## 4 Numerical Results
More details on errors are given in Table 2 and Figure 3.  
The runtime and error from (21) are computed for POD-reduced system as shown in Table 2 and for POD-DEIM and POD-GPOD reduced systems as shown in Table 3 for different dimension \(m\) of nonlinear basis, and number of selected rows \(q\). Figure 1: Solution of the FD full-order system for Allen–Cahn equation with \(\epsilon=0.01\) and the corresponding singular values of the snapshot matrix \(\mathbf{S}\) from SVD using POD basis. Figure 2: [Test 1] Solution of the POD, POD-DEIM, and POD-GPOD reduced systems with \(\epsilon=0.01\) (from left to right). It can be observed in Figure 3 that the error decreases as the dimension of POD increases as also shown in Table 2. For a fixed POD dimension of \(k=50\), the error of POD-DEIM reduced system can be reduced by increasing the dimension of DEIM basis as shown in Table 3 and Figure 3. Similarly for POD-GPOD reduced system, when POD dimension is fixed to be \(k=50\), the error can be decreased by increasing both the dimension of nonlinear basis and the number of selected rows as shown in Table 3. From Figure 3 and Table 3, for each \(m\), the error of POD-GPOD approach is clearly smaller than the error of POD-DEIM approach. Note that, from Figure 3, although the accuracy of the POD approach is higher (less error) than the POD-DEIM and POD-GPOD approaches, the reduction of the simulation time seems to be far less than these last two approaches, as shown in Tables 2 and 3. To compare the speedup of the reduced order approaches, the runtimes given in Tables 2 and 3 are all scaled with the simulation time for the full-order system, which is considered to be 100%. From Table 2, the runtime of POD approach is shown to be approximately 20-30% of the original full order system, i.e., around 3.3-5 times faster than the full order system, when the POD dimensions \(k=10,20,...,70\) are used. From Table 3, the POD-DEIM \begin{table}
\begin{tabular}{c c c} \hline \hline
**POD Basis (\(k\))** & **Error** & **Runtime (Scaled \%)** \\ \hline
10 & \(1.0178\times 10^{-3}\) & 20.93\% \\
20 & \(1.2202\times 10^{-7}\) & 21.14\% \\
30 & \(4.9045\times 10^{-11}\) & 21.63\% \\
40 & \(7.2924\times 10^{-14}\) & 22.25\% \\
50 & \(7.3258\times 10^{-14}\) & 22.61\% \\
60 & \(7.3410\times 10^{-14}\) & 24.46\% \\
70 & \(7.3256\times 10^{-14}\) & 31.61\% \\ \hline Full (100) & - & 100\% \\ \hline \hline \end{tabular}
\end{table} 
Table 2: [Test 1] Runtime and error of the POD reduced system. Figure 3: [Test 1] Error plot of the approximation the POD reduced system (left plot) and the POD-DEIM, and POD-GPOD reduced systems (right plot) with dimension of DEIM or nonlinear basis \(m\). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**DEIM** & **Error** & **Runtime (Scaled \%)** & **GPOD:** & & **Error** & **Runtime (Scaled \%)** \\ \hline
10 & \(4.3498\times 10^{-3}\) & 2.51\% & 10 & 40 & \(2.5195\times 10^{-3}\) & 5.49\% \\
20 & \(7.4722\times 10^{-4}\) & 3.10\% & 20 & 50 & \(1.9981\times 10^{-4}\) & 5.53\% \\
30 & \(4.3474\times 10^{-6}\) & 3.19\% & 30 & 60 & \(1.0603\times 10^{-7}\) & 5.92\% \\
40 & \(7.8929\times 10^{-10}\) & 4.28\% & 40 & 70 & \(1.7405\times 10^{-11}\) & 5.98\% \\
50 & \(1.7124\times 10^{-10}\) & 4.61\% & 50 & 80 & \(1.2332\times 10^{-11}\) & 6.19\% \\
60 & \(2.4349\times 10^{-11}\) & 4.54\% & 60 & 90 & \(3.8857\times 10^{-12}\) & 6.31\% \\
70 & \(2.0649\times 10^{-11}\) & 4.88\% & 70 & 100 & \(1.2603\times 10^{-13}\) & 6.57\% \\ \hline Full & - & 100\% & Full & - & - & 100\% \\ \hline \hline \end{tabular}
\end{table} 
Table 3: [Test 1] Runtime and error of the POD-DEIM reduced system (right table) with POD basis with dimension \(k=50\). approach used only 2.51-4.88% of the full order system's runtime, i.e., roughly 20-40 time faster, when POD dimension \(k=50\) with DEIM dimension \(m=10,...,70\) are used. For POD-GPOD approach, as shown in the second part of Table 3, the runtime is reduced to approximately 5.49-6.57% of the original system, i.e., roughly 15-19 times faster than the original system.