Substructuring Waveform Relaxation Methods for Parabolic Optimal Control Problems

We study in this paper Dirichlet-Neumann and Neumann-Neumann waveform relaxation methods for the parallel solution of linear-quadratic parabolic optimal control problems, originating from the examples of transient optimal heating with distributed control. Unlike in the case of single linear or nonlinear parabolic problem, we need to solve here two coupled parabolic problems that arise as a part of optimality system for the optimal control problem. We present the detail algorithms for the case of two non-overlapping subdomains and show conditional convergence properties in few special cases. We illustrate our findings with numerical results.