## 3 Numerical Results
### Parameter Uncertainty Analysis of Burgers' Equation  
#### 3.2.1 Estimate Analysis  
Let \(\Omega=[0,1]\subset R\) be the spatial domain and the time be \(t\in[0,1]\subset R\). We consider the [PERSON] partial differential equation [25,38]:  
\[\left\{ \begin{array}{l}\frac{\partial x}{\partial t}+x\frac{\partial x}{ \partial\omega}-\mu\frac{\partial^{2}x}{\partial\omega^{2}}=0\\ x_{0}(\mu)=0\end{array} \right., \tag{24}\]  
where \(x\) is the state at spatial coordinate \(\omega\in\Omega\), and \(\mu\) is the input uncertainty parameter, which obeys the uniform distribution in the domain \([0.1,1]\) for uncertainty analysis. The boundary conditions are \(x(0,t;\mu)=1\) and \(x(1,t;\mu)=-1\) for \(t:[0,1]\subset R\). We use the finite difference method to discretize the Burgers' Equation (24) in an iso-spacing grid with 129 points of the spatial domain \(\Omega\). We obtain parameterized nonlinear ordinary differential equations:  
\[\frac{d}{dt}\mathbf{y}(t;\mu)=\mathbf{A}(\mu)\mathbf{y}(t;\mu)+\mathbf{F}(\mu )\mathbf{y}(t:\mu)+\mathbf{B}(\mu)u(t) \tag{25}\]  
Figure 4: (**a,b**) Comparison of the MFMC and standard MC estimates of main or total effect sensitivities for the NLS Equation (18) with computational budgets \(p=100\). (**c,d**) Comparison of the MFMC and standard MC estimates of main or total effect sensitivities for the NLS Equation (21) with computational budgets \(p=1000\). where \(\mathbf{A}(\mu)\in R^{N\times N}\) is the operator of Burgers' equation linear term, \(\mathbf{F}(\mu)\in\mathbb{R}^{N\times N(N+1)/2}\) is the operator corresponding to the nonlinear term, and \(\mathbf{B}(\mu)\in R^{N\times P}\) is the operator of boundary condition. To develop a multi-fidelity estimator based on the data-driven low-fidelity model of nonlinear partial differential equations. The numerical solutions of Equation (24) are used as a high-fidelity model \(f^{(1)}\). Then, the data-driven low-fidelity models are obtained by using operator reduction coupled with DEIM. Two types of low-fidelity models are constructed, respectively. The values of random parameter space for one of the low-fidelity model \(f^{(2)}\) are calculated by DEIM of high fidelity models. We construct other reduction model from 129 to 20 dimensions by using the operator reduction of the full-model. The value of random parameter space is calculated by DEIM for reduction model, and we obtain the low-fidelity model \(f^{(3)}\). The information of each multi-fidelity model, calculated using 20,000 samples, is reported in Table 4.  
In the uncertainty quantification analysis of nonlinear partial differential equation, we are interested in the expected \(E[f(\mu)]\) of random parameter. The sample output function is \(f(\mu)=\int_{0}^{1}x(w,T;\mu)dw\), and the time is \(T=0.5\). The MC estimate \(\overline{y}_{m}^{(1)}\) of \(E[f^{(1)}(\mu)]\) is computed from \(n=2\times 10^{4}\) and is used to estimate the MSE. The MC method is compared with the MFMC estimator under the given calculation budgets. It is hoped that the optimal MFMC method can obtain satisfactory estimation. We use the high-fidelity and the low-fidelity models as shown in Table 5 to estimates the mean, variance and sensitivity of output function for the Burgers' equation, where the low-fidelity models use the operator reduction method and the DEIM for the parameterization Burgers' equation. \(\alpha\omega\) High-fidelity (FD) \(f^{(1)}\text{Low}-\text{fidelity}\) (DEIM) \(f^{(2)}\text{Low}-\text{fidelity}\) (\(\text{OR}+\text{DEIM}\)) \(f^{(3)}1\times 10^{-4}\) The high-fidelity model \(f^{(1)}\), low-fidelity models \(f^{(2)}\) and \(f^{(3)}\) are divided into three groups for comparative study. Figure 5 displays the estimated MSE (23) of those three models, respectively. The MSE of the MFMC estimator achieves a speedup of up to two orders of magnitude compared to the standard MC estimator.