## 2 Model Problem of the Allen-Cahn Equation and Finite Difference Approximation
### POD-GPOD Reduced System  
This subsection provides an extension of DEIM based on Gappy POD to improve the accuracy of the POD-DEIM reduced system while still reducing the computational cost of nonlinear term in a PODsystem. This method uses a dimension of the nonlinear basis less than the number of the indices used for selecting row in the approximation. Recall the approximation in (11):  
\[\mathbf{F}(\mathbf{V}_{k}\mathbf{\tilde{u}}(t_{j}))\approx\mathbf{ Zc}(t_{j});\qquad\mathbf{c}(t_{j})\in\mathbb{R}^{m}.\]  
Instead of using \(m\) components of the equation above to determine \(\mathbf{c}(t_{j})\), we can use \(q\) components where \(q>m\). These \(q\) components can be determined from Algorithm 2 with the input basis vectors \(\mathbf{z}_{1},...,\mathbf{z}_{m},\mathbf{z}_{m+1},...,\mathbf{z}_{q}\) obtained from the left singular vectors of the nonlinear snapshot matrix \(\mathbf{F}=\{F(\mathbf{u}(t_{1})),...,F(\mathbf{u}(t_{n_{s}}))\}\) corresponding to the \(q\) largest singular values. In this case the maximum value of \(q\) is the rank of the matrix \(\mathbf{F}\). Suppose \(\widetilde{\mathbf{P}}\in\mathbb{R}^{n\times q}\) be the output from Algorithm 2 from using \(\{\mathbf{z}_{j}\}_{j=1}^{q}\). Then, the goal is to obtain \(\mathbf{c}(t_{j})\) from  
\[\widetilde{\mathbf{P}}^{T}\mathbf{F}(\mathbf{V}_{k}\mathbf{\tilde{u}}(t_{j})) \approx(\widetilde{\mathbf{P}}^{T}\mathbf{Z})\mathbf{c}(t_{j}) \tag{18}\]  
such that  
\[\min_{\mathbf{c}(t_{j})}\left\|\widetilde{\mathbf{P}}^{T}\mathbf{ F}(\mathbf{V}_{k}\mathbf{\tilde{u}}(t_{j}))-(\widetilde{\mathbf{P}}^{T} \mathbf{Z})\mathbf{c}(t_{j})\right\|_{2}^{2}. \tag{19}\]  
The optimal solution is given by  
\[\mathbf{c}(t_{j}) =[(\widetilde{\mathbf{P}}^{T}\mathbf{Z})^{T}(\widetilde{\mathbf{ P}}^{T}\mathbf{Z})]^{-1}(\widetilde{\mathbf{P}}^{T}\mathbf{Z})^{T} \widetilde{\mathbf{P}}^{T}\mathbf{F}(\mathbf{V}_{k}\mathbf{\tilde{u}}(t_{j}))\] \[=(\widetilde{\mathbf{P}}^{T}\mathbf{Z})^{+}\widetilde{\mathbf{P} }^{T}\mathbf{F}(\mathbf{V}_{k}\mathbf{\tilde{u}}(t_{j})),\]  
where \((\widetilde{\mathbf{P}}^{T}\mathbf{Z})^{+}=[(\widetilde{\mathbf{P}}^{T} \mathbf{Z})^{T}(\widetilde{\mathbf{P}}^{T}\mathbf{Z})]^{-1}(\widetilde{ \mathbf{P}}^{T}\mathbf{Z})^{T}\) is the pseudoinverse of \(\widetilde{\mathbf{P}}^{T}\mathbf{Z}\). Since \(\mathbf{F}\) is a componentwise function, \(\mathbf{c}(t_{j})=(\widetilde{\mathbf{P}}^{T}\mathbf{Z})^{+}\mathbf{F}( \widetilde{\mathbf{P}}^{T}\mathbf{V}_{k}\mathbf{\tilde{Z}}(t_{j}))\), and we obtain the following POD-GPOD reduced system  
\[\mathbf{\tilde{u}}(t_{j+1})=\underbrace{\mathbf{C}}_{\mathbf{ precomputed}:k\times k}\underbrace{\mathbf{\tilde{u}}(t_{j})}_{k\times 1}+ \underbrace{\mathbf{N}}_{\mathbf{precomputed}:k\times q}\underbrace{\mathbf{F}( \widetilde{\mathbf{P}}^{T}\mathbf{V}_{k}\mathbf{\tilde{u}}(t_{j}))}_{q\times 1}, \tag{20}\]  
where \(\mathbf{N}=\mathbf{V}_{k}^{T}\mathbf{A}^{-1}\mathbf{Z}(\widetilde{\mathbf{P}} ^{T}\mathbf{Z})^{+}\in\mathbb{R}^{k\times q}\), can be precomputed and used in every iteration. Note that, similar to the DEIM approximation, pre-multiplying \(\widetilde{\mathbf{P}}^{T}\) is equivalent to selecting the rows \(\varrho_{1},\ldots,\varrho_{q}\) and, therefore, there is no need to perform the matrix multiplication directly. Notice that the computational complexity for each time step depends on only the dimensions \(k\) and \(q\), In general, when \(k,q\ll n\), the POD-GPOD reduced system can significantly speed up the simulation time for solving the Allen-Cahn equation. The steps for constructing the POD-GPOD reduced system are summarized in Algorithm 3. Table 1 provides the comparison of computational complexity for solving the original full-order Allen-Cahn Equation (5), the POD reduced system (9), the POD-DEIM reduced system (17), and the POD-GPOD reduced system (20) in each time step. From Table 1, the computational cost of the FD full system in each time step is dominated by \(\mathcal{O}(n^{3})\) where \(n\) is the dimension of spatial domain, which is generally _large_.