# An adaptive Newton algorithm for optimal control problems with application to optimal electrode design

## TL;DR

This paper introduces an adaptive Newton algorithm for nonlinear PDE-constrained optimization, balancing discretization and iteration errors with goal-oriented error estimation, demonstrated through efficient electrode design in neuroscience.

## Contribution

It presents a novel adaptive Newton method that integrates goal-oriented error estimation for efficient PDE-constrained optimization.

## Key findings

- Efficient balancing of discretization and iteration errors.
- Local mesh refinement improves computational efficiency.
- Successful application to optimal electrode design in neuroscience.

## 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 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.