# Multiple-Penalty-Weighted Regularization Inversion for Dynamic Light Scattering
## 3 Numerical Simulation and Analysis  
In the simulations, both bimodal particles and trimodal particles were involved by using [PERSON]'s \(S_{\text{B}}\) function [PERSON] (1964)  
\[f(d)=\frac{\sigma}{(d_{\text{max}}-d_{\text{min}})\sqrt{2\pi}}[t(1-t)]^{-1}\exp \left[-0.5[\mu+\sigma\ln(\frac{t}{1-t})]^{2}\right] \tag{13}\]  
where \(t=(d-d_{\text{min}})/(d_{\text{max}}-d_{\text{min}})\) is the normalized particle size. \(d_{max}\) and \(d_{min}\) are the maximum and the minimum particle size, respectively. \(\mu\) and \(\sigma\) are distribution parameters. The intensity ACF data can be obtained by Equation (1). Two performance indices were introduced for examining the inversion performance:  
\[\text{V1}=\left(\sum_{i=1}^{N}\Bigl{(}\mathbf{f}(D_{i})-\hat{\mathbf{f}}(D_{i })\Bigr{)}^{2}\right)^{1/2} \tag{14}\]  
\[\text{V2}=\frac{|p_{\text{true}}-p_{\text{meas}}|}{p_{true}} \tag{15}\]  
where \(f(D_{i})\) is the simulated \"true\" PSD, \(\hat{f}(D_{i})\) the retrieved PSD, \(P_{\text{true}}\) the peak position value of the simulated \"true\" PSD, and \(P_{\text{meas}}\) the peak position value of the retrieved PSD. The smaller the V1 and V2 values, the better the performance of the inversion. The parameters of simulated PSD are shown in Table 1. d1, d2, and d3 are the particle size corresponding to the peak position value of PSD. Simulations were conducted with the conditions: \(k_{\text{B}}\) = 1.3807 \(\times\) 10\({}^{-23}\) J/K, \(T\) = 298.15 K, \(\eta\) = 0.89 cP, \(n_{\text{m}}\) = 1.3316, \(\lambda_{0}\) = 632.8 nm, \(\theta_{r}\) = 30\({}^{\circ}\), 50\({}^{\circ}\), 70\({}^{\circ}\), 90\({}^{\circ}\), 110\({}^{\circ}\), 130\({}^{\circ}\). AS, BS, and CS represent far bimodal 466/915 nm, near bimodal 316/470 nm, and trimodal 324/601/871 nm simulated PSD, respectively. The size distribution ranges of AS, BS, and CS are [2.01 nm, 1200.01 nm], [2.01 nm, 800.01 nm], and [2.01 nm, 1200.01 nm], respectively, and the discrete points of all simulated distributions are 160. The inversion results are shown in Figures 1-3, and the corresponding performance indices are shown in Tables 2-4. In the Figures, the subtitles (a), (b), (c), and (d) are the inversion results with noise levels of 0, 10\({}^{-4}\), 10\({}^{-2}\), and 0.8, respectively. 'True PSD', 'WR' and 'WMR' represent the true PSD, the inversed PSD by weighted regularization, and the inversed PSD by multiple-penalty-weighted regularization. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline
**Sample** & **d1** & **d2** & **d3** & \(\mu\)**1** & \(\mu\)**2** & \(\mu\)**3** & \(\sigma\)**1** & \(\sigma\)**2** & \(\sigma\)**3** & **a** & **b** & **c** \\ \hline AS & 466 & 915 & - & 3 & \(-6\) & - & 6.76 & 5.1 & - & 0.5 & 0.5 & - \\ BS & 316 & 470 & - & 4.1 & \(-3.1\) & - & 9.7 & 8.5 & - & 0.5 & 0.5 & - \\ CS & 324 & 601 & 871 & 7.0 & \(-1\) & \(-7\) & 7.2 & 9.05 & 7.2 & 1/3 & 1/3 & 1/3 \\ Unit & \(\mu\)m & \(\mu\)m & \(\mu\)m & - & - & - & - & - & - & - & - & - \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of simulated polydisperse PSD data sets. Figure 1: The inversion results with WR and WMR from 466/915 nm bimodal 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. Figure 2: The inversion results with WR and WMR from 316/470 nm bimodal 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.