TY - JOUR
T1 - A path-following inexact Newton method for PDE-constrained optimal control in BV
AU - Hafemeyer, D.
AU - Mannel, F.
N1 - Publisher Copyright:
© 2022, The Author(s).
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PY - 2022/7
Y1 - 2022/7
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=85129826516&partnerID=8YFLogxK
UR - https://www.mendeley.com/catalogue/1bd59428-854b-3abc-9bd5-a8a9e98e834c/
U2 - 10.1007/s10589-022-00370-2
DO - 10.1007/s10589-022-00370-2
M3 - Journal articles
AN - SCOPUS:85129826516
SN - 0926-6003
VL - 82
SP - 753
EP - 794
JO - Computational Optimization and Applications
JF - Computational Optimization and Applications
IS - 3
M1 - 3
ER -
