above probability partition is due to the additivity of probability measures. In addition, a more intuitive and geometric illustration can be seen for Case (1) and Case (2) in Figure A1 of Appendix A.2. (3) For the case that \(g(\cdot)\) is non-monotonic and \(s\geq 2\), similarly, we have  
\[ \begin{split} p_{X}(x;\mathbf{\xi})&=\lim_{\lambda( \Omega_{X}^{*})\to 0}\frac{1}{\lambda(\Omega_{X}^{*})}\int_{\Omega_{\Theta}^{*}}p_{ \Theta}(\theta;\mathbf{\xi})\mathrm{d}\theta\\ &=\lim_{\lambda(\Omega_{X}^{*})\to 0}\frac{1}{\lambda(\Omega_{X}^{*} )}\int_{\Omega_{\Theta_{-1}}^{*}}\int_{\Omega_{\Theta_{0}}^{*}}p_{\Theta}( \theta_{1},\theta_{\sim 1};\mathbf{\xi})\mathrm{d}\theta_{1}\mathrm{d}\theta_{\sim 1}\\ &=\sum_{k=1}^{r}\int_{\Omega_{\Theta_{-1}}^{*}}\lim_{\Lambda( \Omega_{X}^{*})\to 0}\frac{\lambda(\Omega_{\Theta_{1},k}^{*})}{\lambda(\Omega_{X}^{*} )}p_{\Theta}\Big{(}\theta_{1}=g_{k}^{-1}(x,\theta_{\sim 1}),\theta_{\sim 1};\mathbf{\xi} \Big{)}\mathrm{d}\theta_{\sim 1}\\ &=\sum_{k=1}^{r}\int_{\Omega_{\Theta_{-1}}^{*}}|J_{k}|p_{\Theta} \Big{(}\theta_{1}=g_{k}^{-1}(x,\theta_{\sim 1}),\theta_{\sim 1};\mathbf{\xi} \Big{)}\mathrm{d}\theta_{\sim 1},\end{split}  \tag{7}\]  
where \(|J_{k}|=|\partial g_{k}^{-1}(x,\theta_{\sim 1})/\partial x|\) and \(g_{k}^{-1}(x,\theta_{\sim 1})\) is the \(k\)-th inverse function by fixing \(\theta_{\sim 1}=(\theta_{2},\theta_{3},\cdots,\theta_{s})^{\top}\). By considering the sifting property of the Dirac delta function \(\delta_{\mathrm{D}}[\cdot]\), we have  
\[ \begin{split} p_{X}(x;\mathbf{\xi})&=\int_{\Omega_{ \Theta}}p_{X\Theta}(x,\mathbf{\theta};\mathbf{\xi})\mathrm{d}\theta\\ &=\int_{\Omega_{\Theta}}\delta_{\mathrm{D}}[x-g(\mathbf{\theta})]p_{ \Theta}(\theta;\mathbf{\xi})\mathrm{d}\theta,\end{split}  \tag{8}\]  
where \(p_{X\Theta}(x,\mathbf{\theta};\mathbf{\xi})\) is the joint PDF of \((X,\mathbf{\Theta})\). The aforementioned Equations (5)-(8) indicate that \(p_{X}(x;\mathbf{\xi})\) (output PDF) is strictly determined by Equation (1) (physical equation) and \(p_{\Theta}(\mathbf{\theta};\mathbf{\xi})\) (input PDF), which can be expressed in an operator form, i.e.,  
\[p_{X}(x;\mathbf{\xi})=\psi\circ p_{\Theta}(\mathbf{\theta};\mathbf{\xi}), \tag{9}\]  
where \(\psi\) is an operator defined by \(g\). It is beneficial to understand stochastic systems by delineating specific operators. For instance, Equation (9) indicates that the uncertainty propagation (UP) of a stochastic system, i.e., the transformation of probability information from input space to output space, is determined by the physical mechanisms of the system. From a functional perspective, the implementation of this transformation is achieved through the action of the operator determined by physical mechanisms. In this sense, \(\psi\) can be called an UP operator [29]. We will soon see that it can be very convenient to explain the four GSIs by adopting the concept of an UP operator in Equation (9).