# 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  
In the above, the reconstruction calculation process uses the method of randomly generating the measurement matrix to obtain the temperature measurement value. There is a big difference between the reconstruction errors calculated by the combination of different points' locations. This phenomenon shows that the different arrangement of measuring points has a great influence on the reconstruction accuracy. In order to quantify this effect, test condition 5 is taken as the research object. Combined with the previous analysis conclusions, the core tensor of 9 \(\times\) 4 \(\times\) 5 \(\times\) 3 and 13 temperature sensors' placement are used to reconstruct the temperature profile. Five analysis conditions are set up. Under the condition that other temperature distribution  
Figure 8: Comparison of reconstruction results with different sensor numbers (10–25). reconstruction calculation processes are consistent, five groups of different measuring point arrangements are randomly selected. The reconstruction results are as follows. The difference in the reconstruction error in Table 2 fully illustrates the great influence of measuring point arrangement on reconstruction accuracy. Therefore, optimizing the arrangement of measuring points has a positive effect on improving the accuracy of the temperature distribution reconstruction and the stability of the calculation. It can be seen that the coefficient 1-mode decomposition factor vector \(\mathbf{f}_{1x}^{\text{T}}\) is obtained by the Formula (8) and applied to the calculation of the reconstruction results. From Formula (8), \(\mathbf{B}\) is calculated from the measurement matrix \(\mathbf{M}\) and the reconstruction coefficient tensor \(\mathbf{A}_{\text{lunfold}}^{\text{T}}\) along the 1-mode direction. Based on the temperature distribution reconstruction algorithm proposed in this paper, only a small number of temperature measuring points are needed in the calculation process. Therefore, the measurement matrix \(\mathbf{M}\) is a sparse matrix composed of '0'and '1', and the calculated \(\mathbf{B}\) is prone to ill-posed problems. In the numerical analysis, the condition number is introduced to characterize the ill-conditioned degree of matrix \(\mathbf{B}\). In Formula (8), the larger the condition number is, the more sensitive the solution of the coefficient vector \(\mathbf{f}_{1x}^{\text{T}}\) is to the uncertainty in \(k\). The huge deviation of the reconstruction error under different arrangements is derived from this. Based on the above analysis, the optimal placement problem of measuring points is transformed into the condition number minimization problem of Equation (8). The mathematical model is expressed as follows:  
\[\text{min}(\text{cond}(\mathbf{B}))=\text{min}(\|(\mathbf{M}\mathbf{A}_{\text{lunfold}}^{ \text{T}})^{\text{T}}(\mathbf{M}\mathbf{A}_{\text{lunfold}}^{\text{T}})\|\|(\big{(} \mathbf{M}\mathbf{A}_{\text{lunfold}}^{\text{T}}\big{)}^{\text{T}}(\mathbf{M}\mathbf{A}_{\text {lunfold}}^{\text{T}}))^{-1}\|), \tag{11}\]  
where \(\text{cond}(\mathbf{B})\) is the condition number of matrix \(\mathbf{B}\). For the reconstruction problem in this section, a total of 6400 locations can be used as measuring points. When using 13 measuring points for the reconstruction calculation, there are \(\text{C}_{6400}^{13}\) layouts of measuring points. In this case, it is obviously unrealistic to calculate the condition number of each measuring point layout one by one and obtain the global minimum value of the condition number. Therefore, in the process of solving Formula (11), the greedy algorithm is introduced to realize the global optimization calculation of measuring point arrangement [15]. The calculation steps of applying the greedy algorithm to solve the optimal layout of the measuring points are as follows:  
1. All positions are traversed to calculate the number of conditions, and find the location with the smallest number of conditions as the first optimized measurement point;
2.