@misc{KlapprothSchielaDeuflhard2009, author = {Klapproth, Corinna and Schiela, Anton and Deuflhard, Peter}, title = {Consistency Results for the Contact-Stabilized Newmark Method}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-11164}, number = {09-06}, year = {2009}, abstract = {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 St{\"o}rmer-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.}, language = {en} } @misc{SchielaWeiser2009, author = {Schiela, Anton and Weiser, Martin}, title = {Barrier methods for a control problem from hyperthermia treatment planning}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-11567}, number = {09-36}, year = {2009}, abstract = {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.}, language = {en} } @misc{SchielaGuenther2008, author = {Schiela, Anton and G{\"u}nther, Andreas}, title = {Interior Point Methods in Function Space for State Constraints - Inexact Newton and Adaptivity}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-11007}, number = {09-01}, year = {2008}, abstract = {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.}, language = {en} } @misc{PruefertSchiela2008, author = {Pr{\"u}fert, Uwe and Schiela, Anton}, title = {The minimization of an L^{\infty}-functional subject to an elliptic PDE and state constraints}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-10714}, number = {08-17}, year = {2008}, abstract = {We study the optimal control of a maximum-norm objective functional subject to an elliptic-type PDE and pointwise state constraints. The problem is transformed into a problem where the non-differentiable L^{\infty}-norm in the functional will be replaced by a scalar variable and additional state constraints. This problem is solved by barrier methods. We will show the existence and convergence of the central path for a class of barrier functions. Numerical experiments complete the presentation.}, language = {en} } @misc{KlapprothDeuflhardSchiela2008, author = {Klapproth, Corinna and Deuflhard, Peter and Schiela, Anton}, title = {A Perturbation Result for Dynamical Contact Problems}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-10793}, number = {08-27}, year = {2008}, abstract = {This paper is intended to be a first step towards the continuous dependence of dynamical contact problems on the initial data as well as the uniqueness of a solution. Moreover, it provides the basis for a proof of the convergence of popular time integration schemes as the Newmark method. We study a frictionless dynamical contact problem between both linearly elastic and viscoelastic bodies which is formulated via the Signorini contact conditions. For viscoelastic materials fulfilling the Kelvin-Voigt constitutive law, we find a characterization of the class of problems which satisfy a perturbation result in a non-trivial mix of norms in function space. This characterization is given in the form of a stability condition on the contact stresses at the contact boundaries. Furthermore, we present perturbation results for two well-established approximations of the classical Signorini condition: The Signorini condition formulated in velocities and the model of normal compliance, both satisfying even a sharper version of our stability condition.}, language = {en} } @misc{SchielaWollner2008, author = {Schiela, Anton and Wollner, Winnifried}, title = {Barrier Methods for Optimal Control Problems with Convex Nonlinear Gradient Constraints}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-11014}, number = {08-47}, year = {2008}, abstract = {In this paper we are concerned with the application of interior point methods in function space to gradient constrained optimal control problems, governed by partial differential equations. We will derive existence of solutions together with first order optimality conditions. Afterwards we show continuity of the central path, together with convergence rates depending on the interior point parameter.}, language = {en} } @misc{GoetschelWeiserSchiela2010, author = {G{\"o}tschel, Sebastian and Weiser, Martin and Schiela, Anton}, title = {Solving Optimal Control Problems with the Kaskade 7 Finite Element Toolbox}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-11909}, number = {10-25}, year = {2010}, abstract = {This paper presents concepts and implementation of the finite element toolbox Kaskade 7, a flexible C++ code for solving elliptic and parabolic PDE systems. Issues such as problem formulation, assembly and adaptivity are discussed at the example of optimal control problems. Trajectory compression for parabolic optimization problems is considered as a case study.}, language = {en} } @misc{SchielaWachsmuth2011, author = {Schiela, Anton and Wachsmuth, Daniel}, title = {Convergence Analysis of Smoothing Methods for Optimal Control of Stationary Variational Inequalities}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-13125}, number = {11-23}, year = {2011}, abstract = {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.}, language = {en} } @misc{SchielaHintermueller2011, author = {Schiela, Anton and Hinterm{\"u}ller, Michael}, title = {On the Length of the Primal-Dual Path in Moreau-Yosida-based Path-following for State Constrained Optimal Control: Analysis and Numerics}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-14071}, number = {11-37}, year = {2011}, abstract = {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.}, language = {en} } @misc{MeyerPanizziSchiela2010, author = {Meyer, Christian and Panizzi, Lucia and Schiela, Anton}, title = {Uniqueness criteria for solutions of the adjoint equation in state-constrained optimal control}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-11933}, number = {10-28}, year = {2010}, abstract = {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.}, language = {en} }