In this paper, elliptic PDE-constrained optimization problems with box constraints on the control are considered. To numerically solve the problems, we apply the ‘optimize-discretize-optimize’ strategy. Specifically, the alternating direction method of multipliers (ADMM) algorithm is applied in function space first, and then, the standard piecewise linear finite-element approach is employed to discretize the subproblems in each iteration. Finally, some efficient numerical methods are applied to solve the discretized subproblems based on their structures. Motivated by the idea of the multi-level strategy, instead of fixing the mesh size before the computation process, we propose the strategy of gradually refining the grid. Moreover, the subproblems in each iteration are solved by appropriate inexact methods. Based on the strategies above, an efficient convergent multi-level ADMM (mADMM) algorithm is proposed. We present the convergence analysis and the iteration complexity results o(1/k) for the mADMM algorithm. Some numerical experiments are done and the numerical results show the high efficiency of the mADMM algorithm.

在本文中，考虑了椭圆型PDE约束优化问题，该问题带有控制盒约束。为了从数值上解决问题，我们应用了“优化-离散化-优化”战略。具体而言，首先在函数空间中应用乘数交变方向法（ADMM），然后采用标准分段线性有限元方法离散化每次迭代中的子问题。最后，应用一些有效的数值方法来解决离散子问题的结构。基于多级策略的思想，我们提出了逐步细化网格的策略，而不是在计算过程之前固定网格大小。此外，通过适当的不精确方法可以解决每次迭代中的子问题。在上述策略的基础上，提出了一种高效的收敛多级ADMM算法。我们给出收敛性分析和迭代复杂度结果o（1 / k）用于mADMM算法。进行了一些数值实验，数值结果表明该算法具有很高的效率。