# A Forward\(-\)Backward Splitting Equivalent Source Method Based on S\(-\)Difference
## 2 Forward-Backward Equivalent Source Method Based on S\(-\)Difference
### Forward-Backward Splitting (FBS) Algorithm  
First, under the framework of sparse regularization theory, the \(l_{1}\) norm is used to conduct sparse constraints on the equivalent source intensity vector, and the optimization process can be expressed as:  
\[\min_{q}\text{\emph{imize}}\left\|q\right\|_{1}\text{\emph{subject}}\quad to \quad\left\|P-\text{G}q\right\|_{2}\leq\delta \tag{8}\]  
The above equation can be converted into a function minimization problem:  
\[\min_{x\in P^{N}}\left\{F(q)=\phi(q)+\rho P(q)\right\} \tag{9}\]  
Based on this, we will use FBS to approximate the unconstrained minimization problem and derive the closed-form solution for the proximal operator of \(P(q)\)[20; 21]; this makes the FBS more effective. Each iteration of forward\(-\)backward splitting applies gradient descent of \(\phi(q)\). \[\text{q}^{[k+1]}=prox_{\beta\rho P}(\text{q}^{[k]}-\beta\
abla\phi(\text{q}^{[ k]})) \tag{10}\]\(\beta>0\) is the search step size. Order \(\mathrm{x}^{*}\) is the optimal solution of the near\(-\)end operator, namely \(\mathrm{x}^{*}=prox_{\rho}(y)\), when \(R(q)=\left\|q\right\|_{1}\), \(P(q)=\left\|q\right\|_{1}-\left\|q^{s}\right\|_{1}\). Then, solve \(q^{*}\) as:  
\[q_{i}^{*}= \begin{cases}y_{i},&i\in\Gamma_{y}^{s}\\ shrink(y_{i},\rho\beta)&i\in\Gamma_{y}^{N}\backslash\Gamma_{y}^{s}\end{cases}  \tag{11}\]  
where \(shrink(y_{i},\rho\beta)\) represents a soft contraction operator. \(\Gamma_{y}^{s}=\left\{\pi_{y}(1),\pi_{y}(2),\ldots,\pi_{y}(s)\right\}\) and \(\pi_{y}(j)\) is the jth maximum amplitude of y, namely \(\left|y_{\pi_{y}(1)}\right|\geq\left|y_{\pi_{y}(2)}\right|\geq\cdots\geq \left|y_{\pi_{y}(N)}\right|\). \[shrink(y_{i},\rho\beta)=sign(y_{i})\max\{\left|y_{i}\right|-\rho\beta,0\} \tag{12}\]  
The WBH algorithm utilizes an iterative solver to approximate a sparse solution for the equivalent source strength vector. First of all, the residual value \(\mathbf{r}\) is defined. \[\mathbf{r}(q)=\mathrm{p}-\mathrm{G}q \tag{13}\]  
Secondly, the minimized residual quadratic function \(F\) is defined as:  
\[F(q)=\frac{1}{2}\left\|\mathbf{r}(q)\right\|_{2}^{2} \tag{14}\]  
Because function \(F\) is quadratic differentiable, the optimal solution of the residual function \(F(q)\) is calculated by a two\(-\)step method, including the steepest descent step and the threshold step in each iteration. We will use a certain proportion of the output values from WBH as the regularization parameter for threshold filtering in FBS. According to a large amount of simulation data, when the value of the regularization parameter for threshold filtering in the FBS is taken to be between 0.05 times and 0.5 times the maximum value of the WBH, \(\rho\in[0.05\times\max(|q_{\mathrm{tr}}|),0.5\times\max(|q_{\mathrm{tr}}|)]\), the effect of reconstruction is good and stable, where \(\max(|q_{\mathrm{tr}}|)\), the regularization parameter, is chosen by taking the absolute maximum value of the equivalent source amplitude vector calculated by WBH. The main reason for using this parameter selection method is that the WBH algorithm is efficient and accurately outputs the amplitude of the equivalent source strength over a wide frequency range. When the prior sparse range parameter \(s\) is unknown, introduce a new adjustment strategy to adapt to each iteration: \(s^{[k+1]}=size(find(\left|q^{[k]}\right|\geq\min\left\{\left|q_{\pi_{x}(s^{ [k-1]})}^{[k-1]},\delta\right\}))\), where \(\delta>0\). In summary, the LMS algorithm consists of two main processes: filtering and adaptive adjustment.