## 5 Conclusions  
In this paper, we consider model reduction techniques for Allen-Cahn equation. We introduced a modification of DEIM that is based on the concept of GPOD, called POD-GPOD approach, which compromises between the accuracy of POD and computational efficiency of DEIM. We compared the POD-GPOD approach with the standard POD method and POD-DEIM method on the parametrized Allen-Cahn equation. From the numerical results, as expected, the POD reduced system is more accurate than the POD-DEIM reduced system. However, the POD-DEIM reduced system can be used to substantially reduced the computational complexity in the nonlinear terms, and therefore the runtime for solving is significantly less than the POD reduced system. We extend the POD-DEIM reduced system to the POD-GPOD reduced system by increasing the selected rows used in the nonlinear approximation to improve the accuracy. The POD-GPOD reduced system can further reduce the approximation error when compare with the POD-DEIM reduced system with a small increase in simulation time that is still much less than the one used in POD approach. Therefore, POD-GPOD approach is an efficient method that can balance between the accuracy and efficiency for parametrized nonlinear model reduction. This approach can also readily be applied to other nonlinear dynamical  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**System** & **Average Error** & **Ratio of Average Runtime** \\ \hline The full discretized system & - & 1 \\ The POD reduced system & \(\mathcal{O}(10^{-9})\) & 1/3.0 \\ The POD-DEIM reduced system & \(\mathcal{O}(10^{-7})\) & 1/534 \\ The POD-GPOD reduced system & \(\mathcal{O}(10^{-8})\) & 1/261 \\ \hline \hline \end{tabular}
\end{table}
Table 9: [Test 3] The ratio of the average runtime of these systems to the runtime of the full discretized system, and average error of these systems with parameter values \(\epsilon=0.0105\), \(0.01\), and \(0.0095\), using \(k=50\), \(m=60\), and \(q=90\).  
Figure 9: [Test 3] Average error plot of the approximation from POD reduced systems with parameter values \(\epsilon=0.0105\), \(0.01\), and \(0.0095\), using different dimensions of POD basis \(k\) (left) and from the POD-DEIM and POD-GPOD reduced systems with different dimensions of nonlinear basis \(m\) for a fixed POD dimension \(k=50\) (right).  
systems and it is expected to outperform the standard POD approach in term of simulation time and outperform POD-DEIM approach in term of accuracy.  
Conceptualization, S.C.; methodology, S.C.; software, C.D. and S.C.; validation, S.C. and C.D.; formal analysis, S.C. and C.D.; investigation, C.D. and S.C.; resources, S.C. and C.D.; data curation, C.D. and S.C.; writing--original draft preparation, S.C. and C.D.; writing--review and editing, S.C. and C.D.; visualization, C.D.; supervision, S.C.; project administration, S.C.; funding acquisition, S.C. All authors have read and agreed to the published version of the manuscript.  
This research was funded by Development and Promotion of Science and Technology talents project (DPST) Funding Contract No. 039/2558.  
The authors would like to thank [PERSON] for suggestions on this manuscript.  
The authors declare no conflict of interest.  
The following abbreviations are used in this manuscript:  
\begin{tabular}{l l} DEIM & Discrete Empirical Interpolation Method \\ GPOD & Gappy Proper Orthogonal Decomposition \\ MOR & Model Order Reduction \\ POD & Proper Orthogonal Decomposition \\ SVD & Singular Value Decomposition \\ \end{tabular}