Analysis of Block Parareal Preconditioners for Parabolic Optimal Control Problems

In this paper, we describe block matrix algorithms for the iterative solution of large scale linear-quadratic optimal control problems arising from the optimal control of parabolic partial differential equations over a finite control horizon. We describe three iterative algorithms. The first algorithm employs a CG method for solving a symmetric positive definite reduced linear system involving only the unknown control variables. This system can be solved using the CG method, but requires double iteration. The second algorithm is designed to avoid double iteration by introducing an auxiliary variable. It yields a symmetric indefinite system and a positive definite block preconditioner. The third algorithm uses a symmetric positive definite block diagonal preconditioner for the saddle point system and is based on the parareal algorithm. Theoretical results show that the preconditioned algorithm has optimal convergence properties and parallel scalability. Numerical experiments are provided to confirm the theoretical results.