New Time Domain Decomposition Methods for Parabolic Optimal Control Problems I: Dirichlet–Neumann and Neumann–Dirichlet Algorithms

SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 2048-2070, August 2024.
Abstract. We present new Dirichlet–Neumann and Neumann–Dirichlet algorithms with a time domain decomposition applied to unconstrained parabolic optimal control problems. After a spatial semidiscretization, we use the Lagrange multiplier approach to derive a coupled forward-backward optimality system, which can then be solved using a time domain decomposition. Due to the forward-backward structure of the optimality system, three variants can be found for the Dirichlet–Neumann and Neumann–Dirichlet 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.