## 2 Theory
### Solutions to the Optimization Problem Using ADMM  
First, two auxiliary variables \(y=\
abla\rho-v\) and \(z=\varepsilon(v)\) are introduced, and then Equation (8) is transformed into a constrained problem:  
\[ \begin{split}\underset{\rho,\rho}{\text{argmin}}\frac{1}{2}\|A \rho-b\|_{2}^{2}+\alpha_{1}\|\
abla\rho-v\|_{1}+\alpha_{2}\|\varepsilon(v)\|_ {1}\\ s.t.y=\
abla\rho-v,z=\varepsilon(v)\end{split}  \tag{9}\]  
The corresponding augmented Lagrangian function of Equation (9) is:  
\[\underset{\rho,y,z,v,\overline{y},\overline{z}}{\text{argmin}}\frac{1}{2}\|A \rho-b\|_{2}^{2}+\alpha_{1}\|y\|_{1}+\alpha_{2}\|z\|_{1}+\frac{\beta_{1}}{2}\| \
abla\rho-v-y+\overline{y}\|_{2}^{2}+\frac{\beta_{2}}{2}\|\varepsilon(v)-z+ \overline{z}\|_{2}^{2}, \tag{10}\]  
where \(\beta_{1}\), \(\beta_{2}\) are penalty parameters, and \(\overline{z}\), \(\overline{y}\) are Lagrange multipliers. Then, Equation (10) can be separated into sub-problems according to the variables, respectively. The sub-problems are as follows:  
\[\rho^{n+1}=\underset{\rho}{\text{argmin}}\frac{1}{2}\|A\rho-b\|_{2}^{2}+\frac{ \beta_{1}}{2}\|\
abla\rho-v^{n}-y^{n}+\overline{y}^{n}\|_{2}^{2}, \tag{11}\]\[y^{n+1}=\underset{y}{\mathrm{argmin}}\alpha_{1}\|y\|_{1}+\frac{\beta_{1}}{2}\| \
abla p^{n+1}-v^{n}-y+\overline{y}^{n}\|_{2}^{2}, \tag{12}\]  
\[z^{n+1}=\underset{z}{\mathrm{argmin}}\alpha_{2}\|z\|_{1}+\frac{\beta_{2}}{2}\| \varepsilon(v^{n})-z+\overline{z}^{n}\|_{2}^{2}, \tag{13}\]  
\[v^{n+1}=\underset{p}{\mathrm{argmin}}\frac{\beta_{1}}{2}\|\
abla p^{n+1}-v-y^ {n+1}+\overline{y}^{n}\|_{2}^{2}+\frac{\beta_{2}}{2}\|\varepsilon(v)-z^{n+1}+ \overline{z}^{n}\|_{2}^{2}. \tag{14}\]  
Given that Equation (11) is differentiable, then the optimality condition for \(p^{k+1}\) is easily derived as follows:  
\[\big{(}A^{\prime}A+\beta_{1}\
abla^{T}\
abla\big{)}p^{n+1}=A^{\prime}b+\beta_{ 1}\
abla^{T}(v^{n}+y^{n}-\overline{y}^{n}). \tag{15}\]  
Equation (15) can be calculated through the CG method or Gauss-Seidel method. The \(y,z\) sub-problems are similar, and the solutions are given explicitly by a generalized shrinkage formula [45]. Specifically, the solution to the \(y\) sub-problem is:  
\[y^{n+1}=\max\bigg{\{}\|v^{n}-\
abla p^{n+1}-\overline{y}^{n}\|_{2}-\frac{ \alpha_{1}}{\beta_{1}},0\bigg{\}}\cdot\ast\frac{v^{n}-\
abla p^{n+1}-\overline {y}^{n}}{\|v^{n}-\
abla p^{n+1}-\overline{y}^{n}\|_{2}}. \tag{16}\]  
Likewise, we obtain the solution to the \(z\) sub-problem as:  
\[z^{n+1}=\max\bigg{\{}\|\overline{z}^{n}+\varepsilon(v^{n})\|_{2}-\frac{\alpha_ {2}}{\beta_{2}},0\bigg{\}}\cdot\ast\frac{\overline{z}^{n}+\varepsilon(v^{n})} {\|\overline{z}^{n}+\varepsilon(v^{n})\|_{2}}. \tag{17}\]  
For the sub-problem \(v\), we use the first-order necessary conditions for optimality as follows:  
\[ \begin{array}{l}\big{(}\beta_{1}+\beta_{2}D_{1}^{T}D_{1}+\frac{1}{2}\beta_{2 }D_{2}^{T}D_{2}+\frac{1}{2}\beta_{2}D_{3}^{T}D_{3}\big{)}v_{1}^{n+1}=\frac{1} {2}\beta_{2}\big{(}D_{2}^{T}D_{1}v_{2}^{n}+D_{3}^{T}D_{1}v_{3}^{n}\big{)}\\ +\beta_{1}\big{(}D_{1}p^{n+1}-y_{1}^{n+1}+\overline{y}_{1}^{n}\big{)}+\beta_{ 2}D_{1}^{T}\big{(}z_{1,1}^{n+1}-\overline{z}_{1,1}^{n}\big{)}+\beta_{2}D_{2}^{ T}\big{(}z_{1,2}^{n+1}-\overline{z}_{1,2}^{n}\big{)}+\beta_{2}D_{3}^{T}\big{(}z_{1,3}^{n+1 }-\overline{z}_{1,3}^{n}\big{)},\end{array}