Distributed Solution of Optimal Control Problems
Governed by Parabolic Equations
M. Heinkenschloss
Department of Computational and Applied Mathematics
Rice University
M. Herty
Fachbereich Mathematik,
Technische Universität Kaiserslautern
In Robust Optimization: Directed Design,
A. J. Kurdila and P. M. Pardalos and M. Zabarankin (eds.),
pages 71-90, 2006,
Springer-Verlag, Nonconvex Optimization and Its Applications, Vol.~81.
Abstract
We present a spatial domain decomposition (DD) method
for the solution of discretized parabolic 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 elliptic partial differential equations and of
linear--quadratic optimal control problems governed by elliptic equations.
We use a decomposition of the
spatial domain into non-overlapping subdomains.
The optimality conditions for the parabolic linear quadratic optimal
control problem is split into smaller problems restricted to
spatial subdomain-time cylinders. These subproblems correspond
to parabolic linear--quadratic optimal control problems on
subdomains with Dirichlet data on interfaces.
The coupling of these subdomain problems leads to a Schur complement
system in which the unknowns are the state and adjoint variables
on the subdomain interfaces in space and time.
The Schur complement system is solved using a preconditioned GMRES.
The preconditioner is obtained from the solution of appropriate
subdomain parabolic linear--quadratic optimal control problems.
The dependence of the performance of these preconditioners on
mesh size and subdomain size is studied numerically. Our tests indicate
that their dependence on mesh size and subdomain size is similar
to that of its counterpart applied to elliptic equations only.
Our tests also suggest that the preconditioners
are insensitive to the size of the control regularization parameter.
Keywords
Optimal control, parabolic equations, domain decomposition, Neumann-Neumann methods