## 4 Numerical Results
### Numerical Test 2 (Parameter Variation): Homogeneous Boundary Conditions  
The goal of this section is to construct reduced systems that have different values of a given parameter by using one basis set for each POD, DEIM, and GPOD. We consider Allen-Cahn Equation (1) with \(\Omega=[\frac{2\pi}{n},2\pi]\), \(n\in\mathbb{Z}^{+}\), \(n>1\), the initial condition : \(u(x,0)=0.25\sin(x)\), and the homogeneous boundary conditions : \(u(0,t)=0,\quad u(2\pi,t)=0,\quad t>0\). In our numerical tests, the number of internal point is \(n=600\) in \([2\pi,\frac{2\pi}{n}]\), the number of time steps is \(n_{t}=700\) on \([0,5]\) and \(\epsilon=0.01\). We used a space step-size is \(\Delta x=\frac{2\pi}{n-1}\), time step-size is \(\Delta t=\frac{5}{n_{t}-1}\) and the basis sets used in POD and DEIM approximations were constructed from \(n_{s}=700\) snapshots. We used snapshots from the full systems with \(\epsilon=0.01\) and \(\epsilon=0.99\), as shown in Figure 4, respectively, to construct snapshot matrices \(\mathbf{S_{1}}\), \(\mathbf{F_{1}}\) and \(\mathbf{S_{2}}\), \(\mathbf{F_{2}}\). These snapshot matrices were used to construct basis sets for constructing reduced models with different parameter values \(\epsilon\), i.e., \(\epsilon=0.2,0.5,0.8\) as shown in Figure 5. The singular values corresponding to the combined snapshot matrix \([\mathbf{S_{1}},\mathbf{S_{2}}]\), which is used for computing POD basis, is shown in the last plot of Figure 4. The combined nonlinear snapshot matrix \([\mathbf{F_{1}},\mathbf{F_{1}}]\) is used for constructing the basis for DEIM and GPOD approximations. Notice from the plots in Figures 4 and 5 that the behaviors of the solutions are clearly different as the value of parameter \(\epsilon\) changes. given in these tables are scaled with the runtime of the full order system, whose average runtime is considered to be 100%. Note that in Table 5, the dimension \(k\) is fixed to be 50 with different dimension \(m\) of nonlinear basis and different dimension \(q\) of selected rows. It can be observed that the decrease of the average error in Figure 6 corresponds to the increase of dimension of POD basis, as also shown in Table 4. At a fixed POD dimension of \(k=50\), the decrease of the average error corresponds to the increase of dimension of DEIM basis, as also shown in Table 5. Similarly, at a fixed POD dimension of \(k=50\), the decrease of the average error corresponds to the increase of both dimension of nonlinear basis and the number of selected rows in POD-GPOD approach. In Figure 6, the average error for the POD-GPOD reduced system is decreased more than the average error from the POD-DEIM reduced system. The average runtime of the reduced systems by POD approach was ranging between 39.37% to 46.91% of the average runtime of the full-order system, which is roughly 2.1-2.5 times speedup as shown in Table 4. \begin{table}
\begin{tabular}{c c c} \hline \hline
**POD Basis (\(k\))** & **Average Error** & **Average Runtime (Scaled \%)** \\ \hline
10 & \(2.9347\times 10^{-3}\) & 39.37\% \\
20 & \(2.1171\times 10^{-4}\) & 41.61\% \\
30 & \(2.1341\times 10^{-5}\) & 43.69\% \\
40 & \(3.8210\times 10^{-7}\) & 46.12\% \\
50 & \(5.0148\times 10^{-8}\) & 46.79\% \\
60 & \(4.9060\times 10^{-8}\) & 46.87\% \\
70 & \(3.7179\times 10^{-9}\) & 46.91\% \\ \hline Full (100) & - & 100\% \\ \hline \hline \end{tabular}
\end{table} 
Table 4: [Test 2] Average runtime and average error of the POD reduced systems with parameter values \(\epsilon=0.2,0.5\) and 0.8. Figure 6: [Test 2] Average error plot of the approximation the POD, POD-DEIM, and POD-GPOD reduced systems with dimension of DEIM or nonlinear basis \(m\).