# Multiple-Penalty-Weighted Regularization Inversion for Dynamic Light Scattering
## 5 Discussion and Conclusions  
The main factors limiting PSD recovery in DLS are insufficient information and inevitable noise in ACF data. Multiangle measurement can increase the information in ACF data, and the information-weighting method can improve the information extraction ability of the inversion algorithm and effectively suppress the noise effect in the noise-intensive segments of ACF data. However, information weighting cannot solve the problem of the noise mixed in the information-intensive segments of ACF data. With the increase of the noise in the data, the retrieved PSDs would be seriously affected by the burrs, which leads to the variation of the distribution and even the false peaks. This situation is shown in the simulation data inversions (Figures 1-3), and the suppression effect of multiple-penalty inversion on noise by reducing and restraining burrs and jumps can also be seen in the figures. This is especially evident in the inversion of far bimodal 466/915 nm particles, in which the noise level is up to 0.8, and while the single-penalty-weighting method fails to give the acceptable distribution, the multiple-penalty method still gives the approximate reasonable bimodal distribution (Figure 1d). The difference between the multiple-penalty regularization and the single-penalty regularization is that an additional penalty term is added to the objective function, and the flatness model is used for the second regularization matrix. By using the flatness model to constrain the solution, only a gentle change can be made between the components of each solution. Of course, the acceptability of the multiple-penalty-weighting inversion results is still limited by the noise level, for 316/470 nm near-bimodal particles, the inversion results are still unsatisfactory when the noise level reaches \(10^{-2}\)  
Figure 4: The inversion results with WR and WMR method from experimental DLS data: (**a**) 306/974 nm bimodal particles; (**b**) 300/502 nm bimodal particles. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Size/nm**} & \multicolumn{2}{c}{**WR**} & \multicolumn{2}{c}{**WMR**} \\ \cline{2-5}  & \(P_{\text{meas}}\)/nm & **V2** & \(P_{\text{meas}}\)/nm & **V2** \\ \hline
306/974 & 313/874 & 0.0228/0.1027 & 301/886 & 0.0196/0.0903 \\
300/502 & 280/490 & 0.0667/0.0239 & 290/500 & 0.0333/0.004 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The performance indices of the inversion for experimental DLS data. In addition to burrs and jumps, the inversion errors of the small particles are significant in 316/470 nm bimodal and 324/601/871 nm trimodal particles (Figures 2d and 3d), and the multiple-penalty regularization does not reduce the errors. This happens in the multiangle DLS measurement, usually only when the particle size is less than 350 nm, which is consistent with the previous studies [40]: the improvement of multiangle DLS on particle measurements is only effective for large particles, but very little for particles smaller than 350 nm. For small particles less than 350 nm, increasing the number of measuring angles is not only unable to provide more information, but is likely to introduce more measurement noise, which may corrupt the estimate of the PSD in DLS. In this work, L-curve criterion is used to select the regularization parameter \(\alpha\). Theoretically, L-curve criterion is suitable for unimodal distribution and near-bimodal inversion. But in the actual process, there is a deviation between the selected parameter and the true parameter. For the far-bimodal distribution, this deviation often becomes larger, which leads to a larger deviation of the inversion distribution, especially in the far peak. The inversion result of 306/974 experimental data is such a case that the large deviation occurred in the larger peak in the inversed PSD. Nevertheless, compared with the single-penalty inversion, the multiple-penalty-weighted regularization makes the inversed PSD further improve.