# Application of Tucker Decomposition in Temperature Distribution Reconstruction
## 6 Conclusions  
In this paper, a fast and accurate optimization algorithm based on Tucker decomposition has been proposed to reconstruct 3D temperature fields with sparse data. The relevant mathematical derivations from Tucker decomposition have been carried out to determine a 3D temperature field reconstruction algorithm. The dataset used to generate the core tensor plays a very important role in this method. When using CFD results to establish datasets, it is essential to sample large-scale boundary conditions that can cover most of the working conditions in the flow field. Simulations were performed, and numerical results demonstrated that the Tucker decomposition algorithm combined with sparse measurement is capable of reconstructing the 3D temperature field quickly and accurately. To evaluate the performance of this algorithm, a numerical simulation model of methane combustion has been established based on the Flame D of Sandia Laboratory. Simulation experimental results showed that this method can reconstruct temperature distribution rapidly using limited measurement data with the relative of less than 5.08%, which verifies the feasibility of this algorithm. It can greatly reduce the computational complexity and measurement cost. Finally, this paper analyzes the influence of core tensor dimension on reconstruction accuracy in the process of a temperature distribution reconstruction calculation. Core tensor can be extracted from a high-dimensional experience dataset. The first few dimensions carry most of the useful information. With the use of a higher dimension core vector, too much useless information is introduced to the reconstruction process, and the reconstruction results are more sensitive to the noise. Thus, it is necessary to obtain a balance between reconstruction accuracy and noise sensitivity. The results show that the reconstruction error is less than 2% by optimizing the core tensor dimension. Furthermore, the present study lays the groundwork for future research into some interesting fields with several objectives, including the following:  
1. In order to show more object field characteristics, this study uses a wide range of boundary conditions to establish the dataset. This helps to generate the core tensor of Tucker decomposition, but how to construct dataset more effectively remains to be solved. Further research can focus on the source of empirical data, effective dataset sampling and the impact of dataset noise. 2. The proposed method needs little measurement data to yield an acceptable reconstruction result. The sensor number has some effect on the reconstruction result: the greater the number of sensors used, the better result. Therefore, it is necessary to explore the number of appropriate sensors needed to balance the measurement cost and reconstruction accuracy. Figure 17: Optimized effects of core tensor dimension. When only a few sensors are used for reconstruction, sensor placement has a great influence on reconstruction accuracy. Testing the feasibility of finding a sensor placement optimization algorithm that can yield a much better reconstruction result. Conceptualization, Z.L. and S.L. ; methodology, M.C. ; software, P.Y. ; validation, Z.L., M.C., Y.Z. and P.Y. ; formal analysis, Z.L. ; data curation, Z.L. and M.C. ; writing--original draft preparation, Z.L. ; writing--review and editing, Y.Z. ; supervision, S.L. ; project administration, S.L. ; funding acquisition, S.L. All authors have read and agreed to the published version of the manuscript. This work was supported by the National Natural Science Foundation of China [grant number 61871181]. Not applicable. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Working Condition** & **Boundary Condition (a)** & **Boundary Condition (b)** \\ \hline Test Condition 1 & 823 & 1189 \\ Test Condition 2 & 1407 & 1147 \\ Test Condition 3 & 1426 & 1569 \\ Test Condition 4 & 1746 & 1601 \\ Test Condition 5 & 1983 & 2356 \\ \hline \hline \end{tabular}
\end{table}