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\begin{center}
\rule{6in}{1pt} \
{\large Andrei Draganescu \\
{\bf Multigrid preconditioning of linear systems for interior point methods applied to a class of box-constrained optimal control problems}}

Department of Mathematics and Statistics \\ University of Maryland Baltimore County \\ Baltimore MD 21250
\\
{\tt draga@umbc.edu}\\
Cosmin Petra\end{center}

In this work we construct and analyze multigrid preconditioners for
operators of the form \mbox{${\mathcal D}_{\lambda}+{\mathcal
K}_h^*{\mathcal K}_h$}, where $D_{\lambda}$ is the multiplication with a
relatively ``smooth'' discrete function $\lambda>0$ and ${\mathcal K}_h$
is a discretization of a compact linear operator ${\mathcal K}$. These
systems arise when applying interior point methods to the distributed
optimal control problem \mbox{$\min_u\frac{1}{2}(|\!|{\mathcal K}
u-f|\!|^2 +\beta|\!|u|\!|^2)$} with box constraints
$\underline{u}\leqslant u\leqslant\overline{u}$ on the control $u$. The
presented preconditioning technique is related to the one developed by
Dr{\u a}g{\u a}nescu and Dupont in~\cite{dradup} for the associated
unconstrained problem, and is intended for large-scale problems. As
in~\cite{dradup}, the quality of the resulting preconditioners is shown
to increase as the resolution $h\downarrow 0$ at a rate that is optimal
with respect to $h$ if the meshes are uniform, but decreases as the
smoothness of $\lambda$ declines. We test this algorithm first on a
Tikhnov-regularized backward parabolic equation with $[0,1]$ constraints
and then on the elliptic-constrained optimization problem
\begin{eqnarray*}
\begin{array}{cc}\vspace{7pt}
\textnormal{minimize}& \frac{1}{2}|\!|y-f|\!|^2 +
\frac{\beta}{2}|\!|u|\!|^2 \\ \textnormal{subj. to\ \ } &\Delta
y=u\ ,\ y\in H_0^1(\Omega),\ \underline{u} \leqslant u \leqslant \overline{u}\
\ a.e.
\end{array}
\end{eqnarray*}
In both cases it is shown that the number of linear iterations per
optimization step, as well as the total number of fine-scale
matrix-vector multiplications is decreasing with increasing resolution,
thus showing the method to be potentially very efficient for truly
large-scale problems.

This is joint work with Cosmin Petra from the Argonne National Laboratory.

\begin{thebibliography}{99}
\bibitem{dradup} Andrei Dr{\u a}g{\u a}nescu and Todd F.~Dupont.
\emph{Optimal order multilevel preconditioners for regularized ill-posed
problems}, Math. Comp., 77 (2008), pp. 2001--2038.


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