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.
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
First Page Number
Last Page Number
© 2010, SIAM Publications.