## 7 Numerical Results
### Numerical Test 2  
This section considers a test image of size \(807\times 605\) pixels. The corresponding incomplete image has uniform missing pixels equal to \(50\%\), as shown in Figure 4. The goal of this numerical test is to investigate the accuracy and the speedup of the proposed accelerated POD-LS approach when compared to the standard POD-LS approach. Figure 5 illustrates the reconstructed images by the standard POD-LS approach with basis from SVD using dimension \(k=10,50\) and by the accelerated POD-LS approach with basis from rSVD as described in Algorithm 4 case (i) using \(k=10\) with \(m=10,20,50\) and \(k=50\) with \(m=50,80,150\). The corresponding values of the average relative errors and the scaled CPU time of the reconstructed images in Figure 5 are given in Tables 1 and 2, respectively, for the cases of \(k=10\) and \(k=50\). From these tables, the CPU time of the accelerated POD-LS approach is shown to be roughly reduced by 10-50 times when compared to the standard POD-LS approach for the same level of accuracy. Figure 6 demonstrates the trend of the average relative errors for these 2 approaches for different dimensions \(k=10,30,50\) with DEIM dimension \(m\) ranging from 10 to 210. Notice that, as the dimensions \(k\) and \(m\) get larger, the overall errors seem to decrease. Figure 6 also shows that as \(m\) increases, the errors of the accelerated POD-LS approach decrease and get closer to the errors of the standard POD-LS approach. As in the previous numerical test, in Figure 6, we consider the errors of the accelerated POD-LS approach for both cases (i)  
Figure 4: (Numerical Test 2) (**a**) Original image and (**b**) an incomplete image with \(50\%\) missing pixels. Figure 3: (Numerical Test 1) Average relative error of the reconstruction for \(30\%\), \(50\%\), and \(65\%\) missing components using the standard LS method and the accelerated LS with basis from rSVD, denoted by LS-DEIM for case (i) and LS-DEIM-b for case (ii) as described in Algorithm 4 for POD dimension \(k=30\) with DEIM dimension \(m=10,20,30,40,50,60,70,80\). Figure 2: (Numerical Test 1) Reconstructions of incomplete images with \(30\%\), \(50\%\), and \(65\%\) missing pixels by the standard POD-LS accelerated POD-LS approach using \(k=30\) (**Top**) and the proposed POD-LS approach using \(k=30\) and \(m=50\) (**Bottom**). and (ii) in Algorithm 4. The errors for case (ii) seem to be larger in all cases of \(k\) considered in Figure 6. This confirms the result in the previous numerical test that we should use the POD basis with rows extracted from corresponding to the known component directly, instead of using its orthogonalized basis, to select the DEIM points used in the accelerated least-squares approach. \begin{table}
\begin{tabular}{l l c c c} \hline \hline \multirow{2}{*}{**Method**} & \multirow{2}{*}{**dim POD**} & \multirow{2}{*}{**dim DEIM**} & \multirow{2}{*}{**Relative Error**} & \multicolumn{2}{c}{**CPU Time**} \\  & & & & **(Scaled)** \\ \hline Standard POD-LS & \(k=10\) & - & 0.0627 & \(1\) \\ \hline Accelerated POD-LS & \(k=10\) & \(m=10\) & 0.8566 & 0.01382 \\  & \(k=10\) & \(m=20\) & 0.1239 & 0.02425 \\  & \(k=10\) & \(m=50\) & 0.0969 & 0.10185 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: (Numerical Test 2) The average relative error of the reconstructed missing data and the corresponding scaled CPU times for the reconstruction when the standard POD-LS method and the accelerated POD-LS method with rSVD are used. Dimensions for POD: \(k=10\), DEIM: \(m=10,20,50\). Figure 5: (Numerical Test 2) Reconstructed image from standard POD-LS with basis from SVD using dimension \(k=10,50\) and from the accelerated POD-LS with basis from rSVD using \(k=10\) with \(m=10,20,50\) and using \(k=50\) with \(m=50,80,150\).