Efficient multigrid methods for optimal control of partial differential equations

Abstract

This work is concerned with designing optimal order multigrid preconditioners for optimal control problems constrained by partial differential equations (PDEs). Two different optimal control problems are discussed in the dissertation. For the first problem, the PDE constraint is a linear parabolic equation and the control is the forcing term which is distributed in space and time, while for the second problem, the PDE constraint is an elliptic equation and the controls lie on the boundary. For the first problem (distributed optimal control problem constrained by a linear parabolic equation), standard space-time finite element discretizations (e.g., Crank-Nicolson discretization) lead to suboptimal results. For the boundary control of elliptic equations there is a clear distinction in terms of quality of the preconditioning between Dirichlet and Neumann boundary control, namely we observed what appear to be optimal order results for Neumann boundary control problem, while for Dirichlet boundary control the preconditioners appear to be suboptimal. In addition to the analysis of the multigrid preconditioners, the main contribution of this work for the first problem is to point out a discretization that leads to preconditioners that are of provably optimal order.