New time domain decomposition methods for parabolic optimal control problems II: Neumann-Neumann algorithms

Abstract

We present new Neumann-Neumann algorithms based on a time domain decomposition applied to unconstrained parabolic optimal control problems. After a spatial semi-discretization, the Lagrange multiplier approach provides a coupled forward-backward optimality system, which can be solved using a time domain decomposition. Due to the forward-backward structure of the optimality system, nine variants can be found for the Neumann-Neumann algorithms. We analyze their convergence behavior and determine the optimal relaxation parameter for each algorithm. Our analysis reveals that the most natural algorithms are actually only good smoothers, and there are better choices which lead to efficient solvers. We illustrate our analysis with numerical experiments.