Neumann-Neumann Domain Decomposition Preconditioners for Linear-Quadratic Elliptic Optimal Control Problems

We present a class of domain decomposition (DD) preconditioners for the solution of elliptic linear-quadratic optimal control problems. Our DD preconditioners are extensions of Neumann–Neumann DD preconditioners, which have been successfully applied to the solution of single PDEs. The DD preconditioners are based on a decomposition of the optimality conditions for the elliptic linear-quadratic optimal control problem into smaller subdomain optimality conditions with Dirichlet boundary conditions for the states and the adjoints on the subdomain interfaces. These subdomain optimality conditions are coupled through Neumann interface conditions for the states and the adjoints. This decomposition leads to a Schur complement system in which the unknowns are the state and adjoint variables on the subdomain interfaces. The Schur complement operator is the sum of subdomain Schur complement operators, the application of which is shown to correspond to the solution of subdomain elliptic linear-quadratic optimal control problems, which are essentially smaller copies of the original optimal control problem. We show that, under suitable conditions, the application of the inverse of the subdomain Schur complement operators requires the solution of a subdomain elliptic linear-quadratic optimal control problem with Neumann interface conditions for the state. The subdomain Schur complement operators are analyzed in the variational setting of the problem as well as the algebraic setting obtained after a finite element discretization of the problem. Definiteness properties of the algebraic form of the (subdomain) Schur complement operator(s) are studied. Numerical tests show that the dependence of these preconditioners on mesh size and subdomain size is comparable to its counterpart applied to elliptic equations only. These tests also show that the preconditioners are insensitive to the size of the control regularization parameter.