## 4 Numerical Results of the Proposed Reconstruction Method
### Presentation of Reconstruction Results of Velocity Field and Pressure Field
Finally, by exploring the influence of different noise levels and the number of basis vectors on the reconstruction error, the optimal number of basis vectors is determined for reconstruction calculation. This section increases the noise level from 1% to 30% with an interval of 1%. Referring to Figures 7 and 8, the number of eigenvectors of the velocity field is set to 2, 4, 6, 8 and 10, and the number of eigenvectors of the pressure field is set to 8, 9, 10, 11 and 12. With a fixed number of sensors and arrangement, the corresponding reconstruction error results are shown in Figures 9 and 10. Figure 8: The “energy proportion” of each eigenvalue in all eigenvalues. Figure 9: Joint effect of the number of basis vectors and the noise level (\(\%\)) on the reconstruction error (velocity field). The combined effect of eigenvectors and noise levels on reconstruction errors can be visually seen from the two preceding figures. By comparing the results of velocity and pressure fields, the following conclusions can be drawn:  
Firstly, for velocity field reconstruction, when the number of eigenvectors is two, the error changes most smoothly with the increase of the noise level, although the reconstruction error is the largest. This indicates that the error is less sensitive to noise when the number of eigenvectors is small. As the eigenvector increases, the error is significantly affected by the noise level. These two features show that the increase of the eigenvector is beneficial to the improvement of the reconstruction error, but the excessive increase will make the error more sensitive to noise. Therefore, referring to Figures 7-9, the number of basis vectors used for velocity field reconstruction is determined to be six. Secondly, for pressure field reconstruction, the trend of error with an increasing number of basis vectors is different from that of a velocity field. As the number of basis vectors increases, the reconstruction errors increase monotonically, which indicates that for databases with large data gradients, the reconstruction accuracy is more susceptible to measurement errors. In this case, the selection of the number of basis vectors should comprehensively consider the accuracy of feature extraction and the impact of the noise level. Therefore, referring to Figures 7, 8 and 10, the number of basis vectors used for pressure field reconstruction is determined to be eight. After determining the number of basis vectors, the reconstruction results of the velocity and pressure fields with the inlet boundary condition of 90\({}^{\circ}\), 15 m/s are obtained in this section. Figure 11 shows the position of five planes, i.e., the 3D velocity field to be reconstructed. Figures 12 and 13 show the simulated and reconstructed velocity nephograms, which show that the reconstructed velocity distribution is reasonably similar to the real distribution, and the relative error is only 0.21%. Figures 14 and 15 show the simulated and reconstructed relative pressure nephograms, and the relative error is 6.46%. Figure 15 shows that the low-pressure area is mainly distributed around obstacles, where the velocity is relatively high, which is corresponding to the velocity distribution characteristics in Figure 13, indicating that the reconstructed pressure can accurately restore the real distribution characteristics. The minimal reconstruction error (0.21%) reflects the efficiency of this method in velocity field reconstruction. Meanwhile the relatively high pressure field reconstruction error (6.46%) seems to indicate that the method is not effective for the reconstruction of data with a large gradient. Significantly, the boundary condition of 90\({}^{\circ}\), 15 m/s is not contained in the snapshot matrix, while the reconstruction results are satisfactory,  
Figure 10: Joint effect of the number of basis vectors and the noise level (%) on the reconstruction error (pressure field). which further validates the feasibility of the proposed reconstruction algorithm based on sparse sensors.