Solving Semi-Linear Elliptic Optimal Control Problems with $L^1$-Cost via Regularization and RAS-Preconditioned Newton Methods

TL;DR

This paper introduces a parallel computational framework combining regularization, continuation, and RAS-preconditioned Newton methods to efficiently solve semi-linear elliptic optimal control problems with $L^1$-cost, addressing nonlinearity and non-smoothness.

Contribution

It develops a novel combination of regularization, continuation, and domain-decomposition preconditioning for efficient, parallel solution of $L^1$-regularized PDE-constrained optimal control problems.

Findings

Demonstrates convergence of the regularized solutions.

Shows improved robustness and parallel efficiency of the Newton method.

Provides extensive numerical experiments validating the framework.

Abstract

We present a new parallel computational framework for the efficient solution of a class of $L^2$/$L^1$-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 $L^1$-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.