# Multiple-Penalty-Weighted Regularization Inversion for Dynamic Light Scattering
## 2 Dynamic Light Scattering and Multiple-Penalty-Weighted Regularization
\right]\mathbf{f}= \mathbf{A}_{W}\mathbf{f} \tag{8}\]  
where \(A_{W}\) is the kernel matrix of Equation (8), \(A_{W}=[k_{1}\mathbf{W}_{1}\mathbf{A}_{\theta_{1}};k_{2}\mathbf{W}_{2}\mathbf{ A}_{\theta_{2}};\ldots\) ; \(k_{m}\mathbf{W}_{m}\mathbf{A}_{\theta_{m}}]_{\text{J}}\) and \(\mathbf{g}_{W}^{(1)}\) is a vector set containing the weighted electric field ACF data at scattering angles \(\theta_{1}\), \(\theta_{2}\), \(\ldots\), \(\theta_{m}\). \(\mathbf{g}_{\theta_{r}}^{(1)}\) is the electric field ACF data at scattering angle \(\theta_{r}\), \(\mathbf{A}_{\theta_{r}}\) the kernel matrix corresponding to \(\mathbf{g}_{\theta_{r}}^{(1)}\), \(\mathbf{W}_{\theta r}\) the weighted matrix corresponding to \(\mathbf{g}_{\theta_{r}}^{(1)}\), \(k_{r}\) the constant related to the scattering angle \(\theta_{r}\), and \(r\) an integer from \(1\) to \(m\). \[\mathbf{W}_{\theta_{m}}=diag\big{[}\mathbf{w}_{\theta_{r}j}\big{]} \tag{9}\]  
\[\mathbf{w}_{\theta_{r}j}=\left|\mathbf{g}_{\theta}^{(1)}\big{(}\tau_{j}\big{)} \right|^{\text{Pr}/2} \tag{10}\]  
where Pr is the weight-adjusting parameter and \(\mathbf{g}_{\theta}^{(1)}\big{(}\tau_{j}\big{)}\) is the ACF data from the actual measurement. Adding a new penalty item \(\beta\|L_{1}f\|^{2}\) in the objective function of the weighted regularization, then  
\[M^{\alpha,\beta}(f,g_{W}^{(1)})=\|A_{W}W_{\theta}f-g_{W}^{(1)}\|^{2}+\alpha\| Lf\|^{2}+\beta\|L_{1}f\|^{2} \tag{11}\]  
where \(\beta\) is the newly increased regularization parameter. \(L\), the regular matrix in the first penalty term, is a zero-order matrix (identity matrix), and \(L_{1}\), the regular matrix in the second penalty term, is a first-order matrix [36]. Regularization parameters can be selected by fixed-point iteration method or model function method [37]. In this work, fixed-point iteration method is used and, by optimizing minimum distance function (MDF) [38], the iterative model for obtaining the second regularization parameters is derived as  
\[\beta^{(k+1)}=\frac{\|\mathbf{A}_{W}f(\alpha,\beta^{(k)})-g_{W}^{(1)}\|^{2}_{ 2}}{\|\mathbf{L}f(\alpha,\beta^{(k)})\|^{2}_{2}}\times\frac{\log\big{[}\| \mathbf{L}f(\alpha,\beta^{(k)})\|^{2}_{2}\big{]}-\log\big{[}\|\mathbf{L}f( \alpha,\beta_{b})\|^{2}_{2}\big{]}}{\log\Big{[}\|\mathbf{A}_{W}f(\alpha, \beta^{(k)})-g_{W}^{(1)}\|^{2}_{2}\Big{]}-\log\Big{[}\|\mathbf{A}_{W}f(\alpha, \beta_{a})-g_{W}^{(1)}\|^{2}_{2}\Big{]}},\,(k=0,1,2,\ldots) \tag{12}\]  
where \(\beta_{a}\) and \(\beta_{b}\) are the squares of the smallest and the largest singular values of the \(A_{W}\) matrix, respectively. The steps of selecting regularization parameters are as follows: Firstly, the regularization parameter \(\alpha\) is selected through the \(L\)-curve criterion, and \(\beta_{a}\) and \(\beta_{b}\) are calculated. By assuming the initial regularization parameter \(\beta^{(0)}\) with \(\beta_{a}<\beta^{(0)}<\beta_{\text{b}}\), \(f(\alpha,\beta^{(0)})\) is calculated. Secondly, \(\beta^{(0)}\), \(\beta_{\text{a}}\), \(\beta_{\text{b}}\), and \(f(\alpha,\beta^{(0)})\) are substituted into the Equation (12), and the second parameter \(\beta^{(1)}\) and its corresponding solution \(f(\alpha,\beta^{(1)})\) are obtained. Thirdly, it is determined whether the iteration stop condition, \(\left|\left(\beta^{(k+1)}-\beta^{(k)}\right)/\beta^{(k+1)}\right|\leq 10^{-4}\), is satisfied. Repeat the last two steps in turn if the stop condition is not met, until the condition is satisfied. \(\beta^{(k)}\), which satisfies the iteration stop condition, is the optimal regularization parameter.