## 2 Multi-Fidelity Monte Carlo Method
### Multi-Fidelity Monte Carlo Method
#### 2.2.2 Multi-Fidelity Monte Carlo Framework  
The MSE of MFMC method is estimated by the number of model calculations \(\mathbf{m}\) and the coefficients \(\alpha_{2},\ldots,\alpha_{k}\), and the cost \(p\ \in\ R_{+}\) of estimation \(\hat{s}\) is equal to the calculated budget. Since \(e(\hat{s})=Var[\hat{s}]\), it is sufficient to minimize the variance \(Var[\hat{s}]\) of the MFMC estimator. Therefore, the objective of the MFMC method is to minimize the variance of multi-fidelity estimation \(\hat{s}\) under a given computational budget \(p^{*}\in R_{+}\), which can be achieved by solving of the optimization problem:  
\[\begin{array}{c}\underset{\mathbf{m}\in R^{k},\alpha_{2},\ldots,\alpha_{k}\in R }{\arg\min}\;Var[\hat{s}]\\ \mathrm{s.}\mathrm{t}m_{i-1}-m_{i}\leq 0\\ -m_{1}\leq 0\\ \mathbf{w^{T}m}=p\end{array} \tag{7}\]  
where the control variable coefficients \(\alpha_{2},\ldots,\alpha_{k}\in R\) and the number of model calculations \(m_{1},\ldots,m_{k}\) are variables The inequality constraints \(m_{i-1}-m_{i}\leq 0\) for \(i=2,\ldots,k\) and \(-m_{1}\leq 0\) ensure \(0<m_{1}\leq m_{2}\leq,\ldots,\leq m_{k}\). Note that we will show \(m_{1}^{*}>0\) for the optimal \(m_{1}^{*}\) and therefore \(\mathbf{m}^{*}\in R_{+}^{k}\). The equality constraint \(\mathbf{w^{T}m}=p\) guarantees that the cost of the corresponding MFMC estimator is equal to the computation budget \(p\in R_{+}\). [PERSON] et al. [8] proved that the optimization problem in Equation (7) has unique approximate optimal solution under certain conditions of the high-fidelity and low-fidelity models. Let the \(k\) models \(f^{(1)},\ldots,f^{(k)}\) satisfy \(\left|\rho_{1,1}\right|>\ldots>\left|\rho_{1,k}\right|\) and the costs \(w_{1},\ldots,w_{k}\) satisfy the ratios \(\frac{w_{i-1}}{w_{i}}>\frac{\rho_{1,i-1}^{2}-\rho_{1,i}^{2}}{\rho_{1,i}^{2}- \rho_{1,i+1}^{2}}\), for \(i=2,\ldots,k\). The optimal control variable coefficients are:  
\[\alpha^{*}_{i}=\frac{\rho_{1,i}\sigma_{1}}{\sigma_{i}},i=2,\ldots,k \tag{8}\]  
The optimal numbers of model calculations are:  
\[m_{1}^{*}=\frac{p}{\mathbf{w^{T}r^{*}}},m_{i}^{*}=m_{1}^{*}r_{i}^{*} \tag{9}\]  
where \(\mathbf{r^{*}}=[r_{1}^{*},\ldots,r_{k}^{*}]^{T}\in R_{+}^{k},r_{i}^{*}=\sqrt{ \frac{w_{1}(\rho_{1,i}^{2}-\rho_{1,i+1}^{2})}{w_{1}(1-\rho_{1,i}^{2})}}\), and \(p\in R_{+}\) is the calculation budget. The global minimum solution of Equation (7) is \(\left(\mathbf{m}^{*},\alpha_{2}^{*},\ldots,\alpha_{k}^{*}\right)\). The MFMC framework is shown in Figure 1. Using the number of model calculations \(\mathbf{m}^{*}\) and the coefficients \(\alpha_{2}^{*},\ldots,\alpha_{k}^{*}\), the MSE \(e(s^{*})\) of the multi-fidelity estimation \(s^{*}\) is  
\[e(s^{*})=\frac{\sigma_{1}^{2}}{p}\left(\sum_{i=1}^{k}\sqrt{w_{i}(\rho_{1,i}^{2 }-\rho_{1,i+1}^{2})}\right)^{2} \tag{10}\]  
Figure 1: The MFMC framework.  
Since the cost of MC estimate \(\overline{y}_{m}^{(1)}\) is \(p=mw_{1}\), \(m\) is the numbers of model calculations and \(w_{1}\) is the input calculation cost. The MSE of the MC estimator \(\overline{y}_{m}^{(1)}\) is rewritten as:  
\[e(\overline{y}_{m}^{(1)})=\frac{\sigma_{1}^{2}}{m}=\frac{\sigma_{1}^{2}}{p} \omega_{1} \tag{11}\]  
If we compare the MSE of the MFMC with the MSE of the standard MC estimator, it holds that \(e(\tilde{s}^{*})<e(\overline{y}_{m}^{(1)})\), if and only if  
\[\sum_{i=1}^{k}\sqrt{\frac{w_{i}}{w_{1}}(\rho_{1,i}^{2}-\rho_{1,i+1}^{2})}<1 \tag{12}\]  
The low-fidelity models lead to a MFMC estimator with a smaller MSE than the MSE of the standard MC estimator with the same budget \(p\).