Properties of linear systems in PDE-constrained optimization. Part I: Distributed control 1

Optimization problems with constraints that contain a partial differential equation arise widely in many areas of science. In this paper, we consider distributed control problems in which the 2- and 3-dimensional Poisson problem is the PDE. If a discretize-then-optimization approach is used to solve the optimization problem, then a large dimensional, symmetric and indefinite linear system must be solved. In general, distributed control problems include a regularization term, the size of which is determined by a real value known as the regularization parameter. The spectral properties and, hence, the condition number of the linear system are highly dependent on the size of this regularization parameter. We derive intervals that contain the eigenvalues of the linear systems and, using these, we are able to show that if the regularization parameter is larger than a certain value, then backward-stable direct methods will compute solutions to the discretized optimization problem that have relative errors of the order of machine precision: changing the value of the regularization parameter within this interval will have negligible effect on the accuracy but the condition number of the system may have dramatically changed. We also analyse the spectral properties of the Schur complement and reduced systems derived via the nullspace method. Throughout the paper, we complement the theoretical results with numerical results.