## 3 Results
### DMD Mode-Shapes Reconstruction  
As the first test, the vorticity of the double-gyre flow (as represented in [10]) was taken and used. The vorticity \(w\) is given as  
\[w=\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}= \pi A\cos\left(\pi f(x,t)\right)\sin\left(\pi y\right)\frac{ \partial^{2}f}{\partial x^{2}}-\pi^{2}A\sin\left(\pi f(x,t)\right)\sin\left( \pi y\right)\left(\frac{\partial f}{\partial x}\right)^{2} \tag{18}\] \[-\pi^{2}A\sin\left(\pi f(x,t)\right)\sin\left(\pi y\right)\]  
where \(A=0.1\), \(\omega=\frac{2\pi}{10}\), \(\epsilon=0.25\), and  
\[f(x,t)=\epsilon\sin\left(\omega t\right)x^{2}+x-2\epsilon\sin\left(\omega t \right)x \tag{19}\]  
The equation was evaluated over the bounded region \([0,2]\times[0,1]\) for 10 s with time intervals of 0.05 s which resulted in 201 snapshots. The region was discretized as a \(512\times 256\) mesh. The number of sampling points was 2500 and they were randomly spread over the region. The same sampling mask was used for both methods. Due to the very few numbers of nonzero Fourier coefficients, only 10 DCT basis vectors along each spatial direction were used (\(s_{x}=s_{y}=10\)). The csDMD method directly resulted in the reconstruction of DMD mode shapes whereas DMDct resulted in the reconstruction of the fully-sampled dataset. After the fully-sampled dataset was reconstructed by DMDct, the exact DMD method was applied to get the DMD mode shapes which were further used for comparison.  
All DMDs were performed with nine modes. Since the complex eigenvalues come in pairs of conjugate numbers, only those having non-negative imaginary parts are represented here. Note that, similar to an eigenvector, a mode shape may be multiplied by any non-zero scalar without making a difference. Thus, to compare the mode shapes, they should be aligned with each other prior to making any comparison. Given \(\boldsymbol{\phi}_{i}\) and \(\boldsymbol{\psi}_{i}\) are the vectorized mode shapes corresponding to the \(i\)-th eigenvalue resulted from DMD and csDMD, respectively, the complex scalar \(c_{i}\) that results in the best alignment of the vector \(\boldsymbol{\psi}_{i}\) with the vector \(\boldsymbol{\phi}_{i}\) is found by solving the following minimization problem  
\[c_{i}=\operatorname*{argmin}_{c}\|\boldsymbol{\phi}_{i}-c\boldsymbol{\psi}_{ i}\|_{2}^{2}=\frac{\boldsymbol{\phi}_{i}^{T}\boldsymbol{\bar{\psi}}_{i}}{ \boldsymbol{\psi}_{i}^{T}\boldsymbol{\bar{\psi}}_{i}} \tag{20}\]  
Similarly, for the vectorized mode shape \(\boldsymbol{\theta}_{i}\) resulted from DMDct, the alignment factor \(d_{i}\) is found as  
\[d_{i}=\operatorname*{argmin}_{d}\|\boldsymbol{\phi}_{i}-d\boldsymbol{\theta}_{ i}\|_{2}^{2}=\frac{\boldsymbol{\phi}_{i}^{T}\boldsymbol{\bar{\theta}}_{i}}{ \boldsymbol{\theta}_{i}^{T}\boldsymbol{\bar{\theta}}_{i}} \tag{21}\]  
Thus, the comparison was made between the vectors \(\boldsymbol{\phi}_{i}\) and the corresponding aligned vectors \(c_{i}\boldsymbol{\psi}_{i}\) and \(d_{i}\boldsymbol{\theta}_{i}\).  
Figure 2 shows the real parts of the mode shapes of the first five DMD modes related to the eigenvalues with non-negative imaginary parts and the reconstruction of their mode shapes. The top row shows the mode shapes obtained by applying exact DMD on the fully-sampled dataset, whereas the second and third rows show the aligned csDMD and DMDct reconstructions, respectively. Each column is titled with the corresponding eigenvalue. Both csDMD and DMDct resulted in the reconstruction of the mode shapes with correlation coefficients of approximately 1 which means the reconstructed mode shapes almost identically resembled the references.  
Five sample snapshots of the fully-sampled dataset reconstruction are shown in Figure 3. Both methods resulted in reconstruction RMSE of 0.002. The top row of Figure 3 shows the reference snapshots. The samples are shown in the second row. The third and fourth rows show the reconstruction of csDMD and DMDct, respectively.