A multi-level ADMM algorithm for elliptic PDE-constrained optimization problems (Q2027683)

In this paper, a multi-level alternating direction method of multipliers ADMM (mADMM) algorithm is employed for solving optimization problems with PDE constraints applying the `optimize-discretize-optimize' strategy. In this way, the subproblems of the inexact ADMM algorithm are discretized by different discretization schemes. Following the multi-level strategy, the grid is refined gradually and the subproblems are solved inexactly with Krylov-based methods. Overall a fast convergent mADMM algorithm is designed, which significantly reduces the computational cost. The authors show that the iteration complexity is of order \({\mathcal O} (1/k) \), where \(k\) is the number of iterations. Numerical results demonstrate the efficiency of the proposed mADMM algorithm.