## 2 Method
### Formulation of DMDct
The unknown matrices \(\mathbf{A}_{j}\) and \(\mathbf{B}_{j}\) are found by solving the \(l_{1}\)-regularization problem  
\[\underset{\mathbf{A}_{j},\mathbf{B}_{j}}{\text{argmin}}\left(E+\beta\sum_{j=1 }^{r}\left(\left\|\mathbf{A}_{j}\right\|_{1}+\left\|\mathbf{B}_{j}\right\|_{1 }\right)\right),\ E\triangleq\frac{1}{2}\sum_{k=0}^{m-1}E_{k} \tag{13}\]  
Some \(l_{1}\)-regularization methods rely on derivatives of \(E\) with respect to the unknown matrices \(\mathbf{A}_{j}\) and \(\mathbf{B}_{j}\). The derivatives are given as  
\[\frac{\partial E}{\partial\mathbf{A}_{j}}=\mathcal{D}_{y}\left(\mathcal{D}_{x }\left(\mathbf{F}_{j}^{T}\right)^{T}\right),\ \mathbf{F}_{j}=\left[f_{ii^{\prime}}^{(j)}\right]_{n_{y}\times n_{x}},\ f_{ii^{ \prime}}^{(j)}=\sum_{k=0}^{m-1}e_{ii^{\prime}}^{(k)}\alpha_{jk} \tag{14}\]  
\[\frac{\partial E}{\partial\mathbf{B}_{j}}=\mathcal{D}_{y}\left(\mathcal{D}_{x }\left(\mathbf{G}_{j}^{T}\right)^{T}\right),\ \mathbf{G}_{j}=\left[\hat{s}_{ii^{\prime}}^{(j)}\right]_{n_{y}\times n_{x}},\ \hat{s}_{ii^{ \prime}}^{(j)}=\sum_{k=0}^{m-1}e_{ii^{\prime}}^{(k)}\beta_{jk} \tag{15}\]  
The implementation steps of DMDct are listed as Algorithm 2. ```
Data:
* \(\mathbf{s}_{k}\): the \(m\) randomly-sampled real-valued data vectors of length \(N_{s}\) (see Figure 1)
* \(M\): the sampling mask (set of \(N_{s}\) pairs of indices \((i,i^{\prime})\) for which the samples are taken)
* \(r\): the number of modes to pick
* \(s_{x},s_{y}\): the respective lengths of DCT transforms along the rows and columns of each snapshot
* \(\beta\): \(l_{1}\)-regularization parameter
* \(\mathbf{A}_{j},\mathbf{B}_{j}\)\((1\leq j\leq r)\): initial values of the \(s_{y}\times s_{x}\) unknown matrices of coefficients (optional, can be zero) Result:
* \(\mathbf{\hat{S}}_{k}\)\((0\leq k<m)\): the \(m\) reconstructed snapshots
1 Form the sampled data matrix \(\mathbf{Z}_{s}=[\mathbf{s}_{0}\ \dots\ \mathbf{s}_{m-1}]\);
2 Apply Exact DMD to the matrix \(\mathbf{Z}_{s}\) to get the vector of DMD eigenvalues \(\boldsymbol{\lambda}\)( see Algorithm 1);
3 Initialize the unknown matrices \(\mathbf{A}_{j},\mathbf{B}_{j}\)\((1\leq j\leq r)\);
4 repeat
5\(E\gets 0\);
6 for\(k=0\)to\(m-1\)do
7\(\mathbf{T}\leftarrow\mathbf{0}\);
8 for\(j=1\)to\(r\)do
9\(\alpha_{jk}\leftarrow\Re\left(\lambda_{j}^{k}\right)\), \(\beta_{jk}\leftarrow\Im\left(\lambda_{j}^{k}\right)\);
10\(\mathbf{T}\leftarrow\mathbf{T}+\alpha_{jk}\mathbf{A}_{j}-\beta_{jk}\mathbf{ B}_{j}\);
11 end for
12\(\mathbf{\hat{S}}_{k}\leftarrow\mathcal{D}_{y}^{-1}\left(\mathcal{D}_{x}^{-1} \left(\mathbf{T}^{T}\right)^{T}\right)\);
13\(\mathbf{\hat{s}}_{k}\leftarrow\) vectorized sample of \(\mathbf{\hat{S}}_{k}\) according to the sampling mask \(M\);
14\(\mathbf{s}_{k}\leftarrow\) vectorized sample of \(\mathbf{S}_{k}\) according to the sampling mask \(M\);
15\(E\gets E+\frac{1}{2}\left\|\mathbf{\hat{s}}_{k}-\mathbf{s}_{k}\right\|_{2}^ {2}\);
16
17 end for
18 ifconvergence not achievedthen
19 for\(j=1\)to\(r\)do
20 Update \(\mathbf{A}_{j}\) and \(\mathbf{B}_{j}\) according to the chosen \(l_{1}\)-regularization method (use Equations (14) and (15) to find the Jacobians \(\frac{\partial E}{\partial\mathbf{A}_{j}}\) and \(\frac{\partial E}{\partial\mathbf{B}_{j}}\) if needed);
21 end for
22
23 end for
24
25 until convergence achieved; returnall\(\mathbf{\hat{S}}_{k}\)\((0\leq k<m)\)
```  
**Algorithm 2**The implementation steps of DMDCt.