# Application of Tucker Decomposition in Temperature Distribution Reconstruction
## 5 Effect of Core Tensor Dimension on Reconstruction Accuracy  
On the basis of the validity of the reconstruction algorithm, the influence of the core tensor dimension on the reconstruction accuracy during the temperature distribution reconstruction is analyzed to further improve the algorithm accuracy. The temperatures of 1600 positions were selected on the three planes of z = 30 d, 45 d and 60 d in the  
Figure 14: Temperature deviations of reconstruction results in test case 5 (Three-dimensional). Figure 15: Reconstruction errors of simulation data (Three-dimensional). methane combustion simulation model as the research object of 3D temperature distribution reconstruction. The three-dimensional temperature distribution reconstruction algorithm based on Tucker decomposition uses the key information of the prior dataset carried by the core tensor to realize the reconstruction. The core tensor of the prior dataset has four dimensions, so it is necessary to analyze the values of the four dimensions one by one. The prior dataset of 3D temperature distribution reconstruction is a tensor of 24 \(\times\) 3 \(\times\) 40 \(\times\) 40. The dimension in the 2-modular direction in the prior data is small, so the calculation amount of optimizing the dimension value in the 2-modular direction is also small. Therefore, the effect of the core tensor dimension on the reconstruction accuracy is explored from the 2-modular direction. Set up the 2-module direction dimension influence analysis experiment, and the experimental steps are as follows:  
1. In order to avoid too few measurement data points affecting the analysis results, 50 measurement data points were used in the reconstruction. In order to avoid the influence of the dimension of other modes, the dimension of other modes is fixed to 10;
2. Perform Tucker decomposition on the prior dataset to obtain a core tensor with a set dimension of 10 \(\times\) \(n\)\(\times\) 10 \(\times\) 10, \(n\) = 1,2,3;
3. Use the core tensor, decomposition factor matrix and measured temperature to reconstruct the 3D temperature field, and calculate the reconstruction error. Add noise data with SNR = 40 dB to the measured temperature and compare the reconstruction results in the presence of measurement error;
4. Repeat the above steps 1000 times to obtain the average reconstruction error in order to eliminate the influence of randomness in the temperature measurement so a universal conclusion can be obtained;
5. Repeat the above steps until the dimension n has traversed all the values. Taking the test case 5 as an example, the different 2-mode dimensions reconstruction results are shown in Table 3.  
As can be seen from Table 3, as the dimensionality in the 2-mode direction increases, the reconstruction error decreases whether using the original temperature measurement data or using the noise-added measurement data. Therefore, the 2-modular dimension of the core tensor is set to be three in the follow-up research work. When the 2-module dimension is three, the effect of the 1-module dimension on the reconstruction accuracy is analyzed. The 3-module and 4-module dimension are set to 10, and the value range of the 1-module dimension n is 1 to 10. Other parameters and calculation steps are consistent with the above analysis and will not be repeated here. Taking the test case 5 as an example, the reconstruction results of different 1-module dimensions are shown in the table below:  
In Table 4, it can be seen that the reconstruction error using the original temperature measurement data will gradually decrease with the dimension increase; the reconstruction error using noise-added measurement data increases with the increase of dimension. From the perspective of balancing the reconstruction accuracy and noise sensitivity, the 1-module dimension of the core tensor is set to two in the subsequent reconstruction work. Similarly,