## 2 Inverse Solution Algorithms
### Overdetermined or Square Systems
#### 2.1.2 Choice of Regularization Parameter \(\beta\)  
In traditional solution strategies for inverse problems, methods are available for choosing regularization parameters, such as Morozov's discrepancy principle, the Generalized Cross-Validation (GCV) method, and the L-curve method [19; 20]. The first two methods have been discussed extensively in the literature. Therefore, we subsequently focus on only the L-curve method for brevity. The L-curve method is widely used for choosing regularization parameters in inverse solutions. In the curve, the solution norm is plotted versus the residual norm by varying the regularization parameter. From Equations (9) and (11), it is straightforward to find the solution norm  
\[\left\|\mathbf{\hat{s}}\right\|_{2}^{2}=\sum\limits_{i=1}^{N}{f_{i}^{2}\sigma_ {i}^{-2}|\alpha_{i}|^{2}} \tag{14}\]  
and the residual norm  
\[\left\|\mathbf{p}-\mathbf{A}\mathbf{\hat{s}}\right\|_{2}^{2}=\sum\limits_{i=1 }^{N}{(1-f_{i})^{2}|\alpha_{i}|^{2}}+\sum\limits_{i=n+1}^{M}{|\mathbf{r}_{ \perp}|^{2}} \tag{15}\]  
Regularization helps to improve robustness against system perturbation and measurement noise. Insights can be gained by writing the solution error as  
\[\mathbf{s}-\mathbf{\hat{s}}=\mathbf{s}-\mathbf{A}^{\#}\mathbf{b}=(\mathbf{s}- \mathbf{A}^{\#}\mathbf{\overline{p}})-\mathbf{A}^{\#}\mathbf{e}=\left[\sum \limits_{i=1}^{N}{(1-f_{i})\frac{\mathbf{u}_{i}^{H}\mathbf{\overline{p}}}{ \sigma_{i}}\mathbf{v}_{i}}\right]-\sum\limits_{i=1}^{N}{f_{i}\frac{\mathbf{u}_ {i}^{H}\mathbf{e}}{\sigma_{i}}\mathbf{v}_{i}}, \tag{16}\]  
where \(\mathbf{A}^{\#}=(\mathbf{A}^{H}\mathbf{A}+\beta^{2}\mathbf{I})^{-1}\mathbf{A} ^{H}\), \((\mathbf{s}-\mathbf{A}^{\#}\mathbf{\overline{p}})\) is the regularization error and \(\mathbf{A}^{\#}\mathbf{e}\) is the perturbation error. Hence, when \(\beta\to 0,f_{i}\to 1\) for very well-conditioned problems with high signal-to-noise ratio (SNR) measurements, the solution error is dominated by the perturbation error and a few high-order modes are filtered out. In this case, the solution norm is sensitive to the choice of \(\beta\). The solution tends to be undersmoothed and susceptible to measurement noise. Conversely, when \(\beta\rightarrow\infty,f_{i}\to 0\) for very ill-conditioned problems with low SNR measurements, the solution error is dominated by regularization error and numerous high-order modes are filtered out. The solution tends to be oversmoothed and fine details such as resolution are thus lost. In this case, the residual norm is sensitive to the choice of \(\beta\). The parameter \(\beta\) acts as a weighting factor between the residual norms and the solution norm. Choosing an appropriate \(\beta\) to strike the balance between these two terms is vital. However, conventional approaches such as GCV and the L-curve method do not always provide satisfactory results in this situation. In this paper, a new method is proposed to facilitate the choice of the regular parameter \(\beta\) for the TIKR method. Setting the gradient of the cost function of the TIKR method, \(J=\left\|\mathbf{p}-\mathbf{A}\mathbf{s}\right\|_{2}^{2}+\beta^{2}\|\mathbf{ s}\|_{2}^{2}\), to zero leads to the normal equation  
\[(\mathbf{A}^{H}\mathbf{A}+\beta^{2}\mathbf{I})\mathbf{s}=\mathbf{A}^{H} \mathbf{p} \tag{17}\]Without loss of generality, assume **A** is of full-column rank. Note that  
\[\textbf{A}^{H}\textbf{A}+\beta^{2}\textbf{I}=\textbf{V}\Big{[}\Sigma^{H}\Sigma+ \beta^{2}\textbf{I}\Big{]}\textbf{V}^{H}=\textbf{V}diag\Big{[}(\sigma_{1}^{2}+ \beta^{2}),\ldots,(\sigma_{N}^{2}+\beta^{2})\Big{]}\textbf{V}^{H}\]  
which has an effective condition number \(\sqrt{(\sigma_{1}^{2}+\beta^{2})/(\sigma_{N}^{2}+\beta^{2})}\). Therefore, if we want the condition number to be \(\tau\) after regularization, we must require  
\[\tau^{2}=\frac{\sigma_{1}^{2}+\beta^{2}}{\sigma_{N}^{2}+\beta^{2}} \tag{18}\]  
Let \(\kappa\) be the condition number of **A**; that is, \(\kappa=\sigma_{1}/\sigma_{M}\).