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Domain Decomposition Methods for
Advection Dominated Linear-Quadratic
Elliptic Optimal Control Problems

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Roscoe A. Bartlett

Optimization/Uncertainty Est Dept (9211)

Sandia National Laboratories

M. Heinkenschloss

Department of Computational and Applied Mathematics

Rice University

D. Ridzal

Department of Computational and Applied Mathematics

Rice University

Bart G. van Bloemen Waanders

Optimization/Uncertainty Est Dept (9211)

Sandia National Laboratories

Sandia Technical Report 2005-2895.

A shortened version is published in
Computer Methods in Applied Mechanics and Engineering,
to appear.

### Abstract

We present an optimization-level domain decomposition (DD) preconditioner
for the solution of advection dominated elliptic linear-quadratic optimal control problems.
The DD preconditioner is 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 Robin transmission conditions for the states
and the adjoints. The parameters in the Robin transmission condition
depend on the advection.
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
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 Robin boundary conditions for the state.

Numerical tests for problems with distributed and with boundary control
show that the dependence of the preconditioners on
mesh size and subdomain size is comparable to its counterpart applied
to a single advection dominated equation. These tests also show that the preconditioners
are insensitive to the size of the control regularization parameter.
#### Keywords

Optimal control, domain decomposition, Neumann-Neumann methods
SAND2005-2895 PDF file.

A shortened version is published in
Computer Methods in Applied Mechanics and Engineering,
to appear.