follows: $$\label{eqn:lagrangian function}
L_{\sigma}(u,z,\lambda;\sigma)=\hat{J}(u)+\delta_{U_{ad}}(z)+\langle\lambda,u-z\rangle_{L^{2}(\Omega)}+\dfrac{\sigma}{2}\|u-z\|_{L^{2}(\Omega)}^{2},$$ where $\lambda \in L^{2}(\Omega)$ denotes the Lagrangian multiplier, $\sigma>0$ is the penalty parameter.

The classical ADMM in finite dimensional spaces can be extended directly in Hilbert space for problem (\[eqn:reduced problem with linear constraint\]). Given $(u^{0},z^{0},{\lambda}^{0})\in L^{2}(\Omega)\times {\rm dom}(\delta_{U_{ad}}(\cdot))\times L^{2}(\Omega)$, parameters $\sigma>0$, $\tau \in \left(0,\dfrac{1+\sqrt{5}}{2}\right)$, the iterative scheme is presented as follows:

$$\left\{\begin{aligned}
\bar u^{k+1}&={\rm argmin} \hat{J}(u)+\langle \bar \lambda^{k},u-\bar z^{k}\rangle_{L^{2}(\Omega)}+\dfrac{\sigma}{2}\|u-\bar z^{k}\|_{L^{2}(\Omega)}^{2},\\
\bar z^{k+1}&={\rm argmin} \delta_{U_{ad}}(z)+\langle \bar \lambda^{k},\bar u^{k+1}-z\rangle_{L^{2}(\Omega)}+\dfrac{\sigma}{2}\|\bar u^{k+1}-z\|_{L^{2}(\Omega)}^{2},\\
\bar \lambda^{k+1}&=\bar \lambda^{k}+\sigma(\bar u^{k+1}-\bar z^{k+1}).
\end{aligned} \right.$$

Usually, it is expensive and unnecessary to exactly compute the solution of each subproblem even if it is feasible. Thus it is natural to use some iterative methods such as Krylov-based methods to solve the subproblems which are equivalent to large scale or ill-conditioned linear systems. The inexact ADMM algorithm in finite dimension space and its convergence results under certain error criterion have been studied extensively recently (see [@Chen2017; @Li2015]). Taking the inexactness of the solutions in the function space into account, Song et al. applied the inexact ADMM algorithm to Hilbert space for PDE-constrained optimization problems and presented the convergence results in [@Song2017Fe]. Given $(u^{0},z^{0},{\lambda}^{0})\in L^{2}(\Omega)\times {\rm dom}(\delta_{U_{ad}}(\cdot))\times L^{2}(\Omega)$, parameters $\sigma>0$, $\tau \in \left(0,\dfrac{1+\sqrt{5}}{2}\right)$. Let$\lbrace\epsilon_{k}\rbrace_{k=0}^{\infty}$ be a sequence satisfying 