Balancing Neumann-Neumann Methods
for Elliptic Optimal Control Problems
Matthias Heinkenschloss
Hoang Nguyen
Department of Computational and Applied Mathematics
Rice University
In
Proceedings of the 15th International Conference on
Domain Decomposition,
R. Kornhuber, R. H. W. Hoppe, J. Periaux, O. Pironneau,
O. B. Widlund, and J. Xu (eds.),
Lecture Notes in Computational Science and Engineering Vol. 40,
Springer-Verlag, Heidelberg,
2004, pp. 589-596.
Abstract
We present Neumann-Neumann domain decomposition (DD) preconditioners
for the solution of elliptic linear quadratic optimal control problems.
The preconditioner is applied to the optimality system.
A Schur complement formulation is derived that
reformulates the original optimality system as a system in the
state and adjoint variables restricted to the subdomain boundaries.
The application of the Schur complement matrix requires the solution
of subdomain optimal control problems with Dirichlet boundary
conditions on the subdomain interfaces.
The application of the inverses of the subdomain Schur complement matrices
require the solution of subdomain optimal control problems with
Neumann boundary conditions on the subdomain interfaces.
Numerical tests show that the dependence of this preconditioner on
mesh size and subdomain size is comparable to its counterpart applied
to elliptic equations only.
PDF file (125 KB).