Block Diagonal Parareal Preconditioner for Parabolic Optimal Control Problems
Christian Schaerer ,
Tarek Mathew ,
Marcus Sarkis
Keywords:
Preconditioner, Finite Element, Parareal, Parabolic Optimal Control, KKT System
Abstract:
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.
MSC 2000:
65M60 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods
65J10 Equations with linear operators (do not use
65Fxx)
Notes:
Submitted
