A Spatial Domain Decomposition Method
for Parabolic Optimal Control Problems
Department of Computational and Applied Mathematics
Technische Universität Kaiserslautern
CAAM TR 05-03, May 2005 (Revised Jan 2006)
Journal of Computational and Applied Mathematics, to appear.
We present a non-overlapping spatial domain decomposition method for the
solution of linear-quadratic parabolic optimal control problems.
The spatial domain is decomposed into non-overlapping subdomains.
The original parabolic optimal control problem is decomposed into smaller
problems posed on space-time cylinder subdomains with auxiliary state and
adjoint variables imposed as Dirichlet boundary conditions on the
space-time interface boundary. The subdomain problems are coupled through
Robin transmission conditions. This leads to a Schur complement equation
in which the unknowns are the auxiliary state adjoint variables on the
space-time interface boundary. The Schur complement operator is the sum of
space-time subdomain Schur complement operators. The application of these
subdomain Schur complement operators is equivalent to the solution of an
subdomain parabolic optimal control problem. The subdomain Schur complement
operators are shown to be invertible and the application of their inverses is equivalent
to the solution of a related subdomain parabolic optimal control problem.
We introduce a new family of Neumann-Neumann type preconditioners for the Schur complement
system including several different coarse grid corrections.
We compare the numerical performance of our preconditioners with an
alternative approach recently introduced by Benamou.
Optimal control, parabolic equations, domain decomposition, Neumann-Neumann methods