PDE-Constrained Optimization with Local Control and Boundary Observations: Robust Preconditioners

Abstract

We consider PDE-constrained optimization problems with control functions defined on a subregion of the domain of the state equation. The main purpose of this paper is to define and analyze robust preconditioners for KKT systems associated with such optimization tasks, that is, preconditioners that lead to iteration bounds, for the MINRES scheme, that are independent of the regularization parameter $\alpha$ and the mesh size $h$. Our analysis addresses elliptic control problems, subject to Tikhonov regularization, and covers cases with boundary observations only and locally defined control functions. A number of numerical experiments are presented. We also explain how the results can be adapted to an inverse problem arising in connection with electrocardiography.