Reduced order solution of structured linear systems arising in certain PDE‐constrained optimization problems

The solution of PDE‐constrained optimal control problems is a computationally challenging task, and it involves the solution of structured algebraic linear systems whose blocks stem from the discretized first‐order optimality conditions. In this paper we analyze the numerical solution of this large‐scale system: we first perform a natural order reduction, and then we solve the reduced system iteratively by exploiting specifically designed preconditioning techniques. The analysis is accompanied by numerical experiments on two application problems.