@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{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{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{DeuflhardSchielaWeiser2011, author = {Deuflhard, Peter and Schiela, Anton and Weiser, Martin}, title = {Mathematical Cancer Therapy Planning in Deep Regional Hyperthermia}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-14204}, number = {11-39}, year = {2011}, abstract = {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.}, language = {en} } @misc{LubkollSchielaWeiser2012, author = {Lubkoll, Lars and Schiela, Anton and Weiser, Martin}, title = {An optimal control problem in polyconvex hyperelasticity}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-14745}, number = {12-08}, year = {2012}, abstract = {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.}, language = {en} } @misc{WeiserGaenzlerSchiela2004, author = {Weiser, Martin and G{\"a}nzler, Tobias and Schiela, Anton}, title = {A Control Reduced Primal Interior Point Method for PDE Constrained Optimization}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-8138}, number = {04-38}, year = {2004}, abstract = {A primal interior point method for control constrained optimal control problems with PDE constraints is considered. Pointwise elimination of the control leads to a homotopy in the remaining state and dual variables, which is addressed by a short step pathfollowing method. The algorithm is applied to the continuous, infinite dimensional problem, where discretization is performed only in the innermost loop when solving linear equations. The a priori elimination of the least regular control permits to obtain the required accuracy with comparable coarse meshes. Convergence of the method and discretization errors are studied, and the method is illustrated at two numerical examples.}, language = {en} } @misc{Schiela2007, author = {Schiela, Anton}, title = {Barrier Methods for Optimal Control Problems with State Constraints}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-9504}, number = {07-07}, year = {2007}, abstract = {We study barrier methods for state constrained optimal control problems with PDEs. In the focus of our analysis is the path of minimizers of the barrier subproblems with the aim to provide a solid theoretical basis for function space oriented path-following algorithms. We establish results on existence, continuity and convergence of this path. Moreover, we consider the structure of barrier subdifferentials, which play the role of dual variables.}, language = {en} } @article{KlapprothDeuflhardSchiela2009, author = {Klapproth, Corinna and Deuflhard, Peter and Schiela, Anton}, title = {A Perturbation Result for Dynamical Contact Problems}, volume = {2}, journal = {Numer. Math}, pages = {237 -- 257}, year = {2009}, language = {en} } @article{SchielaGuenther2011, author = {Schiela, Anton and G{\"u}nther, Andreas}, title = {An Interior Point Algorithm with Inexact Step Computation in Function Space for State Constrained Optimal Control}, volume = {119}, journal = {Numer. Math.}, number = {2}, publisher = {Springer Berlin / Heidelberg}, doi = {10.1007/s00211-011-0381-4}, pages = {373 -- 407}, year = {2011}, language = {en} } @article{SchielaWeiser2008, author = {Schiela, Anton and Weiser, Martin}, title = {Superlinear convergence of the Control Reduced Interior Point Method for PDE Constrained Optimization}, volume = {39}, journal = {Computational Optimization and Applications}, number = {3}, pages = {369 -- 393}, year = {2008}, language = {en} } @article{HinzeSchiela2011, author = {Hinze, Michael and Schiela, Anton}, title = {Discretization of Interior Point Methods for State Constrained Elliptic Optimal Control Problems: optimal error estimates and parameter adjustment}, volume = {48}, journal = {Comp. Opt. and Appl.}, number = {3}, publisher = {Springer}, doi = {10.1007/s10589-009-9278-x}, pages = {581 -- 600}, year = {2011}, language = {en} } @article{WeiserGaenzlerSchiela2008, author = {Weiser, Martin and G{\"a}nzler, Tobias and Schiela, Anton}, title = {A control reduced primal interior point method for a class of control constrained optimal control problems}, volume = {41}, journal = {Comput. Optim. Appl.}, number = {1}, pages = {127 -- 145}, year = {2008}, language = {en} } @article{WeiserSchiela2004, author = {Weiser, Martin and Schiela, Anton}, title = {Function space interior point methods for PDE constrained optimization}, volume = {4}, journal = {PAMM}, number = {1}, pages = {43 -- 46}, year = {2004}, language = {en} } @article{WeiserSchielaDeuflhard2005, author = {Weiser, Martin and Schiela, Anton and Deuflhard, Peter}, title = {Asymptotic Mesh Independence of Newton's Method Revisited}, volume = {42}, journal = {SIAM J. Num. Anal.}, number = {5}, pages = {1830 -- 1845}, year = {2005}, language = {en} } @article{PruefertSchiela2009, author = {Pr{\"u}fert, Uwe and Schiela, Anton}, title = {The Minimization of a maximum-norm functional subject to an elliptic PDE and state constraints}, volume = {89}, journal = {Z. angew. Math. Mech.}, number = {7}, pages = {536 -- 551}, year = {2009}, language = {en} } @article{Schiela2008, author = {Schiela, Anton}, title = {A Simplified Approach to Semismooth Newton Methods in Function Space}, volume = {19}, journal = {SIAM J. on Optimization}, number = {3}, pages = {369 -- 393}, year = {2008}, language = {en} } @phdthesis{Schiela2006, author = {Schiela, Anton}, title = {The Control Reduced Interior Point Method. A Function Space Oriented Algorithmic Approach}, address = {Fachbereich Mathematik}, year = {2006}, language = {en} } @inproceedings{Schiela2009, author = {Schiela, Anton}, title = {An Extended Mathematical Framework for Barrier Methods in Function Space}, volume = {70}, booktitle = {Domain Decomposition Methods in Science and Engineering XVIII}, year = {2009}, language = {en} } @article{Schiela2009, author = {Schiela, Anton}, title = {Barrier Methods for Optimal Control Problems with State Constraints}, volume = {20}, journal = {SIAM J. on Optimization}, number = {2}, pages = {1002 -- 1031}, year = {2009}, language = {en} } @article{Schiela2009, author = {Schiela, Anton}, title = {State constrained optimal control problems with states of low regularity}, volume = {48}, journal = {SIAM J. on Control and Optimization}, number = {4}, pages = {2407 -- 2432}, year = {2009}, language = {en} } @misc{GoetschelSchielaWeiser2019, author = {G{\"o}tschel, Sebastian and Schiela, Anton and Weiser, Martin}, title = {Kaskade 7 -- a Flexible Finite Element Toolbox}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-74616}, year = {2019}, abstract = {Kaskade 7 is a finite element toolbox for the solution of stationary or transient systems of partial differential equations, aimed at supporting application-oriented research in numerical analysis and scientific computing. The library is written in C++ and is based on the Dune interface. The code is independent of spatial dimension and works with different grid managers. An important feature is the mix-and-match approach to discretizing systems of PDEs with different ansatz and test spaces for all variables. We describe the mathematical concepts behind the library as well as its structure, illustrating its use at several examples on the way.}, language = {en} } @misc{Schiela2008, author = {Schiela, Anton}, title = {State constrained optimal control problems with states of low regularity}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-509}, number = {08-24}, year = {2008}, abstract = {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.}, language = {en} }