## 4 Quasi-Monte Carlo Method
### Quasi-Random Sequence and Box-Muller Transformation for Reliability Sensitivity Analysis  
The axis orthogonal importance sampling method for reliability sensitivity analysis based on quasi-random sequence and Box-Muller transformation (termed as QBM-AOIS) can be simply described as a deterministic version of the axis orthogonal importance sampling method. Considering the unbiased estimate of the failure probability in Equation (8) and its standard deviation in Equation (9), QBM-AOIS method approximates them through  
\[\hat{p}_{f}=\frac{1}{N}\sum_{i=1}^{N}\Phi_{(0,1)}(-b(\Theta_{i})) \tag{29}\]  
\[\sigma\Big{(}\hat{p}_{f}\Big{)}=\sqrt{\frac{1}{N-1}\Bigg{(}\frac{1}{N}\sum_{i =1}^{N}\Phi_{(0,1)}{}^{2}(-b(\Theta_{i}))-\hat{p}_{f}{}^{2}\Bigg{)}} \tag{30}\]  
where \(\Theta_{i}\) is the \(i\)th sample of a quasi-random sequence, \(b(\Theta_{i})\) is the reliability index of the corresponding plane. In contrast to Equations (8) and (9), one can clearly see that the only difference between the two methods is that the random samples in the axis orthogonal importance sampling method are replaced by well-chosen deterministic sequences stemming from the quasi-random sequence and Box-Muller transformation. In the same way, if random variables are all the normal variable, reliability sensitivities in the original space \(X\) are rewritten as:  
\[\frac{\partial\hat{p}_{f}}{\partial\mu_{X_{j}}}=\frac{\partial\hat{p}_{f}}{ \partial\mu_{Y_{j}}}\cdot\frac{\partial\mu_{Y_{j}}}{\partial\mu_{X_{j}}}=\frac {1}{\sigma_{j}}\frac{1}{N\sqrt{2\pi}}{\sum_{i=1}^{N}}e^{-b^{2}(\Theta_{i})/2} a_{j} \tag{31}\]  
\[\frac{\partial\hat{p}_{f}}{\partial\sigma_{X_{j}}}=\frac{\partial\hat{p}_{f}} {\partial\sigma_{Y_{j}}}\cdot\frac{\partial\sigma_{Y_{j}}}{\partial\sigma_{X_{ j}}}=\frac{1}{\sigma_{j}}\frac{1}{N\sqrt{2\pi}}{\sum_{i=1}^{N}}e^{-b^{2}( \Theta_{i})/2}a_{j}^{2}b(\Theta_{i}) \tag{32}\]Its standard deviations are re-expressed as:  
\[\sigma\Bigg{(}\frac{\partial\hat{p}_{f}}{\partial\mu_{X_{j}}}\Bigg{)}=\frac{1}{ \sigma_{j}}\sqrt{\frac{1}{N-1}\Bigg{\{}\frac{1}{N}{\sum_{i=1}^{N}}\bigg{[}\frac {a_{1j}}{\sqrt{2\pi}}e^{-b^{2}(\Theta_{i})/2}\bigg{]}^{2}-\Bigg{(}\frac{ \partial\hat{p}_{f}}{\partial\mu_{y_{j}}}\bigg{)}^{2}\Bigg{\}}} \tag{33}\]  
\[\sigma\Bigg{(}\frac{\partial\hat{p}_{f}}{\partial\sigma_{X_{j}}}\Bigg{)}=\frac{ 1}{\sigma_{j}}\sqrt{\frac{1}{N-1}\Bigg{\{}\frac{1}{N}{\sum_{i=1}^{N}}\bigg{[} \frac{a_{1j}}{\sqrt{2\pi}}b(\Theta_{i})e^{-b^{2}(\Theta_{i})/2}\bigg{]}^{2}- \Bigg{(}\frac{\partial\hat{p}_{f}}{\partial\sigma_{y_{j}}}\bigg{)}^{2}\Bigg{\}}} \tag{34}\]  
If non-normal random variables are involved, reliability sensitivity and its standard deviation are respectively calculated as:  
\[\frac{\partial\hat{p}_{f}}{\partial\theta}\approx\frac{\hat{p}_{f}\Big{|}_{ \theta+\Delta\theta}-\hat{p}_{f}\Big{|}_{\theta}}{\Delta\theta}=\frac{1}{N( \Delta\theta)}{\sum_{i=1}^{N}}[\Phi_{(0,1)}\big{(}-b(\Theta_{i})|_{ \theta+\Delta\theta}\big{)}-\Phi_{(0,1)}(-b(\Theta_{i})|_{\theta})] \tag{35}\]  
\[\sigma\Big{(}\frac{\partial\hat{p}_{f}}{\partial\theta}\Big{)} \approx\frac{\sigma\big{(}\hat{p}_{f}\big{|}_{\theta+\Delta\theta}-\hat{p}_{f }\big{|}_{\theta}\big{)}}{(\Delta\theta)} \tag{36}\] \[=\sqrt{\frac{\mathbb{E}\left[\left(\hat{p}_{f(\theta_{i+\Delta \theta}-\hat{p}_{f})}^{2}\right)-\mathbb{E}\left[\left(\hat{p}_{f(\theta_{i+ \Delta\theta}-\hat{p}_{f})}\right)\right]^{2}}{(N-1)(\Delta\theta)^{2}}}\] \[=\frac{1}{\Delta\theta}\sqrt{\frac{1}{(N-1)N}{\sum_{i=1}^{N}} \Big{(}\Phi_{(0,1)}\big{(}-b(\Theta_{i})|_{\theta+\Delta\theta}\big{)}-\Phi_ {(0,1)}(-b(\Theta_{i})|_{\theta})\Big{)}^{2}-\Big{(}\hat{p}_{f(\theta+\Delta \theta}-\hat{p}_{f(\theta)}\Big{)}^{2}}\]  
The procedure of the axis orthogonal important sampling method for structure reliability and its sensitivity in this paper is in detail described as follows (shown in Figure 3):  
Step 1. The performance function \(g(\mathbf{x})\) is transformed into the standard normal space as \(g(v)\) in terms of Rosenblatt transform. A MPFP \(\mathbf{v}^{*}\) is obtained by the first-order reliability method.