A path-following inexact Newton method for PDE-constrained optimal control in BV (extended version)

TL;DR

This paper introduces a path-following inexact Newton method for PDE-constrained optimal control problems involving functions of bounded variation, demonstrating convergence, efficiency, and robustness through theoretical analysis and numerical experiments.

Contribution

It develops a novel inexact Newton approach for BV control problems with TV regularization, proving convergence and providing a practical finite element implementation.

Findings

Convergence of auxiliary problem solutions to the original problem

Fast local convergence of the inexact Newton method

Numerical results show high accuracy and robustness

Abstract

We study a PDE-constrained optimal control problem that involves functions of bounded variation as controls and includes the TV seminorm of the control in the objective. We apply a path-following inexact Newton method to the problems that arise from smoothing the TV seminorm and adding an $H^1$ regularization. We prove in an infinite-dimensional setting that, first, the solutions of these auxiliary problems converge to the solution of the original problem and, second, that an inexact Newton method enjoys fast local convergence when applied to a reformulation of the optimality system in which the control appears as implicit function of the adjoint state. We show convergence of a Finite Element approximation, provide a globalized preconditioned inexact Newton method as solver for the discretized auxiliary problems, and embed it into an inexact path-following scheme. We construct a two-dimensional test problem with fully explicit solution and present numerical results to illustrate the accuracy and robustness of the approach. This is an extended version of the corresponding journal article appearing in "Computational Optimization and Applications". It contains some proofs that are omitted in the journal's version.