## 3 Numerical Results
### Uncertainty Analysis of NLS Equation with Initial Value  
#### 3.1.1 Estimate Analysis  
NLS equation is a basic equation in quantum mechanics [33; 34]. It can be used to describe the propagation phenomena and principles of optical solitons in fibers, the propagation of optical pulses in dispersion and nonlinear media and the motion of superconducting electrons in electromagnetic fields. We consider the NLS nonlinear partial differential equation:  
\[\left\{ \begin{array}{l}i\frac{\partial u}{\partial t}+\frac{1}{2}\frac{ \partial^{2}u}{\partial x^{2}}+u|u|^{2}=0\\ u(x,0)=u_{0}(x)\end{array} \right. \tag{21}\]  
where \(u\) is a complex number function. Let \(\Omega=(-20,20)\) be the spatial domain and the time be \(t\in[0,2\pi]\subset R\). We assume that the initial value is an uncertain input variable. In the NLS equation, the initial condition satisfies \(u_{0}(x)=N^{*}sech(x)+Z\), \(N=2\) is the double soliton order, \(x\in\Omega\) is the propagation distance and \(Z\) obeys the normal distribution for the uncertainty analysis. Rewriting (21) by Fourier transformation, gives the set of differential equations  
\[\frac{\partial\hat{u}}{\partial t}=-\frac{i}{2}k^{2}\hat{u}+i\widehat{|u|^{2}u} \tag{22}\]  
where \(-\frac{i}{2}k^{2}\hat{u}\) is linear term and \(\widehat{i|u|^{2}u}\) is nonlinear term. To investigate the performance of our multi-fidelity estimation method based on data-driven low-fidelity models. We use the fourth- and fifth-order Runge-Kutta method to solve Equation (22), and the numerical solutions of NLS Equation are regarded as the high-fidelity model \(f^{(1)}\). The data-driven low-fidelity model of NLS equation is used to reduce the dimension for the numerical solutions by POD method and select three POD basis functions. Then, DEIM is used to reduce the dimension of higher order nonlinear terms and we obtain a type of the low-fidelity model \(f^{(2)}\). Table 1 shows the information for calculating these models with 1000 samples. We are interested in the expectation \(E[f(Z)]\) of the NLS equation if the initial value is uncertain. We choose the sample output function is \(f(Z)=\int_{\Omega}\|u(x,T;u_{0}(Z))\|dx\) and the time is \(T=\pi\). The Monte Carlo estimate \(\Psi_{m}^{(1)}\) of \(E[f^{(1)}(Z)]\) is computed from \(n=1\times 10^{3}\) realizations and is used to estimate the MSE. We use the high-fidelity and the low-fidelity models listed in Table 1 to estimates the mean, variance and sensitivity index of output function for the NLS equation. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Models** & \(\mathbf{\alpha}\) & \(\mathbf{\omega}\) & \(\mathbf{m}\) \\ \hline High-fidelity (FD) \(f^{(1)}\) & 1 & 480 & 204 \\ Low-fidelity (POD + DEIM) \(f^{(2)}\) & 0.2082 & 3 & 698 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Number of samples per model, coefficients and costs for NLS equation. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Models** & \(\mathbf{\rho_{i}}\) & \(\mathbf{\sigma_{i}^{2}}\) \\ \hline High-fidelity (FD) \(f^{(1)}\) & 1 & 214.1663 \\ Low-fidelity (POD + DEIM) \(f^{(2)}\) & 0.9989 & 214.3100 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The Pearson correlation coefficient and variances of high- and low-fidelity models for NLS equation. We compare the MC and MFMC expectation estimators in terms of their MSE that is estimated from 20 replicates as  
\[\hat{\varrho}=\frac{1}{20}\sum_{i=1}^{20}\left(E[f^{(1)}(Z)]-\hat{s}_{i}\right)^{2} \tag{23}\]  
where \(E[f^{(1)}(Z)]\) is the reference expectation estimator and \(\hat{s}_{i}\) is either an MFMC estimator (10) or an MC estimator (11). The reference expectation \(E[f^{(1)}(Z)]\) is computed from \(n=1\times 10^{3}\) samples and we obtain \(E[f^{(1)}(Z)]=450.1967\). Figure 2 reports the estimated MSE (23) of the standard MC estimators and the MFMC estimators of budgets from \(p=10^{3}\) to \(10^{5}\). Compared with the standard MC method, the MSE of our MFMC method is smaller. That is, the MFMC method can be significantly better than the standard MC.