## 2 Mathematical Methods
### Reconstruction Algorithm Based on POD
For example, the first element of \(\mathbf{f}_{\mathrm{M}}\) is marked as \(\mathbf{f}_{\mathrm{M}}^{(1)}\), if \(\mathbf{f}_{\mathrm{M}}^{(1)}\) is the measurement data that shows the temperature value of the jth element in \(\mathbf{t}\), then element \(\mathbf{M}_{\mathrm{1 xj}}\) is set as \"1\" and the other elements are set as \"0\". The order of elements in \(\mathbf{t}\) is associated with the location in the objective field; once the k sensors installment is complete, \(\mathbf{M}\) is a known condition. With Equation (3), the reconstruction solution \(\mathbf{f}_{\mathrm{R}}\) is represented by a linear combination, as follows:  
\[\mathbf{f}_{\mathrm{R}}=\mathbf{W}\mathbf{x}_{\mathrm{R}}, \tag{5}\]  
where \(\mathbf{x}_{\mathrm{R}}\) is the reconstruction coefficient vector. The purpose of reconstruction is to find a solution \(\mathbf{x}_{\mathrm{R}}\) that can make \(\mathbf{f}_{\mathrm{R}}\approx\mathbf{t}\). Combining Equations (3) and (4) results in Equation (5), as follows:  
\[\mathbf{t}_{\rm M}\approx\mathbf{M}\mathbf{W}\mathbf{x}_{\rm R}. \tag{6}\]  
Thus, the relationship between the limited measurement data \(\mathbf{t}_{\rm M}\); POD basis vectors \(\mathbf{w}_{1},\mathbf{w}_{2},\mathbf{w}_{3},\cdots,\mathbf{w}_{l}\); and the coefficient vector \(\mathbf{x}_{\rm R}\) is established. As \(\mathbf{M}\mathbf{W}\) is a \(\mathbf{k}\times 1\) dimension matrix, when \(\mathbf{k}<1\), \(\mathbf{x}_{\rm R}\) has infinitely many solutions, and it is not suitable for this study; when \(\mathbf{k}=1\), \(\mathbf{x}_{\rm R}\) has a unique solution; and when \(\mathbf{k}>1\), Equation (5) can be deemed as an overdetermined system, and an optimal solution \(\mathbf{x}_{\rm R}\) can be achieved by least-squares fitting [35]. It could also explain why methods based on POD require a minimum number of sensors, which must always be equal to or larger than the total number of basis vectors used [36]. The objective function \(f\) is obtained as follows:  
\[f=\frac{1}{2}\|\mathbf{t}_{\rm M}-\mathbf{M}\mathbf{W}\mathbf{x}_{\rm R}\|_{2}^{2}. \tag{7}\]  
In order to minimize \(f\), the optimal solution \(\mathbf{x}_{\rm R}\) is determined by differentiating \(f\) with respect to \(\mathbf{x}_{\rm R}\) and setting the value to zero. \[\frac{\partial f}{\partial\mathbf{x}_{\rm R}}=\frac{\partial(\mathbf{x}_{\rm R}{}^{ \rm T}(\mathbf{M}\mathbf{W}){}^{\rm T}(\mathbf{M}\mathbf{W})\mathbf{x}_{\rm R}-2\mathbf{x}_{\rm R}{}^{ \rm T}(\mathbf{M}\mathbf{W}){}^{\rm T}\mathbf{t}_{\rm M}+\mathbf{t}_{\rm M}{}^{\rm T}\mathbf{t}_{ \rm M})}{\partial\mathbf{x}_{\rm R}}=0. \tag{8}\]  
As a result, \((\mathbf{M}\mathbf{W}){}^{\rm T}(\mathbf{M}\mathbf{W})\mathbf{x}_{\rm R}-(\mathbf{M}\mathbf{W}){}^{\rm T} \mathbf{t}_{\rm M}=0\); then, the optimal solution \(\mathbf{x}_{\rm R}\) is determined by the following equation:  
\[(\mathbf{M}\mathbf{W}){}^{\rm T}(\mathbf{M}\mathbf{W})\mathbf{x}_{\rm R}=(\mathbf{M}\mathbf{W}){}^{\rm T} \mathbf{t}_{\rm M}. \tag{9}\]  
For comparing the reconstruction results, the reconstruction error, \(e_{\rm R}\), is defined as  
\[e_{\rm R}=\frac{\|\mathbf{t}_{\rm R}-\mathbf{t}\|_{2}}{\|\mathbf{t}\|_{2}}\times 100\%. \tag{10}\]  
The two-dimensional temperature field reconstruction algorithm procedure can be summarized in the following steps. First, CFD is used to simulate a series of temperature fields under different boundary conditions. These calculation results constitute the dataset for analyzing the objective field. Then, in the POD framework, the POD basis vectors are obtained from the dataset. Finally, temperature field reconstruction is carried out on the basis of the POD basis vectors and real-time measurement data.