# Reconstruction Optimization Algorithm of 3D Temperature Distribution Based on Tucker Decomposition
## 4 Reconstruction Parameters Optimization of Tucker Decomposition
### Effect of Measuring Point Arrangement on Reconstruction Accuracy
Retain the first optimized measurement point, traverse the remaining position, calculate the corresponding condition number together with the first optimized measurement position, and take the position with the smallest condition number as the second optimized measurement position;
3. Repeat step 2 until all measuring points are optimized. The optimal arrangement of measuring points calculated by the greedy algorithm is applied to the temperature distribution reconstruction, and it can improve the accuracy and stability of the algorithm to a certain extent. However, when comparing the reconstruction results, it is found that there is an optimization space to further reduce the condition number. Therefore, after using the greedy algorithm to obtain the preliminary optimization results, \begin{table}
\begin{tabular}{c c} \hline \hline
**Arrangement** & **Reconstruction Error(\%)** \\ \hline
1 & 3.1 \\
2 & 11.7 \\
3 & 6.78 \\
4 & 9.2 \\
5 & 18.7 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Reconstruction errors with different sensor placement. the climbing algorithm is introduced to further search for the local optimal solution [16]. The calculation steps of the algorithm are as follows. 1. The initial optimization condition number obtained by the greedy algorithm is set as the judgment basis, and the corresponding measuring point arrangement is taken as the starting position;
2. Generate a new arrangement in a random step length near the preliminary optimized position of measuring points;
3. Calculate the condition number corresponding to the new measuring point arrangement. If the condition number is less than the judgment basis, the condition number is used as a new judgment basis, the corresponding measuring point position is updated to a new starting position, and the calculation in Step (2) is repeated. If the condition number is greater than the criterion, repeat Step (2) directly;
4. Set the optimization calculation step; if in the calculation step and the judgment basis has remained unchanged, that is, the judgment basis for local optimum, stop the optimization calculation. The core tensor of \(9\times 4\times 5\times 3\) and 13 temperature measuring points are used to build the temperature distribution. The comparison of optimization results is shown in Figure 9.  
As shown in Figure 9, the arrangement of the measuring points obtained by the conditional number method is optimized in the temperature distribution reconstruction of the five test cycles. It can be seen from Figure 9 that the arrangement of the measuring points calculated by the greedy algorithm and the hill climbing algorithm can reduce the error of temperature distribution reconstruction to a certain extent. In the five groups of test conditions, the mountain climbing algorithm is used to compare the original data, and the reconstruction accuracy is improved by 3.2%, 3.5%, 3.8%, 4.0%, and 3.3%, respectively. The mountain climbing algorithm is improved by 0.18%, 0.24%, 0.17%, 0.03%, and 0.13%, respectively, compared with the greedy algorithm. Therefore, in the simulation experiment, the arrangement of measuring points obtained by using the optimal arrangement algorithm based on the greedy algorithm and the hill climbing algorithm can improve the accuracy of the reconstruction algorithm. Figure 9: Comparison of Optimized Effects.