## 5 Optimal Sensor Placement and Validation Results Based on QR Pivoting
### Simulated Reconstruction Results of the Proposed Optimal Sensor Placement  
As shown in the preceding section, we have completed simulation verification of the proposed reconstruction algorithm, proving its efficiency and stability in 3D reconstruction performance. Moreover the ability of the proposed method in reconstruction performance depends critically on the placement of the sensor locations. Thus it is essential to develop an algorithm to determine the optimal sensor placement. An overarching goal of optimal sensor placement is choosing the fewest \(n\) sensors for reconstruction and inherently involves a tradeoff between the number of sensors and reconstruction accuracy. In what follows, the reconstruction performance based on QR pivoting is evaluated in three aspects: reconstructed nephograms, comparison of relative reconstruction errors and the condition number of \(\text{K}_{\lambda}\). In this section, the number of optimized sensors for velocity field reconstruction and pressure field reconstruction is respectively set as 6, 10 and 8, 15 with reference to Figures 12 and 13. In addition, the noise level is 10%; the boundary condition with 90\({}^{\circ}\), 15 m/s and boundary condition 3 in Table 2 are selected for discussion. For ease of representation, the two boundary conditions are named Boundary Condition 1 and Boundary Condition 2, respectively. After optimizing calculation, it is found that the optimal locations of the sensors obtained by QR pivoting are unique for the determined database, which means that only the optimal locations of 10 and 15 sensors need to be determined, so only the location of 10 sensors in the velocity field and the location of 15 sensors in the pressure field need to be shown in this section. Figures 19-22 show the reconstructed nephograms of the velocity and pressure fields with sensor locations (red and black dots) using QR pivoting, which indicate that using limited sensors can still accurately reconstruct a wind field if the placement is reasonable. As can be seen from these figures, the optimized sensor locations are located on the first plane (0.22 m), so in order to more accurately observe the sensor locations, the nephograms of the first plane and the measurement points distribution are plotted separately, as shown in Figures 23-26. Overall, two features of the optimized sensor distribution can be drawn from the nephograms: (1) the optimal locations are entirely distributed on the 0.22 m plane, where the velocity, pressure and their gradient are the largest; (2) most sensors are located near the sinusoidal model, where the gradients of velocity and pressure are relatively large. Both features seem to indicate that better reconstruction results can be obtained by placing the sensor in areas with large gradients of velocity and pressure, where reconstruction errors are also typically larger. Figure 19: Reconstructed velocity nephograms with optimal sensor locations (Boundary Condition 1). Figure 21: Reconstructed velocity nephograms with optimal sensor locations (Boundary Condition 2). Figure 22: Reconstructed pressure nephograms with optimal sensor locations (Boundary Condition 2). Figure 23: Reconstructed velocity nephogram (boundary condition 1) of the first plane with ten-sensor positions (black points). Figure 24: Reconstructed pressure nephogram (boundary condition 1) of the first plane with fifteen-sensor positions (black points). Figure 25: Reconstructed velocity nephogram (boundary condition 2) of the first plane with ten-sensor positions (black points). Given the new error metric, i.e., the matrix condition number presented in Section 4.2, the performance of QR pivoting in reducing the condition number will be further evaluated in this section. Tables 4-7 show the condition number and the relative reconstruction error of two placements in the velocity field and the pressure field with two boundary conditions.