An Adaptive Newton Algorithm for Optimal Control Problems with Application to Optimal Electrode Design

Thomas Carraro (), Simon Dörsam, Stefan Frei and Daniel Schwarz
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Thomas Carraro: Heidelberg University
Simon Dörsam: Heidelberg University
Stefan Frei: University College London
Daniel Schwarz: Behavioural Neurophysiology, Max Planck Institute for Medical Research

Journal of Optimization Theory and Applications, 2018, vol. 177, issue 2, No 11, 498-534

Abstract: Abstract In this work, we present an adaptive Newton-type method to solve nonlinear constrained optimization problems, in which the constraint is a system of partial differential equations discretized by the finite element method. The adaptive strategy is based on a goal-oriented a posteriori error estimation for the discretization and for the iteration error. The iteration error stems from an inexact solution of the nonlinear system of first-order optimality conditions by the Newton-type method. This strategy allows one to balance the two errors and to derive effective stopping criteria for the Newton iterations. The algorithm proceeds with the search of the optimal point on coarse grids, which are refined only if the discretization error becomes dominant. Using computable error indicators, the mesh is refined locally leading to a highly efficient solution process. The performance of the algorithm is shown with several examples and in particular with an application in the neurosciences: the optimal electrode design for the study of neuronal networks.

Keywords: Optimal control; PDE constraints; Adaptive finite elements; DWR method; A posteriori error estimation; Inexact Newton method; Stopping criteria; Electroporation; Neuronal network; 49K20; 49M15; 65N30; 49M05; 65K10 (search for similar items in EconPapers)
Date: 2018
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DOI: 10.1007/s10957-018-1242-4

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