Preconditioning of Active-Set Newton Methods for PDE-constrained Optimal Control Problems
This work addresses the computational bottleneck of solving large-scale PDE-constrained optimal control problems with constraints, providing more robust preconditioners for practitioners in computational science and engineering.
The paper introduces two new preconditioners for saddle point linear systems arising in active-set Newton methods for PDE-constrained optimal control problems with constraints. Numerical experiments on 3D problems demonstrate robustness with respect to problem parameters and show improvements over existing preconditioned conjugate gradient methods.
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.