Preprint A479/2006
Block Diagonal Parareal Preconditioner for Parabolic Optimal Control Problems

Marcus Sarkis | Schaerer, Christian | Mathew , Tarek

**Keywords: **
Preconditioner | Finite Element | Parareal | Parabolic Optimal Control | KKT System

We describe an iterative algorithm for the solution of a large scale
linear-quadratic parabolic optimal control problem. Unlike Ricatti
equation based methods, we determine the control variable by an
iterative procedure which solves a large saddle point system
obtained by an {\it all at once} discretization strategy involving
the state (primal) variables, the control variables and the adjoint
(dual) variables. We derive a reduced symmetric indefinite linear
system involving the control variables and auxiliary variables, and
solve it using a preconditioned MINRES iteration, with a symmetric
positive definite block diagonal preconditioner based on the
parareal algorithm. Theoretical and numerical results show that the
preconditioned algorithm has adequate convergence properties and
parallel scalability.