## 4 Numerical Results  
This section presents the numerical tests that demonstrate the accuracy and efficiency of the resulting POD-GPOD approach on approximating the solution of the parametrized Allen-Cahn equation when compared with the standard POD and POD-DEIM approaches. It mainly considers 3 numerical tests. In Section 4.1, we test these model reduction frameworks on Allen-Cahn equation with non-homogenious boundary conditions and square block initial condition. In Sections 4.2 and 4.3, we test these model reduction approaches with various parameter values for two different initial conditions. In particular, we use only one basis for POD and one basis for DEIM or GPOD to construct several reduced systems for different parameter values that are not previously used in the snapshot sets. In all of these numerical experiments, we use the MATLAB program for solving the numerical solutions. The approximation error of the solution from each of these reduced systems is defined as  
\[e=\frac{\|\mathbf{u}-\tilde{\mathbf{u}}\|_{2}}{\|\mathbf{u}\|_{2}}, \tag{21}\]  
where \(\mathbf{u}\) and \(\tilde{\mathbf{u}}\), respectively, are snapshot matrices from solving the FD full system and reduced system, which could be POD, POD-DEIM, or POD-GPOD reduced system. Numerical Test 1 (Fixed Parameter): Non-Homogenious Boundary Conditions with Square Block Initial Data  
In this subsection, we used a square block initial condition to test model reduction approaches for Allen-Cahn Equation (1). The initial condition is defined by a piecewise constant function with value \(a\) on wave crest and value \(b\) on wave trough. We used the following inputs: internal point \((n=100)\) in \begin{table}
\begin{tabular}{c c} \hline \hline
**System** & **Complexity (One Iteration Step)** \\ \hline The full discretized system & \(\mathcal{O}(n^{3})\) \\ The POD reduced system & \(\mathcal{O}(k^{3}+nk^{2})\) \\ The POD-DEIM reduced system & \(\mathcal{O}(k^{3}+mk^{2})\) \\ The POD-GPOD reduced system & \(\mathcal{O}(k^{3}+qk^{2})\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Computational complexity for computing numerical solutions for Allen-Cahn equation. \([-1,1]\), time steps (\(n_{t}=600\)) on \([0,12]\) and \(\epsilon=0.01\). We used a space step-size \(\Delta x=\frac{2}{n-1}\), and time step-size \(\Delta t=\frac{12}{n_{t}-1}\). First, we selected wave crest \(a=1\) and wave trough \(b=-1\). We solved the numerical solution of the FD full system. When the time increases, the evolution of the Allen-Cahn equation still keeps the phase separation after long time calculation in the left plot of Figure 1. From the right plot of Figure 1, the decreasing of singular values of the snapshot matrix and then beginning to stabilize implies that solution information lies within a subspace whose dimension is significantly lower than the full dimension used in the discretization. That is, we can further reduce the dimensions of the full discretized system and still obtain accurate approximate solution. Next, we solved the numerical solution of the reduced system and used dimensions of POD basis \(k=50\) and DEIM basis \(m=60\). We obtained the approximate evolution of phase function of Allen-Cahn equation from POD, and POD-DEIM approaches for time \(t=0\) to \(t=12\) as shown in the first two plots of Figure 2. Finally, GPOD is used for reducing the complexity of the nonlinear term from the POD reduced system using a dimension of nonlinear basis \(m\) that is less than the number of selected rows \(q\). The dimension of POD basis, nonlinear basis, and the number of selected rows in this numerical test are \(k=50\), \(m=60\), and \(q=90\), respectively. We obtained the evolution of the phase function of Allen-Cahn equation by using POD-GPOD method for time \(t=0\) to \(t=12\) as shown in the last plot of Figure 2. Notice that the approximations from both POD-DEIM and POD-GPOD seem to be visually indistinguishable from the original solution.