Preprint A470/2006
Analysis of block matrix preconditioners for elliptic optimal control problems

Christian E. Schaerer | Mathew, Tarek P. | Sarkis, M.

**Keywords: **
Optimal control | elliptic Neumann problem | augmented Lagrangian | saddle point problem | regularization | preconditioners.

In this paper, we describe and analyze several block matrix iterative algorithms for solving a {\it saddle point} linear system arising from the discretization of a inear-quadratic {\it elliptic control} problem with Neumann boundary conditions. To ensure that the problem is well posed,
a {\it regularization} term with a parameter $\alpha$ is included.
The first algorithm reduces the saddle point system to
a symmetric positive definite Schur complement system
for the control variable and employs CG acceleration,
however, double iteration is required (except in special cases). A preconditioner yielding a rate of convergence
independent of the mesh size $h$ is described for $\Omega
\subset R^{2}$ or $R^{3}$, and a preconditioner
independent of $h$ and $\alpha$ when $\Omega \subset R^{2}$.
Next, two algorithms avoiding double iteration are described using an {\it augmented Lagrangian} formulation.
One of these algorithms solves the augmented saddle point system employing MINRES acceleration, while the other solves a symmetric positive definite reformulation of the augmented saddle point system employing CG acceleration. For both algorithms, a symmetric positive definite preconditioner is described yielding a rate of convergence independent of $h$. In addition to the above algorithms, two {\it heuristic} algorithms are described, one a projected CG algorithm, and the other an indefinite block matrix preconditioner employing GMRES
acceleration. Rigorous convergence results, however, are not known for the heuristic algorithms.