Non-overlapping Schwarz methods in time for parabolic optimal control problems

TL;DR

This paper explores the application of classical Schwarz methods with time domain decomposition to unconstrained parabolic optimal control problems, highlighting different properties and variants based on the system's structure.

Contribution

It introduces novel Schwarz method variants tailored for parabolic optimal control problems, analyzing their properties and potential as smoothers or solvers.

Findings

Different properties based on forward-backward structure

Variants using Dirichlet and Neumann conditions

Some variants serve as effective smoothers or solvers

Abstract

We present here the classical Schwarz method with a time domain decomposition applied to unconstrained parabolic optimal control problems. Unlike Dirichlet-Neumann and Neumann-Neumann algorithms, we find different properties based on the forward-backward structure of the optimality system. Variants can be found using only Dirichlet and Neumann transmission conditions. Some of these variants are only good smoothers, while others could lead to efficient solvers.