strategies of gradually refining the grid and inexactly solving the subproblems, and propose an efficient convergent multi-level ADMM (mADMM) algorithm. Specifically, we apply the ADMM algorithm in function space, then employ the standard piecewise linear finite element approach and implement the strategy of gradually refining the grid to the related subproblems appearing in each iteration. For the discretized subproblems, we use inexact strategies to solve them. For example, to solve the smooth subproblems, we can use some iterative methods such as Krylov-based methods. Thanks to the above strategies, we can solve the problem more efficiently and reduce the computation cost significantly. Moreover, we give the convergence analysis as well as the iteration complexity results $o(1/k)$ for the mADMM algorithm.

The paper is organized as follows. In section 2, we apply the inexact ADMM algorithm in function space first, then use finite element method to discretize the associated subproblems and propose the multi-level ADMM (mADMM) algorithm. Section 3 gives the convergence analysis and the iteration complexity of the proposed mADMM algorithm. The numerical results are given in section 4. Section 5 contains a brief summary of this paper.

A multi-level ADMM algorithm {#sec:2}
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In this section, we apply the *‘optimize-discretize-optimize’* 