## 4 Numerical Results
### Numerical Test 3 (Parameter Variation): Non-Homogeneous Boundary Conditions  
This section considers parametrized Allen-Cahn Equation (1) with non-homogeneous boundary conditions. We used \(\Omega=[-1,1]\), with the initial condition:  
\(u(x,0)=0.53x+0.47\sin(1.5\pi x)\), and the non-homogeneous boundary conditions:  
\(u(-1,t)=-1,\quad u(1,t)=1,\quad t>0\). In this numerical tests, the number of internal points is \(n=600\) in \([-1,1]\), the number of time steps is \(n_{t}=700\) on \([0,60]\) and \(\epsilon=0.01\). We used a space step-size is \(\Delta x=\frac{2}{n-1}\), time step-size is \(\Delta t=\frac{60}{n_{t}-1}\). The value of parameter \(\epsilon\) is varying in this numerical test. We used snapshots from the full systems with \(\epsilon=0.011\), and \(\epsilon=0.009\), from time \(t=0\) to \(t=60\), as shown in Figure 7, to construct matrices of solution snapshots and nonlinear snapshots. The decay of singular values of the matrix of solution snapshots is shown in the last plot of Figure 7. These snapshot matrices were used to construct basis sets for constructing reduced models of full systems with different parameter values \(\epsilon\), i.e., \(\epsilon=0.0095,0.01,0.0105\), as shown in Figure 8. Notice from Figures 7 and 8 that the behaviors of the solutions are clearly different as the parameter changes, i.e., as the value of parameter \(\epsilon\) gets bigger, the solution reaches the steady state faster. \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 & \(\mathcal{O}(10^{-8})\) & \(1/2.1\) \\ The POD-DEIM reduced system & \(\mathcal{O}(10^{-5})\) & \(1/411\) \\ The POD-GPOD reduced system & \(\mathcal{O}(10^{-6})\) & \(1/254\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: [Test 2] The ratio of the average runtime of these systems to the runtime of the full discretized system with parameter values \(\epsilon=0.2\), \(0.5\), and \(0.8\), using \(k=50\), \(m=40\) and \(q=70\). Figure 7: [Test 3] Solution of the full discretized systems with \(\epsilon=0.009\) and \(\epsilon=0.011\), and the corresponding singular values these snapshots (from left to right). In Tables 7 and 8, we present average runtimes and average errors from the parameter variation using \(\epsilon=0.0105\), \(0.01\), and \(0.0095\), for POD, POD-DEIM, and POD-GPOD approaches, respectively. We computed the runtime and error from (4.1). Table 8 shows the average error of the POD-GPOD reduced systems with POD basis of dimension \(k=50\), where \(m\) is the dimension of nonlinear basis, and \(q\) is the number of selected rows. The results in this section illustrate the same trends as in the previous numerical tests. From Tables 7 and 8, the POD approach is more accurate, but it is much less efficient in term of computational time than the POD-DEIM and POD-GPOD approaches. From Table 7, when the POD approach uses \(k=50\), the average error is of order \(\mathcal{O}(10^{-9})\) and the average runtime is shown to be \(32.957\%\) of the full-order system, which is approximately three times faster than the full-order system. From Table 8, when the POD-DEIM reduced system uses \(k=50\) and \(m=50\), the average error is of order \(\mathcal{O}(10^{-6})\) and the average runtime is \(0.191\%\) of the full-order system, which is approximately \(524\) times faster than the full-order system. From Table 8, when the POD-GPOD reduced system uses \(k=50\), \(m=50\) and \(q=80\), the the average error is of order \(\mathcal{O}(10^{-7})\) and the average runtime is \(0.379\%\)