A FE-ADMM algorithm for Lavrentiev-regularized state-constrained elliptic control problem

TL;DR

This paper introduces a novel FE-ADMM algorithm for Lavrentiev-regularized elliptic control problems with state constraints, combining error estimation, efficient solution strategies, and a two-phase approach for improved accuracy.

Contribution

It develops a new hADMM algorithm with weighted norms and a two-phase strategy incorporating PDAS for better solutions in state-constrained elliptic control problems.

Findings

Error estimates confirm the effectiveness of the discretization.

The hADMM algorithm demonstrates high efficiency in numerical tests.

The two-phase strategy improves solution accuracy significantly.

Abstract

In this paper, elliptic control problems with pointwise box constraints on the state is considered, where the corresponding Lagrange multipliers in general only represent regular Borel measure functions. To tackle this difficulty, the Lavrentiev regularization is employed to deal with the state constraints. To numerically discretize the resulted problem, since the weakness of variational discretization in numerical implementation, full piecewise linear finite element discretization is employed. Estimation of the error produced by regularization and discretization is done. The error order of full discretization is not inferior to that of variational discretization because of the Lavrentiev-regularization. Taking the discretization error into account, algorithms of high precision do not make much sense. Utilizing efficient first-order algorithms to solve discretized problems to moderate accuracy is sufficient. Then a heterogeneous alternating direction method of multipliers (hADMM) is proposed. Different from the classical ADMM, our hADMM adopts two different weighted norms in two subproblems respectively. Additionally, to get more accurate solution, a two-phase strategy is presented, in which the primal-dual active set (PDAS) method is used as a postprocessor of the hADMM. Numerical results not only verify error estimates but also show the efficiency of the hADMM and the two-phase strategy.