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

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.