@misc{SchielaWeiser, author = {Schiela, Anton and Weiser, Martin}, title = {Superlinear Convergence of the Control Reduced Interior Point Method for PDE Constrained Optimization}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-8490}, number = {05-15}, abstract = {A thorough convergence analysis of the Control Reduced Interior Point Method in function space is performed. This recently proposed method is a primal interior point pathfollowing scheme with the special feature, that the control variable is eliminated from the optimality system. Apart from global linear convergence we show, that this method converges locally almost quadratically, if the optimal solution satisfies a function space analogue to a non-degeneracy condition. In numerical experiments we observe, that a prototype implementation of our method behaves in compliance with our theoretical results.}, language = {en} } @misc{WeiserSchiela, author = {Weiser, Martin and Schiela, Anton}, title = {Function space interior point methods for PDE constrained optimization}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-8027}, number = {04-27}, abstract = {A primal-dual interior point method for optimal control problems with PDE constraints is considered. The algorithm is directly applied to the infinite dimensional problem. Existence and convergence of the central path are analyzed. Numerical results from an inexact continuation method applied to a model problem are shown.}, language = {en} } @misc{WeiserGaenzlerSchiela, 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}, 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{WeiserSchielaDeuflhard, author = {Weiser, Martin and Schiela, Anton and Deuflhard, Peter}, title = {Asymptotic Mesh Independence of Newton's Method Revisited}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-7352}, number = {03-13}, abstract = {The paper presents a new affine invariant theory on asymptotic mesh independence of Newton's method in nonlinear PDEs. Compared to earlier attempts, the new approach is both much simpler and more natural from the algorithmic point of view. The theory is exemplified at collocation methods for ODE boundary value problems and at finite element methods for elliptic PDE problems.}, language = {en} } @misc{KlapprothSchielaDeuflhard, 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}, 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{Schiela, author = {Schiela, Anton}, title = {An Extended Mathematical Framework for Barrier Methods in Function Space}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-10593}, number = {08-07}, abstract = {An extended mathematical framework for barrier methods for state constrained optimal control compared to [Schiela, ZIB-Report 07-07] is considered. This allows to apply the results derived there to more general classes of optimal control problems, in particular to boundary control and finite dimensional control.}, language = {en} } @misc{SchielaGuenther, 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}, 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{PruefertSchiela, 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}, 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{KlapprothDeuflhardSchiela, 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}, 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{SchielaWollner, 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}, 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{Schiela, author = {Schiela, Anton}, title = {A continuity result for Nemyckii Operators and some applications in PDE constrained optimal control}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-9354}, number = {06-41}, abstract = {This work explores two applications of a classical result on the continuity of Nemyckii operators to optimal control with PDEs. First, we present an alternative approach to the analysis of Newton's method for function space problems involving semi-smooth Nemyckii operators. A concise proof for superlinear convergence is presented, and sharpened bounds on the rate of convergence are derived. Second, we derive second order sufficient conditions for problems, where the underlying PDE has poor regularity properties. We point out that the analytical structure in both topics is essentially the same.}, language = {en} } @misc{Schiela, 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}, 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} } @misc{Schiela, author = {Schiela, Anton}, title = {Convergence of the Control Reduced Interior Point Method for PDE Constrained Optimal Control with State Constraints}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-9099}, number = {06-16}, abstract = {We propose a variant of the control reduced interior point method for the solution of state constrained problems. We show convergence of the corresponding interior point pathfollowing algorithm in function space. Morever, we provide error bounds for the iterates.}, language = {en} } @misc{Schiela, 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}, 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} } @misc{HinzeSchiela, 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}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-10414}, number = {07-40}, abstract = {An adjustment scheme for the relaxation parameter of interior point approaches to the numerical solution of pointwise state constrained elliptic optimal control problems is introduced. The method is based on error estimates of an associated finite element discretization of the relaxed problems and optimally selects the relaxation parameter in dependence on the mesh size of discretization. The finite element analysis for the relaxed problems is carried out and a numerical example is presented which confirms our analytical findings.}, language = {en} } @misc{Schiela, author = {Schiela, Anton}, title = {An Interior Point Method in Function Space for the Efficient Solution of State Constrained Optimal Control Problems}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-10471}, number = {07-44}, abstract = {We propose and analyse an interior point path-following method in function space for state constrained optimal control. Our emphasis is on proving convergence in function space and on constructing a practical path-following algorithm. In particular, the introduction of a pointwise damping step leads to a very efficient method, as verified by numerical experiments.}, language = {en} } @misc{Schiela, author = {Schiela, Anton}, title = {Optimality Conditions for Convex State Constrained Optimal Control Problems with Discontinuous States}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-10342}, number = {07-35}, abstract = {We discuss first order optimality conditions for state constrained optimal control problems. Our concern is the treatment of problems, where the solution of the state equation is not known to be continuous, as in the case of boundary control in three space dimensions or optimal control with parabolic partial differential equations. We show existence of measure valued Lagrangian multipliers, which have just enough additional regularity to be applicable to all possibly discontinuous solutions of the state equation.}, language = {en} } @misc{MeyerPanizziSchiela, 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}, 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} } @misc{Schiela, author = {Schiela, Anton}, title = {A cubic regularization algorithm for nonconvex optimization in function space (in preparation)}, issn = {1438-0064}, abstract = {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.}, language = {en} } @misc{LubkollSchielaWeiser, 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}, 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} }