Preconditioning of Active-Set Newton Methods for PDE-constrained Optimal Control Problems

TL;DR

This paper introduces two novel preconditioners for saddle point systems in PDE-constrained optimal control problems, improving robustness and efficiency in active-set Newton methods through block matrix factorization.

Contribution

The paper presents new preconditioners based on block matrix factorization that enhance the solution of saddle point systems in PDE-constrained optimal control problems.

Findings

Preconditioners show robustness across various problem parameters.

Numerical experiments demonstrate improved convergence in 3D problems.

Comparison with existing methods highlights efficiency gains.

Abstract

We address the problem of preconditioning a sequence of saddle point linear systems arising in the solution of PDE-constrained optimal control problems via active-set Newton methods, with control and (regularized) state constraints. We present two new preconditioners based on a full block matrix factorization of the Schur complement of the Jacobian matrices, where the active-set blocks are merged into the constraint blocks. We discuss the robustness of the new preconditioners with respect to the parameters of the continuous and discrete problems. Numerical experiments on 3D problems are presented, including comparisons with existing approaches based on preconditioned conjugate gradients in a nonstandard inner product.