15:00
Iterative regularization of nonsmooth equations
Prof. Dr. Christian Clason | Universität Duisburg-Essen | Germany
Show details
Authors:
Prof. Dr. Christian Clason | Universität Duisburg-Essen | Germany
Dr. Huu Nhu Vu | Universität Duisburg-Essen
We consider inverse problems for nonlinear forward models that are directionally but not Fréchet differentiable; examples include solution mappings for nonsmooth partial differential equations or variational inequalities. In this setting, standard derivative-based regularization methods based on successive linearization such as Landweber or Levenberg--Marquardt iteration are inapplicable. We show that using elements of the Bouligand subdifferential for the linearization still leads to a convergent regularization scheme. The proof rests on an asymptotic stability property that follows from a generalized tangential cone condition, which can be verified for some nonsmooth model problems. Numerical results demonstrate the performance of these nonsmooth iterative regularization methods.
15:20
On Second-Order Optimality Conditions for Optimal Control Problems Governed by the Classical Obstacle Problem
Dr. Constantin Christof | TU München
Show details
Authors:
Dr. Constantin Christof | TU München
Prof. Dr. Gerd Wachsmuth | Technische Universität Cottbus-Senftenberg | Germany
This talk is concerned with second-order optimality conditions
for Tikhonov-regularized optimal control problems governed by the classical obstacle problem.
Using a simple observation that allows to identify precisely
the structure of optimal controls on the active set,
we derive various conditions that guarantee the
local/global optimality of first-order stationary points and
the local/global quadratic growth of the reduced objective function.
Our analysis extends and refines existing results from the literature, and
also covers those situations where the problem at hand
involves additional box-constraints on the control.
As a byproduct, our approach shows that Tikhonov-regularized
optimal control problems for the classical obstacle problem
can be reformulated as state-constrained optimal control problems for the Poisson equation.