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We consider first order optimality conditions for state constrained optimal control problems. In particular we study the case where the state equation has not enough regularity to admit existence of a Slater point in function space. We overcome this difficulty by a special transformation. Under a density condition we show existence of Lagrange multipliers, which have a representation via measures and additional regularity properties.

We consider an optimal control problem from hyperthermia treatment planning and its barrier regularization. We derive basic results, which lay the groundwork for the computation of optimal solutions via an interior point path-following method. Further, we report on a numerical implementation of such a method and its performance at an example problem.

Convergence Analysis of Smoothing Methods for Optimal Control of Stationary Variational Inequalities
(2011)

In the article an optimal control problem subject to a stationary variational inequality
is investigated. The optimal control problem is complemented with pointwise control constraints.
The convergence of a smoothing scheme is analyzed. There, the variational inequality
is replaced by a semilinear elliptic equation. It is shown that solutions of the regularized optimal
control problem converge to solutions of the original one. Passing to the limit in the
optimality system of the regularized problem allows to prove C-stationarity of local solutions of the original problem.
Moreover, convergence rates with respect to the regularization parameter for the error in the control are obtained.
These rates coincide with rates obtained by numerical experiments, which are included in the paper.

We derive a-priori estimates on the length of the primal-dual path that results from a
Moreau-Yosida approximation of the feasible set for state constrained optimal control problems. These bounds depend on the regularity of the state and the dimension of the
problem. Comparison with numerical results indicates that these bounds are sharp and
are attained for the case of a single active point.

Uniqueness criteria for solutions of the adjoint equation in state-constrained optimal control
(2010)

The paper considers linear elliptic equations with regular Borel measures as inhomogeneity. Such equations frequently appear in state-constrained optimal control
problems. By a counter-example of Serrin, it is known that, in the presence of
non-smooth data, a standard weak formulation does not ensure uniqueness for such
equations. Therefore several notions of solution have been developed that guarantee
uniqueness. In this note, we compare different definitions of solutions, namely the ones
of Stampacchia and the two notions of solutions of Casas and Alibert-Raymond, and show that they are the same. As side results, we reformulate the solution
in the sense of Stampacchia, and prove the existence and uniqueness of solutions in
in case of mixed boundary conditions.

This paper surveys the required mathematics for a typical challenging problem from computational medicine, the cancer therapy planning in deep regional hyperthermia. In the course of many years of close cooperation with clinics, the medical problem gave rise to quite a number of subtle mathematical problems, part of which had been unsolved when the common project started. Efficiency of numerical algorithms, i.e. computational speed and monitored reliability, play a decisive role for the medical treatment. Off-the-shelf software had turned out to be not sufficient to meet the requirements of medicine. Rather, new mathematical theory as well as new numerical algorithms had to be developed. In order to make our algorithms useful in the clinical environment, new visualization software, a virtual lab, including 3D geometry processing of individual virtual patients had to be designed and implemented. Moreover, before the problems could be attacked by numerical algorithms, careful mathematical modelling had to be done. Finally, parameter identification and constrained optimization for the PDEs had to be newly analyzed and realized over the individual patient's geometry. Our new techniques had an impact on the specificity of the individual patients' treatment and on the construction of an improved hyperthermia applicator.

A cubic regularization algorithm for nonconvex optimization in function space (in preparation)
(2012)

We propose a cubic regularization algorithm that is constructed to deal with nonconvex minimization
problems in function space. It allows for a flexible choice of the regularization term and thus accounts for the fact
that in such problems one often has to deal with more than one norm. Global and local convergence results are established
in a general framework. Moreover, several variants of step computations are compared. In the context of nonlinear
elasticity it turns out the a cg method applied to an augmented Hessian is more robust than truncated cg.

We consider a shape implant design problem that arises in the context of facial surgery.
We introduce a reformulation as an optimal control problem, where the control acts
as a boundary force. The state is modelled as a minimizer of a polyconvex
hyperelastic energy functional. We show existence of optimal solutions and
derive - on a formal level - first order optimality conditions. Finally, preliminary numerical results
are presented.

The paper considers the time integration of frictionless dynamical contact problems between viscoelastic bodies in the frame of the Signorini condition. Among the numerical integrators, interest focuses on the contact-stabilized Newmark method recently suggested by Deuflhard et al., which is compared to the classical Newmark method and an improved energy dissipative version due to Kane et al. In the absence of contact, any such variant is equivalent to the Störmer-Verlet scheme, which is well-known to have consistency order 2. In the presence of contact, however, the classical approach to discretization errors would not show consistency at all because of the discontinuity at the contact. Surprisingly, the question of consistency in the constrained situation has not been solved yet. The present paper fills this gap by means of a novel proof technique using specific norms based on earlier perturbation results due to the authors. The corresponding estimation of the local discretization error requires the bounded total variation of the solution. The results have consequences for the construction of an adaptive timestep control, which will be worked out subsequently in a forthcoming paper.

Interior Point Methods in Function Space for State Constraints - Inexact Newton and Adaptivity
(2008)

We consider an interior point method in function space for PDE constrained optimal control problems with state constraints. Our emphasis is on the construction and analysis of an algorithm that integrates a Newton path-following method with adaptive grid refinement. This is done in the framework of inexact Newton methods in function space, where the discretization error of each Newton step is controlled by adaptive grid refinement in the innermost loop. This allows to perform most of the required Newton steps on coarse grids, such that the overall computational time is dominated by the last few steps. For this purpose we propose an a-posteriori error estimator for a problem suited norm.