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 H1 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 auxiliary optimality systems 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.
| Original language | English |
|---|---|
| Article number | 3 |
| Journal | Computational Optimization and Applications |
| Volume | 82 |
| Issue number | 3 |
| Pages (from-to) | 753-794 |
| Number of pages | 42 |
| ISSN | 0926-6003 |
| DOIs | |
| Publication status | Published - 07.2022 |
Funding
Dominik Hafemeyer acknowledges support from the graduate program TopMath of the Elite Network of Bavaria and the TopMath Graduate Center of TUM Graduate School at Technische Universität München. He received a scholarship from the Studienstiftung des deutschen Volkes. He receives support from the IGDK Munich-Graz. Funded by the Deutsche Forschungsgemeinschaft, grant no 188264188/GRK1754. Dominik Hafemeyer acknowledges support from the graduate program TopMath of the Elite Network of Bavaria and the TopMath Graduate Center of TUM Graduate School at Technische Universität München. He received a scholarship from the Studienstiftung des deutschen Volkes. He receives support from the IGDK Munich-Graz. Funded by the Deutsche Forschungsgemeinschaft, grant no 188264188/GRK1754.
