## 4 Discussion
As shown in Figure 8a, as the standard deviation increases, the PVNR values drop which is as expected since higher noise standard deviation means a lower signal-to-noise ratio. For the case of high noise (SD = 0.50), csDMD resulted in a very low PVNR value (<10 dB) in all snapshots (not shown in the figure). This was even lower than the PVNR values of the noisy dataset which means csDMD failed to denoise the data in that case. The second parameter was the sampling ratio. Figure 8b shows higher sampling ratio results in higher PVNR and so, better reconstruction. This is expected as well since higher sampling ratio means more information is provided. In contrast to the first and second cases, the results of changing the number of basis vectors are interesting and unexpected. As shown in Figure 8c, the highest PVNR values correspond to the case of the fewest number of basis vectors (\(45\times 20\)). We initially expected to observe an improvement in the results as the number of basis vectors increased which did not happen. The reason is that the number of unknowns is determined by the number of basis vectors, i.e., for the case of \(45\times 20\) basis vectors, there is a total of 900 unknowns, whereas, for the case of \(90\times 40\) basis vectors, the number of unknowns is 3600. Increasing the number of unknowns affects the performance of the \(l_{1}\)-regularization method and makes it more difficult to find the proper non-zero subset of coefficients. Thus, limiting the number of basis vectors to a reasonable value is the key. The last parameter to study was the number of dynamic modes. The SVHT method suggested five dynamic modes to pick. Picking fewer modes than five resulted in lower PVNR values over the first half of the snapshots, whereas picking more modes did not make any improvement. This shows the number of modes resulted from SVHT is a good choice. In all cases studied, the PVNR values of csDMD over the first few snapshots were too low whereas DMDct resulted in less deviation of PVNR values than csDMD. In addition, in all cases, DMDct almost always resulted in higher PVNR values than csDMD. Even though DMDct was developed for the special case of 2-D problems defined over a rectangular grid of equally-spaced nodes, the method can be extended to the 3D problems as well. It is also possible to adapt the method to an arbitrary grid of nodes. In summary, DMDct outperforms csDMD in terms of reconstructing the whole dataset regarding the defined metrics. One disadvantage of DMDct compared to csDMD is the more computation time it needs. This is because there are more data to fit in DMDct than csDMD. Since DMDct aims at reconstructing the whole dataset, the Exact DMD must be employed at the end if the mode shapes are desired. The results of both DMDct and csDMD are sensitive to the value of sparseness coefficient \(\beta\) in Equation (13). Here, we ran each algorithm with various \(\beta\) values and then picked the best ones for comparison. For a real case, where the actual solution is unknown, this approach is impractical. The proper choice of sparseness coefficient \(\beta\) remains an open question and will be addressed later.