@phdthesis{10594, abstract = {{Multiobjective optimization plays an increasingly important role in modern applications, where several criteria are often of equal importance. The task in multiobjective optimization and multiobjective optimal control is therefore to compute the set of optimal compromises (the Pareto set) between the conflicting objectives. Since – in contrast to the solution of a single objective optimization problem – the Pareto set generally consists of an infinite number of solutions, the computational effort can quickly become challenging. This is even more the case when many problems have to be solved, when the number of objectives is high, or when the objectives are costly to evaluate. Consequently, this thesis is devoted to the identification and exploitation of structure both in the Pareto set and the dynamics of the underlying model as well as to the development of efficient algorithms for solving problems with additional parameters, with a high number of objectives or with PDE-constraints. These three challenges are addressed in three respective parts. In the first part, predictor-corrector methods are extended to entire Pareto sets. When certain smoothness assumptions are satisfied, then the set of parameter dependent Pareto sets possesses additional structure, i.e. it is a manifold. The tangent space can be approximated numerically which yields a direction for the predictor step. In the corrector step, the predicted set converges to the Pareto set at a new parameter value. The resulting algorithm is applied to an example from autonomous driving. In the second part, the hierarchical structure of Pareto sets is investigated. When considering a subset of the objectives, the resulting solution is a subset of the Pareto set of the original problem. Under additional smoothness assumptions, the respective subsets are located on the boundary of the Pareto set of the full problem. This way, the “skeleton” of a Pareto set can be computed and due to the exponential increase in computing time with the number of objectives, the computations of these subsets are significantly faster which is demonstrated using an example from industrial laundries. In the third part, PDE-constrained multiobjective optimal control problems are addressed by reduced order modeling methods. Reduced order models exploit the structure in the system dynamics, for example by describing the dynamics of only the most energetic modes. The model reduction introduces an error in both the function values and their gradients, which has to be taken into account in the development of algorithms. Both scalarization and set-oriented approaches are coupled with reduced order modeling. Convergence results are presented and the numerical benefit is investigated. The algorithms are applied to semi-linear heat flow problems as well as to the Navier-Stokes equations. }}, author = {{Peitz, Sebastian}}, title = {{{ Exploiting structure in multiobjective optimization and optimal control}}}, doi = {{10.17619/UNIPB/1-176}}, year = {{2017}}, } @article{8756, abstract = {{We present a new algorithm for model predictive control of non-linear systems with respect to multiple, conflicting objectives. The idea is to provide a possibility to change the objective in real-time, e.g. as a reaction to changes in the environment or the system state itself. The algorithm utilises elements from various well-established concepts, namely multiobjective optimal control, economic as well as explicit model predictive control and motion planning with motion primitives. In order to realise real-time applicability, we split the computation into an online and an offline phase and we utilise symmetries in the open-loop optimal control problem to reduce the number of multiobjective optimal control problems that need to be solved in the offline phase. The results are illustrated using the example of an electric vehicle where the longitudinal dynamics are controlled with respect to the concurrent objectives arrival time and energy consumption.}}, author = {{Peitz, Sebastian and Schäfer, Kai and Ober-Blöbaum, Sina and Eckstein, Julian and Köhler, Ulrich and Dellnitz, Michael}}, issn = {{2405-8963}}, journal = {{Proceedings of the 20th World Congress of the International Federation of Automatic Control (IFAC)}}, number = {{1}}, pages = {{8674--8679}}, title = {{{A multiobjective MPC approach for autonomously driven electric vehicles}}}, doi = {{10.1016/j.ifacol.2017.08.1526}}, volume = {{50}}, year = {{2017}}, } @inproceedings{29430, author = {{Gail, T. and Ober-Blöbaum, Sina and Leyendecker, S. }}, booktitle = {{ECCOMAS Thematic Conference on Multibody Dynamics}}, title = {{{Variational multirate integration in discrete mechanics and optimal control}}}, year = {{2017}}, } @article{45941, author = {{Kovács, Balázs and Li, Buyang and Lubich, Christian and Power Guerra, Christian A.}}, issn = {{0029-599X}}, journal = {{Numerische Mathematik}}, keywords = {{Applied Mathematics, Computational Mathematics}}, number = {{3}}, pages = {{643--689}}, publisher = {{Springer Science and Business Media LLC}}, title = {{{Convergence of finite elements on an evolving surface driven by diffusion on the surface}}}, doi = {{10.1007/s00211-017-0888-4}}, volume = {{137}}, year = {{2017}}, } @article{45942, author = {{Kovács, Balázs and Lubich, Christian}}, issn = {{0029-599X}}, journal = {{Numerische Mathematik}}, keywords = {{Applied Mathematics, Computational Mathematics}}, number = {{2}}, pages = {{365--388}}, publisher = {{Springer Science and Business Media LLC}}, title = {{{Stability and convergence of time discretizations of quasi-linear evolution equations of Kato type}}}, doi = {{10.1007/s00211-017-0909-3}}, volume = {{138}}, year = {{2017}}, } @article{45940, author = {{Kovács, Balázs and Lubich, Christian}}, issn = {{0029-599X}}, journal = {{Numerische Mathematik}}, keywords = {{Applied Mathematics, Computational Mathematics}}, number = {{1}}, pages = {{91--117}}, publisher = {{Springer Science and Business Media LLC}}, title = {{{Stable and convergent fully discrete interior–exterior coupling of Maxwell’s equations}}}, doi = {{10.1007/s00211-017-0868-8}}, volume = {{137}}, year = {{2017}}, } @article{45946, author = {{Kovács, Balázs and Power Guerra, Christian Andreas}}, issn = {{0749-159X}}, journal = {{Numerical Methods for Partial Differential Equations}}, keywords = {{Applied Mathematics, Computational Mathematics, Numerical Analysis, Analysis}}, number = {{2}}, pages = {{518--554}}, publisher = {{Wiley}}, title = {{{Maximum norm stability and error estimates for the evolving surface finite element method}}}, doi = {{10.1002/num.22212}}, volume = {{34}}, year = {{2017}}, } @article{45943, author = {{Kovács, Balázs}}, issn = {{0272-4979}}, journal = {{IMA Journal of Numerical Analysis}}, keywords = {{Applied Mathematics, Computational Mathematics, General Mathematics}}, number = {{1}}, pages = {{430--459}}, publisher = {{Oxford University Press (OUP)}}, title = {{{High-order evolving surface finite element method for parabolic problems on evolving surfaces}}}, doi = {{10.1093/imanum/drx013}}, volume = {{38}}, year = {{2017}}, } @article{45945, author = {{Kovács, Balázs and Power Guerra, Christian Andreas}}, issn = {{0749-159X}}, journal = {{Numerical Methods for Partial Differential Equations}}, keywords = {{Applied Mathematics, Computational Mathematics, Numerical Analysis, Analysis}}, number = {{2}}, pages = {{518--554}}, publisher = {{Wiley}}, title = {{{Maximum norm stability and error estimates for the evolving surface finite element method}}}, doi = {{10.1002/num.22212}}, volume = {{34}}, year = {{2017}}, } @techreport{55293, author = {{Barth, D. and Beck, M. and Dose, T. and Glaßer, Ch. and Michler, L. and Technau, Marc}}, title = {{{Emptiness problems for integer circuits}}}, year = {{2017}}, } @inproceedings{55292, author = {{Barth, D. and Beck, M. and Dose, T. and Glaßer, Ch. and Michler, L. and Technau, Marc}}, booktitle = {{42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)}}, editor = {{Larsen, Kim G. and Bodlaender, Hans L. and Raskin, Jean-Francois}}, pages = {{33:1–33:14}}, publisher = {{Schloss Dagstuhl–Leibniz-Zentrum für Informatik}}, title = {{{Emptiness problems for integer circuits}}}, doi = {{10.4230/LIPIcs.MFCS.2017.33}}, volume = {{83}}, year = {{2017}}, } @article{55275, author = {{Technau, Marc and Technau, N.}}, journal = {{Comput. Methods Funct. Theory}}, number = {{2}}, pages = {{255–272}}, title = {{{A Loewner equation for infinitely many slits}}}, doi = {{10.1007/s40315-016-0179-6}}, volume = {{17}}, year = {{2017}}, } @article{42791, abstract = {{We describe a practical algorithm to solve the constructive membership problem for discrete two-generator subgroups of SL₂(R) or PSL₂(R). This algorithm has been implemented in Magma for groups defined over real algebraic number fields.}}, author = {{Kirschmer, Markus and Rüther, Marion G.}}, issn = {{0021-8693}}, journal = {{Journal of Algebra}}, keywords = {{Algebra and Number Theory}}, pages = {{519--548}}, publisher = {{Elsevier BV}}, title = {{{The constructive membership problem for discrete two-generator subgroups of SL(2,R)}}}, doi = {{10.1016/j.jalgebra.2017.02.029}}, volume = {{480}}, year = {{2017}}, } @phdthesis{33, abstract = {{Lightweight materials play an ever growing role in today's world. Saving on the mass of a machine will usually translate into a lower energy consumption. However, lightweight applications are prone to develop performance problems due to vibration induced by the operation of the machine. The Fraunhofer Institute for Manufacturing Technology and Advanced Materials in Dresden conducts research into the damping properties of composite materials. They are experimenting with hollow, particle filled spheres embedded in the lightweight material. Such a system is the technical motivation of this thesis. Ultimately, a numerical experiment to derive the coefficient of restitution is required. The simulation developed in this thesis is based on a discrete element method to track the individual particle and sphere trajectories. Based on a potential based approach for the particle interactions deployed in molecular dynamics, the behavior of the particles can be controlled effectively. The simulated volume is using reflecting boundaries and encloses the hollow sphere. In this work, a highly flexible memory structure was used with a linked cell approach to cope with the highly flexible mass of particles. This allows for a linear complexity of the method in regard to the particle number by reducing the computational overhead of the interaction computation. Multiple numerical experiments show the great effect the particles have on the damping behavior of the system.}}, author = {{Steinle, Tobias}}, title = {{{Modeling and simulation of metallic, particle-damped spheres for lightweight materials}}}, year = {{2016}}, } @inproceedings{34, author = {{Dellnitz, Michael and Eckstein, Julian and Flaßkamp, Kathrin and Friedel, Patrick and Horenkamp, Christian and Köhler, Ulrich and Ober-Blöbaum, Sina and Peitz, Sebastian and Tiemeyer, Sebastian}}, booktitle = {{Progress in Industrial Mathematics at ECMI}}, issn = {{2212-0173}}, pages = {{633--641}}, publisher = {{Springer International Publishing}}, title = {{{Multiobjective Optimal Control Methods for the Development of an Intelligent Cruise Control}}}, doi = {{10.1007/978-3-319-23413-7_87}}, volume = {{22}}, year = {{2016}}, } @article{21937, author = {{Nüske, Feliks and Schneider, Reinhold and Vitalini, Francesca and Noé, Frank}}, issn = {{0021-9606}}, journal = {{The Journal of Chemical Physics}}, title = {{{Variational tensor approach for approximating the rare-event kinetics of macromolecular systems}}}, doi = {{10.1063/1.4940774}}, year = {{2016}}, } @inproceedings{8759, abstract = {{In a wide range of applications, it is desirable to optimally control a system with respect to concurrent, potentially competing goals. This gives rise to a multiobjective optimal control problem where, instead of computing a single optimal solution, the set of optimal compromises, the so-called Pareto set, has to be approximated. When it is not possible to compute the entire control trajectory in advance, for instance due to uncertainties or unforeseeable events, model predictive control methods can be applied to control the system during operation in real time. In this article, we present an algorithm for the solution of multiobjective model predictive control problems. In an offline scenario, it can be used to compute the entire set of optimal compromises whereas in a real time scenario, one optimal compromise is computed according to an operator's preference. The results are illustrated using the example of an industrial laundry. A logistics model of the laundry is developed and then utilized in the optimization routine. Results are presented for an offline as well as an online scenario.}}, author = {{Peitz, Sebastian and Gräler, Manuel and Henke, Christian and Molo, Mirko Hessel-von and Dellnitz, Michael and Trächtler, Ansgar}}, booktitle = {{Procedia Technology}}, issn = {{2212-0173}}, pages = {{483--490}}, title = {{{Multiobjective Model Predictive Control of an Industrial Laundry}}}, doi = {{10.1016/j.protcy.2016.08.061}}, year = {{2016}}, } @misc{6569, author = {{Jurgelucks, Benjamin and Claes, Leander}}, title = {{{Increasing the Sensitivity of Impedance with Respect to Material Parameters of Triple-Ring Electrode Piezoelectric Transducers using Algorithmic Differentiation}}}, year = {{2016}}, } @inproceedings{6570, author = {{Jurgelucks, Benjamin and Claes, Leander}}, booktitle = {{AD2016 The 7th International Conference on Algorithmic Differentiation}}, location = {{Christ Church, Oxford, United Kingdom}}, pages = {{99--102}}, title = {{{Optimisation of triple-ring-electrodes on piezoceramic transducers using algorithmic differentiation}}}, year = {{2016}}, } @inbook{16579, author = {{Dellnitz, Michael and Eckstein, Julian and Flaßkamp, Kathrin and Friedel, Patrick and Horenkamp, Christian and Köhler, Ulrich and Ober-Blöbaum, Sina and Peitz, Sebastian and Tiemeyer, Sebastian}}, booktitle = {{Mathematics in Industry}}, isbn = {{9783319234120}}, issn = {{1612-3956}}, title = {{{Multiobjective Optimal Control Methods for the Development of an Intelligent Cruise Control}}}, doi = {{10.1007/978-3-319-23413-7_87}}, year = {{2016}}, }