New time domain decomposition methods for parabolic control problems I: Dirichlet-Neumann and Neumann-Dirichlet algorithms

TL;DR

This paper introduces new time domain decomposition algorithms for parabolic optimal control problems, analyzing their convergence and efficiency through theoretical and numerical studies.

Contribution

It develops novel Dirichlet-Neumann and Neumann-Dirichlet algorithms with a detailed convergence analysis and optimal relaxation parameters for parabolic control problems.

Findings

Identified effective algorithms as smoothers and improved solver choices.

Analyzed convergence behavior and optimal relaxation parameters.

Validated methods with numerical experiments.

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 semi-discretization, 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.