## 2 Model Problem of the Allen-Cahn Equation and Finite Difference Approximation
### POD Reduced System  
In this section, we will reduce the dimension of the full discretized system for the Allen-Cahn equation using POD-Galerkin method. To construct a POD reduced system, we have to first compute POD basis from the \(n_{s}\) snapshots of the solution \(\mathbf{u}(t)\) from the full system (4):  
\[\mathbf{S}=\left[\mathbf{u}_{1},...,\mathbf{u}_{n_{s}}\right],\]  
where \(u_{j}=u(t_{j})\). We will next explain how POD basis can be obtained based on optimal orthogonal approximation. Let \(\mathbf{\Phi}\) be a matrix whose columns consist of orthonormal vectors of rank \(k\leq r\), where \(r=\text{rank}(\mathbf{S})\). Then, recall that the best approximation of \(\mathbf{u}_{j}\) by using this matrix \(\mathbf{\Phi}\) is given by  
\[\mathbf{u}_{j}\approx\mathbf{\Phi\Phi^{T}u}_{j},\qquad j=1,...,n_{s}.\]  
POD basis is the optimal orthonormal basis that minimizes these approximation errors as shown in the following definition. **Definition 1** ([49]). : _The POD basis matrix of dimension \(k<r\), denoted by \(\mathbf{V}_{k}=[\mathbf{v}_{1},...,\mathbf{v}_{k}]\in\mathbb{R}^{n\times k}\), of a given snapshot set \(\{\mathbf{u}_{j}\}_{j=1}^{n_{s}}\subset\mathbb{R}^{n}\) is the solution of_  
\[\min_{\mathbf{\Phi}\in\mathbb{R}^{n\times k}\text{rank}(\mathbf{\Phi})=k}\sum _{j=1}^{n_{s}}\left\|\mathbf{u}_{j}-\mathbf{\Phi\Phi^{T}u}_{j}\right\|_{2}^{2},\qquad\mathbf{\Phi^{T}\Phi}=\mathbf{I}.\]  
Consider the singular value decomposition (SVD) of the snapshot matrix \(\mathbf{S}\) in the form \(\mathbf{S}=\mathbf{V}\mathbf{\Sigma}\mathbf{W}^{T}\), where \(\mathbf{V}=[\mathbf{v}_{1},...,\mathbf{v}_{r}]\in\mathbb{R}^{n\times r}\) and \(\mathbf{W}=[\mathbf{w}_{1},...,\mathbf{w}_{r}]\in\mathbb{R}^{n_{s}\times r}\) are matrices with orthonormal columns, and \(\mathbf{\Sigma}=\)diag\((\sigma_{1},...,\sigma_{r})\in\mathbb{R}^{r\times r}\) is a diagonal matrix with \(\sigma_{1}\geq\sigma_{2}\geq\cdots\geq\sigma_{r}>0\). It can be shown [49] that the POD basis matrix \(\mathbf{V}_{k}=[\mathbf{v}_{1},...,\mathbf{v}_{k}]\) consists of the first \(k\) columns of the left singular vectors in \(\mathbf{V}\) corresponding to the \(k\) largest singular values and this minimum sum of approximation errors can be written in terms of neglected singular values as  
\[\left\|\mathbf{u}_{j}-\mathbf{V}_{k}\mathbf{V}_{k}^{T}\mathbf{u}_{j}\right\|_ {2}^{2}=\sum_{i=k+1}^{r}\sigma_{i}^{2}. \tag{6}\]  
This error can be used to identify an appropriate dimension \(k\) for a given threshold of accuracy. We can construct the POD basis \(\mathbf{V}_{k}\) from the following Algorithm 1[50], derived from solution snapshots of the full system in \(\mathbf{S}\). ```
\(\bullet\) INPUT : \(\mathbf{S}=[\mathbf{u}(t_{1}),...,\mathbf{u}(t_{n_{s}})]\in\mathbb{R}^{n\times n _{s}}\) \(\bullet\) OUTPUT : \(\mathbf{V}_{k}=[\mathbf{v}_{1},...,\mathbf{v}_{k}]\in\mathbb{R}^{n\times k}\)
1. Perform the singular value decomposition (SVD) of \(\mathbf{S}=\mathbf{V}\mathbf{\Sigma}\mathbf{W}^{T}\) to produce orthogonal matrices \(\mathbf{V}=[\mathbf{v}_{1},...,\mathbf{v}_{r}]\in\mathbb{R}^{n\times r}\) and \(\mathbf{W}=[\mathbf{w}_{1},...,\mathbf{w}_{r}]\in\mathbb{R}^{n_{s}\times r}\) and diagonal matrix \(\mathbf{\Sigma}=\)diag\((\sigma_{1},...,\sigma_{r})\in\mathbb{R}^{r\times r}\), where \(r\) is the rank of \(\mathbf{S}\). 2. Set a threshold to select the \(k\) largest modes from the diagonal matrix \(\mathbf{\Sigma}\) based on (6). 3. Select the columns in matrix \(\mathbf{V}\) which correspond to modes selected in 2 to generate the POD basis \(\{\mathbf{v}_{1},...,\mathbf{v}_{k}\}\in\mathbb{R}^{n\times k}\) and construct the POD projection matrix \(\mathbf{V}_{k}=[\mathbf{v}_{1},...,\mathbf{v}_{k}]\). ```  
**Algorithm 1**Proper Orthogonal Decomposition (POD)  
We can use POD basis matrix \(\mathbf{V}_{k}\) for constructing the reduced system for the full discretized system, by the following two steps.