A Robust Preconditioner for the Hessian System in Elliptic Optimal Control Problems

Abstract

We consider an elliptic optimal control problem in two dimensions, in which the control variable corresponds to the Neumann data on a boundary segment, and where the performance functional is regularized to ensure that the problem is well posed. A finite element discretization of this control problem yields a saddle point linear system, which can be reduced to a symmetric positive definite Hessian system for determining the control variables. We formulate a robust preconditioner for this reduced Hessian system, as a matrix product involving the discrete Neumann to Dirichlet map and a mass matrix, and show that it yields a condition number bound which is uniform with respect to the mesh size and regularization parameters. On a uniform grid, this preconditioner can be implemented using a fast sine transform. Numerical tests verify the theoretical bounds.