Neumann-Neumann Domain Decomposition Preconditioners
for Linear-Quadratic Elliptic Optimal Control Problems
Abstract
We present a class of domain decomposition (DD) preconditioners
for the solution of elliptic 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 partial differential equations.
The DD preconditioners are 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 Neumann interface conditions for the states
and the adjoints. 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 elliptic linear quadratic
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 Neumann interface conditions for the state.
The subdomain Schur complement operators are analyzed in the variational
setting of the problem as well as the algebraic setting obtained after
a finite element discretization of the problem. Definiteness properties
of the algebraic form of the (subdomain) Schur complement operator(s)
are studied.
Numerical tests show that the dependence of these preconditioners on
mesh size and subdomain size is comparable to its counterpart applied
to elliptic equations only. These tests also show that the preconditioners
are insensitive to the size of the control regularization parameter.