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.