In this paper, sparse elliptic PDE-constrained optimization problems with \(\varvec{L}^{\varvec{1}}\) -control cost ( \(\varvec{L}^{\varvec{1}}\) -EOCP) are considered. To solve these infinite-dimensional optimization problems numerically, we propose an inexact adaptive multi-level alternating direction method of multipliers (iAm-ADMM). The iAm-ADMM is designed by extending the alternating direction method of multipliers (ADMM) to infinite-dimensional spaces, addressing the challenges associated with the high computational cost of traditional finite element discretization methods. The core idea of the proposed method is to employ a multi-level strategy, gradually refining the mesh during the solution process. Additionally, we introduce an innovative approach for guiding the mesh refinement using a posteriori error estimation, ensuring that the refinement is driven by the true residual. In each iteration, the subproblems are solved inexactly, allowing for greater computational efficiency. Numerical experiments demonstrate the advantages of the iAm-ADMM algorithm, particularly its flexibility in mesh refinement, in efficiently solving infinite-dimensional optimization problems.
