# A multi-level ADMM algorithm for elliptic PDE-constrained optimization problems

## TL;DR

This paper introduces a multi-level ADMM algorithm for efficiently solving elliptic PDE-constrained optimization problems with box constraints, combining mesh refinement, inexact subproblem solutions, and convergence guarantees.

## Contribution

It proposes a novel multi-level ADMM method with mesh refinement and inexact subproblem solving, along with convergence analysis and complexity results.

## Key findings

- The mADMM algorithm converges with a rate of O(1/k).
- Numerical experiments demonstrate high efficiency of the proposed method.
- The multi-level strategy improves computational performance over fixed mesh approaches.

## Abstract

In this paper, the elliptic PDE-constrained optimization problem with box constraints on the control is studied. To numerically solve the problem, we apply the 'optimize-discretize-optimize' strategy. Specifically, the alternating direction method of multipliers (ADMM) algorithm is applied in function space first, 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 inexactly. 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) of the proposed algorithm for the PDE-constrained optimization problems. Numerical results show the high efficiency of the mADMM algorithm.