# Multiple-Penalty-Weighted Regularization Inversion for Dynamic Light Scattering
## 3 Numerical Simulation and Analysis
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Size/nm**} & \multirow{2}{*}{**Noise Level**} & \multicolumn{4}{c}{**WR**} & \multicolumn{4}{c}{**WMR**} \\ \cline{3-8}  & & **V1** & \(P_{\text{meas}}\)/**nm** & **V2** & **V1** & \(P_{\text{meas}}\)/**nm** & **V2** \\ \hline \multirow{4}{*}{466/915} & 0 & 0.0252 & 451/474/923 & 0.0322/0.0172/0.0087 & 0.0195 & 474/923 & 0.0172/0.0087 \\  & \(10^{-4}\) & 0.0231 & 459/923 & 0.0150/0.0087 & 0.0229 & 459/930 & 0.0150/0.0164 \\  & \(10^{-2}\) & 0.0349 & 481/923 & 0.0322/0.0087 & 0.0254 & 459/923 & 0.0150/0.0057 \\  & 0.08 & 0.1551 & -/908 & -/0.0076 & 0.1297 & 425/908 & 0.0880/0.0076 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The performance indices of the inversion for simulated 466/915 nm bimodal DLS data under different noise levels. Figure 3: The inversion results with WR and WMR from 324/601/871 nm trimodal simulated DLS data under different noise levels: (**a**) noise level is 0; (**b**) noise level is \(10^{-4}\); (**c**) noise level is \(10^{-2}\); (**d**) noise level is 0.08. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Size/nm**} & \multirow{2}{*}{**Noise Level**} & \multicolumn{4}{c}{**WR**} & \multicolumn{2}{c}{**WMR**} \\ \cline{3-8}  & & **V1** & \(P_{\text{meas}}\)/**nm** & **V2** & **V1** & \(P_{\text{meas}}\)/**nm** & **V2** \\ \hline \multirow{4}{*}{316/470} & 0 & 0.0422 & 301/475 & 0.0475/0.0106 & 0.0286 & 301/475 & 0.0475/0.0106 \\  & \(10^{-4}\) & 0.0480 & 306/480 & 0.0316/0.0213 & 0.0380 & 311/475 & 0.0158/0.0106 \\  & \(10^{-2}\) & 0.0576 & 301/485 & 0.0475/0.0319 & 0.0500 & 301/475 & 0.0475/0.0106 \\  & 0.08 & 0.2113 The inversion results, with the single-penalty and the multiple-penalty-weighted regularization from 466/915 nm bimodal simulated DLS data, are shown in Figure 1. It is obvious that, with the increase of the added noise, there are obvious burrs in the PSDs inversed by weighted regularization function. When the noise level reaches 0.08, an obvious false peak is found in the PSDs inversed by single-penalty-weighted regularization. After adding another penalty to the objective function, some false peaks and burrs are eliminated, and a bimodal PSD, which is relatively close to the true PSD, is still obtained. From the corresponding performance indices (Table 2), it can be seen that the fitting errors and peak position errors are reduced obviously by using the multiple-penalty-weighted regularization method. The retrieved PSDs, with the single-penalty and the multiple-penalty-weighted regularization from 316/470 nm bimodal simulated DLS data, are shown in Figure 2 and the corresponding performance indices are shown in Table 3. From Figure 2, it can be seen that in addition to the obvious burrs and false peaks in the results inversed by single-penalty-weighted regularization, the jump also occurs in the size distribution as the noises increase, and this situation is improved by adding a penalty term in the weighted inversion. When the noise level reaches 0.08, large peak deviations occur in the inversed PSDs for both methods, while the results obtained by the multiple-penalty-weighted regularization are relatively better than those obtained by the single-penalty one in terms of the peak values and burr occurrence. Table 3 indicates that all the fitting errors and the peak position errors in the PSDs obtained by the multiple-penalty-weighted regularization are obviously less than that by the single-penalty-weighted regularization. Figure 3 shows the inversion results of the single-penalty and the multiple-penalty-weighted regularization for 324/601/871 nm trimodal simulated DLS data, and Table 4 gives the corresponding performance indices. From Figure 3 and Table 4, we can see that the multiple-penalty-weighted regularization effectively eliminates the burr and spurious peaks which appeared in the PSDs inversed by single-penalty-weighted regularization. Both the peak errors and peak position errors of the former are obviously smaller than that of the latter.