Preprint A481/2006
A Robust Preconditioner for the Hessian System in Elliptic Optimal Control Problems

Christian Schaerer | Goncalves, Etereldes | Mathew , Tarek | Sarkis, Marcus

**Keywords: **
Optimal control | elliptic Neumann problem | fast sine transform | saddle point problem | regularization | preconditioners

In this paper, we describe a robust preconditioner for
the symmetric positive definite Hessian system associated
with the finite element discretization of an elliptic
optimal control problem in two dimensions. The Hessian system is obtained by block reduction of the original discretization and determines the control variables, which in our application corresponds to Neumann data on a segment of the boundary of
the elliptic problem. We formulate a Fast Sine Transform based preconditioner for the Hessian matrix, and show that it yields a condition number that is uniformly bounded with respect to the mesh size and regularization terms. Numerical results confirm the theoretical bounds.