multiplier, $\rho>0$ is the penalty parameter.

Given $(x^0,y^0,\lambda^0)\in {\rm dom}f\times{\rm dom}g\times \Lambda$, penalty parameter $\rho>0$ and the step size parameter $\tau\in \left(0,\frac{\sqrt{5}+1}{2}\right)$, then the iterative format of the classical ADMM is as follows: $$\left\{\begin{aligned}
x^{k+1}&={\text{argmin}}\mathcal{L}_{\rho}(x,y^k,\lambda^k;\rho),\\
y^{k+1}&={\text{argmin}} \mathcal{L}_{\rho}(x^{k+1},y,\lambda^k;\rho),\\
\lambda^{k+1}&=\lambda^k+\tau\rho(Ax^{k+1}+By^{k+1}-c).
\end{aligned} \right.$$

The advantage of ADMM is that it separates $f(x)$ and $g(y)$ into two subproblems. In each subproblem, there is only one variable, the other one is fixed. Thus each subproblem could be solved easily and efficiently. For the classical ADMM, the convergence analysis was first conducted by [@Fortin1983Chapter; @Gabay1976A; @Glowinski1980Lectures], and for the recent interesting new developments on the convergence analysis of ADMM-type method, see [@han2017linear; @Sun2014A; @2018arXiv180904249Y].

To apply ADMM-type method to solve the problem (\[eqn:orginal problems\]), we use the solution operator $S$ and introduce an artificial variable $z$, then we equivalently rewrite problem (\[eqn:orginal problems\]) as the following reduced form: $$\label{eqn:reduced problem with linear constraint}
\begin{aligned}
\min \limits_{(u,z)\in U\times Z}^{}\ \ \  &\hat{J}(u)+\delta_{U_{ad}}(z) \\
{\rm s.t.}\qquad \ &u=z, 
\end{aligned} \tag{RP}$$ where $\hat{J}(u):=\dfrac{1}{2}\|S(u+y_{c})-y_{d}\|_{L^{2}(\Omega)}^{2}+\dfrac{\alpha}{2}\|u\|_{L^{2}(\Omega)}^{2}$ denotes the reduced cost function, $\delta_{U_{ad}}(z)$ denotes the indicator function of $U_{ad}$, $$\delta_{U_{ad}}(z)=\left\{\begin{array}{ll}{1,} & {z \in U_{ad}}, \\ {\infty,} & {z \notin U_{ad}}.\end{array}\right.$$ The augmented Lagrangian function of (\[eqn:reduced problem with linear constraint\]) is defined as 