Solving Semi-Linear Elliptic Optimal Control Problems with $L^1$-Cost via Regularization and RAS-Preconditioned Newton Methods
Abstract
We present a new parallel computational framework for the efficient solution of a class of /-regularized optimal control problems governed by semi-linear elliptic partial differential equations (PDEs). The main difficulty in solving this type of problem is the nonlinearity and non-smoothness of the -term in the cost functional, which we address by employing a combination of several tools. First, we approximate the non-differentiable projection operator appearing in the optimality system by an appropriately chosen regularized operator and establish convergence of the resulting system's solutions. Second, we apply a continuation strategy to control the regularization parameter to improve the behavior of (damped) Newton methods. Third, we combine Newton's method with a domain-decomposition-based nonlinear preconditioning, which improves its robustness properties and allows for parallelization. The efficiency of the proposed numerical framework is demonstrated by extensive numerical experiments.
Cite
@article{arxiv.2411.00546,
title = {Solving Semi-Linear Elliptic Optimal Control Problems with $L^1$-Cost via Regularization and RAS-Preconditioned Newton Methods},
author = {Gabriele Ciaramella and Michael Kartmann and Georg Müller},
journal= {arXiv preprint arXiv:2411.00546},
year = {2025}
}