SIAM Journal on Scientific Computing
In this paper, we describe block matrix algorithms for the iterative solution of a large-scale linear-quadratic optimal control problem involving a parabolic partial differential equation over a finite control horizon. We consider an "all at once" discretization of the problem and formulate three iterative algorithms. The first algorithm is based on preconditioning a symmetric positive definite reduced linear system involving only the unknown control variables; however inner-outer iterations are required. The second algorithm modifies the first algorithm to avoid inner-outer iterations by introducing an auxiliary variable. It yields a symmetric indefinite system with a positive definite block preconditioner. The third algorithm is the central focus of this paper. It modifies the preconditioner in the second algorithm by a parallel-in-time preconditioner based on the parareal algorithm. Theoretical results show that the preconditioned algorithms have optimal convergence properties and parallel scalability. Numerical experiments confirm the theoretical results.
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© 2010, SIAM Publications.
Mathew, T. P., Sarkis, M., & Schaerer, C. E. (2010). Analysis of Block Parareal Preconditioners for Parabolic Optimal Control Problems. SIAM Journal on Scientific Computing, 32(3), 1180-1200. http://dx.doi.org/10.1137/080717481