An Alternating Direction Method of Multipliers for the Optimization Problem Constrained with a Stationary Maxwell System

Abstract

This paper mainly focuses on an efficient numerical method for the optimization problem constrained with a stationary Maxwell system. Following the idea of [32], the edge element is applied to approximate the state variable and the control variable, then the continuous optimal control problem is discretized into a finite dimensional one. The novelty of this paper is the approach for solving the discretized system. Based on the separable structure, an alternating direction method of multipliers (ADMM) is proposed. Furthermore, the global convergence analysis is established in the form of the objective function error, which includes the discretization error by the edge element and the iterative error by ADMM. Finally, numerical simulations are presented to demonstrate the efficiency of the proposed algorithm.