Title: An adaptive Newton algorithm for optimal control problems with application to optimal electrode design,
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.