Nonlinear optimized Schwarz preconditioner for elliptic optimal control problems

TL;DR

This paper presents a nonlinear Schwarz preconditioner for elliptic optimal control problems, improving robustness and efficiency over direct semismooth Newton methods through domain decomposition and tailored parameters.

Contribution

It introduces a novel nonlinear preconditioned iteration based on domain decomposition and semismooth Newton for elliptic optimal control problems with nonsmooth terms.

Findings

Significant robustness improvements over direct methods

Enhanced computational efficiency in multi-subdomain tests

Effective parameter and continuation strategies identified

Abstract

We introduce a domain decomposition-based nonlinear preconditioned iteration for solving nonlinear, nonsmooth elliptic optimal control problems, with a nonlinear reaction term, $L^1$ regularization and box constraints on the control function. The method is obtained by applying semismooth Newton to the fixed-point equation of the parallel optimized Schwarz iteration. As a proof of concept, numerical experiments are performed on two subdomains, as well as on a multi-subdomain test case. The results show that it is possible to obtain substantial improvements in robustness and efficiency with the new method, relative to semismooth Newton applied directly to the full optimization problem, provided appropriate Robin parameters and a good continuation strategy are chosen.