## 2 Inverse Solution Algorithms
### Overdetermined or Square Systems
#### 2.1.2 Choice of Regularization Parameter \(\beta\)
In general, for very ill-conditioned systems, \(\kappa\gg\tau\gg 1\), and  
\[\beta=\sqrt{\frac{\sigma_{1}^{2}-\tau^{2}\sigma_{M}^{2}}{\tau^{2}-1}}\approx \sqrt{\frac{\sigma_{1}^{2}-\tau^{2}(\sigma_{1}^{2}/\kappa^{2})}{\tau^{2}}}= \sqrt{\frac{\sigma_{1}^{2}}{\kappa^{2}\tau^{2}}(\kappa^{2}-\tau^{2})}\approx \frac{\sigma_{1}}{\tau} \tag{19}\]  
Therefore, the regularization parameter \(\beta\) can be chosen to be the maximal singular value \(\sigma_{1}\) of **A** divided by the regularized condition number \(\tau\). For example, one may choose that \(\tau=100\), which causes 40-dB potential loss of SNR in the inverse solution. Normally, **A** tends to be ill-conditioned at low frequencies. Choosing a frequency-independent \(\beta\) may suffice for the worst-case scenario. Thus, the regularization parameter \(\beta\) is chosen according to the maximal condition number at a selected low frequency (100 Hz is selected in the following simulation). Next, a coarse search is performed by varying \(\beta\) in orders of 10. A potential interval in which an optimal \(\beta\) may exist is located by observing how an objective function, such as Perceptual Evaluation of Speech Quality (PESQ) [29], varies with \(\beta\). Finally, a fine-grained search is performed in the potential interval by using the Golden Section Search (GSS) algorithm [21]. GSS is an optimization technique suited for finding the extremum of a unimodal function. It is a simple bracketing method that does not require evaluation of the gradient of the cost function. In each search, a probe point must be selected within the left and right brackets according to the golden ratio. The golden ratio can be defined as  
\[\varphi=\frac{1+\sqrt{5}}{2}=1.618\,\varphi\text{ is the golden ratio}\]  
Let \(f(\beta)\) be the objective function (PESQ in our case) for which we wish to find the optimal \(\beta\) that maximizes \(f(\beta)\). First, \(f(\beta)\) has been evaluated at two points, \(\beta_{1}\) and \(\beta_{3}\). The maximizing value is between \(\beta_{1}\) and \(\beta_{3}\). The golden ratio can be used to find \(\beta_{2}\) and \(\beta_{4}\). \(\beta_{2}\) and \(\beta_{4}\) can be written as  
\[\beta_{2}\ =\ \beta_{3}-(\beta_{3}-\ \beta_{1})/\varphi,\beta_{4}\ =\ \beta_{1}+\ (\beta_{3}\ -\ \beta_{1})/\varphi\]  
If \(f(\beta_{2})\) is larger than \(f(\beta_{4})\), a maximum clearly lies in the interval between \(\beta_{1}\) and \(\beta_{4}\). Therefore, the new \(\beta_{3}\) is equal to \(\beta_{4}\). If \(f(\beta_{2})\) is smaller than \(f(\beta_{4})\), a maximum lies in the interval between \(\beta_{2}\) and \(\beta_{3}\). Therefore, the new \(\beta_{1}\) is equal to \(\beta_{2}\). Figure 1 shows the Schematic of golden section search. The process is repeated until the gap between \(\beta_{2}\) and \(\beta_{4}\) is small. The iteration process stops when the beta converges within a prespecified tolerance window (0.0001 in our case). The optimal beta \(\beta_{opt}\) can be written as  
\[\beta_{opt}=(\beta_{2}+\beta_{4})/2\]  
The algorithm is summarized as the following pseudocode:Therefore, this study developed a procedure for choosing optimal regularization parameter beta. The procedure involves four steps as follows:  
* Step 1. Select \(\tau\) according to a condition number threshold. * Step 2. Select a constant \(\beta=\sigma_{1}/\tau\) as an initial guess. For a frequency-domain design, it may be necessary to choose a frequency-independent \(\beta\) for the worst-case scenario. * Step 3. Perform a coarse search by running the simulation forward and backward with 10 s powers of \(\beta\). Locate a potential interval in which an optimal \(\beta\) may exist by observing the trend of an objective function, such as PESQ, with respect to \(\beta\). * Step 4. Perform a fine search by using optimization methods such as the GSS algorithm to find the optimal regularization parameter \(\beta\).