## 3 Model Solution
### Tikhonov Regularization and Solution Error Analysis  
The singular value decomposition (SVD) of \(A\) is  
\[A=U\Sigma V^{*} \tag{12}\]  
where \(V\) is an orthogonal matrix composed of the corresponding eigenvectors \((v_{1},\cdots,v_{2M+1})\) of \(A^{*}A\) with eigenvalues \(\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{2M+1}>0\), and all eigenvalues of \(A^{*}A\) are positive real numbers. \(\Sigma\) is the singular value matrix, satisfying \(\Sigma(i,i)=\sigma_{i},i\leq 2M+1\), where \(\sigma(i)=\sqrt{\lambda_{i}}\) is the singular values of \(A\). \(U\) is an orthogonal matrix of dimension \(N\times N\), and its columns are formed by the vectors\((u_{1},\cdots,u_{N})\).  
Suppose \(S_{\hat{n}}^{0}\) is the least squares solution of (2). Based on [PERSON]'s theorem Picard (1953), it can be expressed as  
\[S_{\hat{n}}^{0}=A^{+}s_{\delta}=\sum_{j=1}^{2M+1}\frac{\langle s_{\delta},\,u_ {j}\rangle}{\sigma_{j}}v_{j} \tag{13}\]  
When there are certain singular values \(\sigma_{j}\) of \(A\) that are very small or close to zero, the solution for Equation (11) becomes ill-posed. In this case, \(\tilde{n}\) has a significant impact on the solution. To solve such problems, a Tikhonov regularization operator \(R_{\alpha}:\mathbb{C}^{N}\to\mathbb{C}^{2M+1}\) is constructed [PERSON] (1953), such that  
\[S_{\hat{n}}^{\alpha}=R_{\alpha}s_{\delta}=\sum_{j=1}^{2M+1}\frac{q(\alpha, \sigma_{j})}{\sigma_{j}}\langle s_{\delta},\,u_{j}\rangle v_{j}\quad\alpha>0 \tag{14}\]  
where \(S_{\hat{n}}^{\alpha}\) is the regularized solution of Equation (13) under the regularization parameter \(\alpha\), and \(q(\alpha,\sigma_{j})\) is the filtering factor [PERSON] (1953). For small singular values \(\sigma_{j}\), \(q(\alpha,\sigma_{j})\) can attenuate the amplification effect of \(1/\sigma_{j}^{2}\) on the error, thereby obtaining a stable solution for Equation (13). The expression for the filtering factor is  
\[q(\alpha,\sigma_{j})=\frac{\sigma_{j}^{2}}{\alpha^{2}+\sigma_{j}^{2}}\quad \alpha>0,0\leq\sigma_{j}\leq\|A\| \tag{15}\]  
Let  
\[z=\sqrt{\sum_{j=1}^{2M+1}\frac{1}{\sigma_{j}^{2}}\big{|}\big{\langle}S,\,v_{ j}\big{\rangle}\big{|}^{2}} \tag{16}\]  
utilizing the Tikhonov regularization method to quantify the solution error of Equation (10) as  
\[\big{\|}S_{\hat{n}}^{\alpha}-S\big{\|}\leq\frac{1}{2\alpha}\|\tilde{n}\|+ \frac{\alpha}{2}z \tag{17}\]  
The specific formula derivation can be found in Appendix A. The result of Equation (17) indicates that, when a non-uniform sampling problem is determined, the solving error is only dependent on \(\alpha\), as the magnitude of \(\|\tilde{n}\|\) and \(z\) can be considered as unknown constants. The solving error caused by \(\tilde{n}\) decreases as \(\alpha\) increases. When \(\alpha\to\infty\), this term tends to zero. The second term on the right-hand side of Equation (17) represents the regularization error caused by filter factor and increases with the increase of \(\alpha\)  
The magnitude of solving error for \(s_{2}(t_{i}^{(2)})\) under different \(\alpha\) is shown in Figure 2. The 500 selected regularization parameters are evenly distributed on a logarithmic scale axis \(\lg\alpha\) with the range of \([\sigma_{1},\sigma_{2M+1}]\). When \(\alpha\) is small, the solving error primarily caused by \(\tilde{n}\) reaches the order of \(10^{5}\), indicating that \(\tilde{n}\) has a significant impact on the solution results for \(s_{2}(t_{i}^{(2)})\). As \(\alpha\) increases, the solving error gradually decreases, reaching its minimum value when \(\alpha=5.885\times 10^{-7}\). However, as \(\alpha\) further increases, the regularization error begins to dominate. Therefore, the role of \(\alpha\) is to balance the two types of errors in Equation (17). In practical solutions to non-uniform sampling problems, it is important to select an appropriate \(\alpha\) in order to minimize the overall solving error as much as possible.