@inbook{16963,
author = {Kuklinski, Christiane and Leis, Elena and Liebendörfer, Michael and Hochmuth, Reinhard},
booktitle = {Beiträge zum {Mathematikunterricht} 2019 53. {Jahrestagung} der {Gesellschaft} für {Didaktik} der {Mathematik}.},
editor = {Frank, Andreas and Krauss, Stefan and Binder, Karin},
publisher = {WTM-Verlag},
title = {{Erklärung von Mathematikleistung im Ingenieursstudium}},
year = {2020},
}
@inbook{17411,
abstract = {Many dynamical systems possess symmetries, e.g. rotational and translational invariances of mechanical systems. These can be beneficially exploited in the design of numerical optimal control methods. We present a model predictive control scheme which is based on a library of precomputed motion primitives. The primitives are equivalence classes w.r.t. the symmetry of the optimal control problems. Trim primitives as relative equilibria w.r.t. this symmetry, play a crucial role in the algorithm. The approach is illustrated using an academic mobile robot example.},
author = {Flaßkamp, Kathrin and Ober-Blöbaum, Sina and Peitz, Sebastian},
booktitle = {Advances in Dynamics, Optimization and Computation},
editor = {Junge, Oliver and Schütze, Oliver and Froyland, Gary and Ober-Blöbaum, Sina and Padberg-Gehle, Kathrin},
isbn = {9783030512637},
issn = {2198-4182},
publisher = {Springer},
title = {{Symmetry in Optimal Control: A Multiobjective Model Predictive Control Approach}},
doi = {10.1007/978-3-030-51264-4_9},
year = {2020},
}
@article{16297,
abstract = {In real-world problems, uncertainties (e.g., errors in the measurement,
precision errors) often lead to poor performance of numerical algorithms when
not explicitly taken into account. This is also the case for control problems,
where optimal solutions can degrade in quality or even become infeasible. Thus,
there is the need to design methods that can handle uncertainty. In this work,
we consider nonlinear multi-objective optimal control problems with uncertainty
on the initial conditions, and in particular their incorporation into a
feedback loop via model predictive control (MPC). In multi-objective optimal
control, an optimal compromise between multiple conflicting criteria has to be
found. For such problems, not much has been reported in terms of uncertainties.
To address this problem class, we design an offline/online framework to compute
an approximation of efficient control strategies. This approach is closely
related to explicit MPC for nonlinear systems, where the potentially expensive
optimization problem is solved in an offline phase in order to enable fast
solutions in the online phase. In order to reduce the numerical cost of the
offline phase, we exploit symmetries in the control problems. Furthermore, in
order to ensure optimality of the solutions, we include an additional online
optimization step, which is considerably cheaper than the original
multi-objective optimization problem. We test our framework on a car
maneuvering problem where safety and speed are the objectives. The
multi-objective framework allows for online adaptations of the desired
objective. Alternatively, an automatic scalarizing procedure yields very
efficient feedback controls. Our results show that the method is capable of
designing driving strategies that deal better with uncertainties in the initial
conditions, which translates into potentially safer and faster driving
strategies.},
author = {Hernández Castellanos, Carlos Ignacio and Ober-Blöbaum, Sina and Peitz, Sebastian},
journal = {International Journal of Robust and Nonlinear Control},
number = {17},
pages = {7593--7618},
title = {{Explicit Multi-objective Model Predictive Control for Nonlinear Systems Under Uncertainty}},
doi = {10.1002/rnc.5197},
volume = {30},
year = {2020},
}
@unpublished{20731,
abstract = {We present a novel algorithm that allows us to gain detailed insight into the effects of sparsity in linear and nonlinear optimization, which is of great importance in many scientific areas such as image and signal processing, medical imaging, compressed sensing, and machine learning (e.g., for the training of neural networks). Sparsity is an important feature to ensure robustness against noisy data, but also to find models that are interpretable and easy to analyze due to the small number of relevant terms. It is common practice to enforce sparsity by adding the ℓ1-norm as a weighted penalty term. In order to gain a better understanding and to allow for an informed model selection, we directly solve the corresponding multiobjective optimization problem (MOP) that arises when we minimize the main objective and the ℓ1-norm simultaneously. As this MOP is in general non-convex for nonlinear objectives, the weighting method will fail to provide all optimal compromises. To avoid this issue, we present a continuation method which is specifically tailored to MOPs with two objective functions one of which is the ℓ1-norm. Our method can be seen as a generalization of well-known homotopy methods for linear regression problems to the nonlinear case. Several numerical examples - including neural network training - demonstrate our theoretical findings and the additional insight that can be gained by this multiobjective approach.},
author = {Bieker, Katharina and Gebken, Bennet and Peitz, Sebastian},
booktitle = {arXiv:2012.07483},
title = {{On the Treatment of Optimization Problems with L1 Penalty Terms via Multiobjective Continuation}},
year = {2020},
}
@article{16964,
author = {Hochmuth, Reinhard and Liebendörfer, Michael and Biehler, Rolf and Eichler, Andreas},
journal = {Neues Handbuch Hochschullehre},
pages = {117--138},
title = {{Das Kompetenzzentrum Hochschuldidaktik Mathematik (khdm)}},
volume = {95},
year = {2020},
}
@article{16965,
author = {Schürmann, Mirko and Schaper, Niclas and Liebendörfer, Michael and Biehler, Rolf and Lankeit, Elisa and Hochmuth, Reinhard and Ruge, Johanna and Kuklinski, Christiane},
journal = {dghd-Newsletter},
pages = {25--29},
title = {{Ein Kurzbericht aus dem Forschungsprojekt WiGeMath-Lernzentren als Unterstützungsmaßnahme für mathematikbezogenes Lernen in der Studieneingangsphase}},
volume = {01/2020},
year = {2020},
}
@inbook{17994,
abstract = {In this work we review the novel framework for the computation of finite dimensional invariant sets of infinite dimensional dynamical systems developed in [6] and [36]. By utilizing results on embedding techniques for infinite dimensional systems we extend a classical subdivision scheme [8] as well as a continuation algorithm [7] for the computation of attractors and invariant manifolds of finite dimensional systems to the infinite dimensional case. We show how to implement this approach for the analysis of delay differential equations and partial differential equations and illustrate the feasibility of our implementation by computing the attractor of the Mackey-Glass equation and the unstable manifold of the one-dimensional Kuramoto-Sivashinsky equation.},
author = {Gerlach, Raphael and Ziessler, Adrian},
booktitle = {Advances in Dynamics, Optimization and Computation},
editor = {Junge, Oliver and Schütze, Oliver and Ober-Blöbaum, Sina and Padberg-Gehle, Kathrin},
isbn = {9783030512637},
issn = {2198-4182},
pages = {55--85},
publisher = {Springer},
title = {{The Approximation of Invariant Sets in Infinite Dimensional Dynamical Systems}},
doi = {10.1007/978-3-030-51264-4_3},
volume = {304},
year = {2020},
}
@article{19938,
abstract = {We show that symplectic integrators preserve bifurcations of Hamiltonian boundary value problems and that nonsymplectic integrators do not. We provide a universal description of the breaking of umbilic bifurcations by nonysmplectic integrators. We discover extra structure induced from certain types of boundary value problems, including classical Dirichlet problems, that is useful to locate bifurcations. Geodesics connecting two points are an example of a Hamiltonian boundary value problem, and we introduce the jet-RATTLE method, a symplectic integrator that easily computes geodesics and their bifurcations. Finally, we study the periodic pitchfork bifurcation, a codimension-1 bifurcation arising in integrable Hamiltonian systems. It is not preserved by either symplectic on nonsymplectic integrators, but in some circumstances symplecticity greatly reduces the error. },
author = {McLachlan, Robert I and Offen, Christian},
journal = {Foundations of Computational Mathematics},
number = {6},
pages = {1363--1400},
title = {{Preservation of Bifurcations of Hamiltonian Boundary Value Problems Under Discretisation}},
doi = {10.1007/s10208-020-09454-z},
volume = {20},
year = {2020},
}
@article{10595,
abstract = {In this article we show that the boundary of the Pareto critical set of an unconstrained multiobjective optimization problem (MOP) consists of Pareto critical points of subproblems where only a subset of the set of objective functions is taken into account. If the Pareto critical set is completely described by its boundary (e.g., if we have more objective functions than dimensions in decision space), then this can be used to efficiently solve the MOP by solving a number of MOPs with fewer objective functions. If this is not the case, the results can still give insight into the structure of the Pareto critical set.},
author = {Gebken, Bennet and Peitz, Sebastian and Dellnitz, Michael},
issn = {0925-5001},
journal = {Journal of Global Optimization},
number = {4},
pages = {891--913},
title = {{On the hierarchical structure of Pareto critical sets}},
doi = {10.1007/s10898-019-00737-6},
volume = {73},
year = {2019},
}
@inproceedings{13107,
abstract = {In this paper, we first outline a Hypothetical Learning Trajectory (HLT), which aims at a formal understanding of the rules for manipulating integers. The HLT is based on task formats, which promote algebraic thinking in terms of generalizing rules from the analysis of patterns and should be familiar to students from their mathematics education experiences in elementary school. Second, we analyze two students' actual learning process based on Peircean semiotics. The analysis shows that the actual learning process diverges from the hypothesized learning process in that the students do not relate the different levels of the diagrams in a way that allows them to extrapolate the rule for the subtraction of negative numbers. Based on this finding, we point out consequences for the design of the tasks.},
author = {Schumacher, Jan and Rezat, Sebastian},
booktitle = {Proceedings of the Eleventh Congress of the European Society for Research in Mathematics Education (CERME11, February 6 – 10, 2019)},
editor = {Jankvist, Uffe Thomas and Van den Heuvel-Panhuizen, Marja and Veldhuis, Michiel},
keyword = {diagrammatic reasoning, hypothetical learning trajectory, induction extrapolatory method, integers, negative numbers, permanence principle, semiotics},
location = {Utrecht},
publisher = {Freudenthal Group & Freudenthal Institute, Utrecht University and ERME},
title = {{A Hypothetical Learning Trajectory for the Learning of the Rules for Manipulating Integers}},
year = {2019},
}
@inbook{16966,
author = {Kuklinski, Christiane and Liebendörfer, Michael and Hochmuth, Reinhard and Biehler, Rolf and Schaper, Niclas and Lankeit, Elisa and Leis, Elena and Schürmann, Mirko},
booktitle = {Proceedings of {CERME} 11},
title = {{Features of innovative lectures that distinguish them from traditional lectures and their evaluation by attending students}},
year = {2019},
}
@book{13139,
editor = {Rezat, Sebastian and Fan, Lianghuo and Hattermann, Mathias and Schumacher, Jan and Wuschke, Holger},
location = {Paderborn},
pages = {392},
publisher = {Universitätsbibliothek Paderborn},
title = {{Proceedings of the Third International Conference on Mathematics Textbook Research and Development: 16-19 September 2019 Paderborn, Germany}},
doi = {10.17619/UNIPB/1-768},
year = {2019},
}
@unpublished{16296,
abstract = {Multiobjective optimization plays an increasingly important role in modern
applications, where several objectives 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 the Pareto set generally consists of an infinite
number of solutions, the computational effort can quickly become challenging
which is particularly problematic when the objectives are costly to evaluate as
is the case for models governed by partial differential equations (PDEs). To
decrease the numerical effort to an affordable amount, surrogate models can be
used to replace the expensive PDE evaluations. Existing multiobjective
optimization methods using model reduction are limited either to low parameter
dimensions or to few (ideally two) objectives. In this article, we present a
combination of the reduced basis model reduction method with a continuation
approach using inexact gradients. The resulting approach can handle an
arbitrary number of objectives while yielding a significant reduction in
computing time.},
author = {Banholzer, Stefan and Gebken, Bennet and Dellnitz, Michael and Peitz, Sebastian and Volkwein, Stefan},
booktitle = {arXiv:1906.09075},
title = {{ROM-based multiobjective optimization of elliptic PDEs via numerical continuation}},
year = {2019},
}
@article{16708,
abstract = { In this work we extend the novel framework developed by Dellnitz, Hessel-von Molo, and Ziessler to
the computation of finite dimensional unstable manifolds of infinite dimensional dynamical systems.
To this end, we adapt a set-oriented continuation technique developed by Dellnitz and Hohmann for
the computation of such objects of finite dimensional systems with the results obtained in the work
of Dellnitz, Hessel-von Molo, and Ziessler. We show how to implement this approach for the analysis
of partial differential equations and illustrate its feasibility by computing unstable manifolds of the
one-dimensional Kuramoto--Sivashinsky equation as well as for the Mackey--Glass delay differential
equation.
},
author = {Ziessler, Adrian and Dellnitz, Michael and Gerlach, Raphael},
issn = {1536-0040},
journal = {SIAM Journal on Applied Dynamical Systems},
pages = {1265--1292},
title = {{The Numerical Computation of Unstable Manifolds for Infinite Dimensional Dynamical Systems by Embedding Techniques}},
doi = {10.1137/18m1204395},
year = {2019},
}
@inproceedings{10597,
abstract = {In comparison to classical control approaches in the field of electrical drives like the field-oriented control (FOC), model predictive control (MPC) approaches are able to provide a higher control performance. This refers to shorter settling times, lower overshoots, and a better decoupling of control variables in case of multi-variable controls. However, this can only be achieved if the used prediction model covers the actual behavior of the plant sufficiently well. In case of model deviations, the performance utilizing MPC remains below its potential. This results in effects like increased current ripple or steady state setpoint deviations. In order to achieve a high control performance, it is therefore necessary to adapt the model to the real plant behavior. When using an online system identification, a less accurate model is sufficient for commissioning of the drive system. In this paper, the combination of a finite-control-set MPC (FCS-MPC) with a system identification is proposed. The method does not require high-frequency signal injection, but uses the measured values already required for the FCS-MPC. An evaluation of the least squares-based identification on a laboratory test bench showed that the model accuracy and thus the control performance could be improved by an online update of the prediction models.},
author = {Hanke, Soren and Peitz, Sebastian and Wallscheid, Oliver and Böcker, Joachim and Dellnitz, Michael},
booktitle = {2019 IEEE International Symposium on Predictive Control of Electrical Drives and Power Electronics (PRECEDE)},
isbn = {9781538694145},
title = {{Finite-Control-Set Model Predictive Control for a Permanent Magnet Synchronous Motor Application with Online Least Squares System Identification}},
doi = {10.1109/precede.2019.8753313},
year = {2019},
}
@article{16709,
author = {Sahai, Tuhin and Ziessler, Adrian and Klus, Stefan and Dellnitz, Michael},
issn = {0924-090X},
journal = {Nonlinear Dynamics},
title = {{Continuous relaxations for the traveling salesman problem}},
doi = {10.1007/s11071-019-05092-5},
year = {2019},
}
@unpublished{16711,
abstract = {Embedding techniques allow the approximations of finite dimensional
attractors and manifolds of infinite dimensional dynamical systems via
subdivision and continuation methods. These approximations give a topological
one-to-one image of the original set. In order to additionally reveal their
geometry we use diffusion mapst o find intrinsic coordinates. We illustrate our
results on the unstable manifold of the one-dimensional Kuramoto--Sivashinsky
equation, as well as for the attractor of the Mackey-Glass delay differential
equation.},
author = {Gerlach, Raphael and Koltai, Péter and Dellnitz, Michael},
booktitle = {arXiv:1902.08824},
title = {{Revealing the intrinsic geometry of finite dimensional invariant sets of infinite dimensional dynamical systems}},
year = {2019},
}
@article{8753,
abstract = {In a wide range of applications it is desirable to optimally control a dynamical 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 the problem under consideration is described by a partial differential equation (PDE), as is the case for fluid flow, the computational cost rapidly increases and makes its direct treatment infeasible. Reduced order modeling is a very popular method to reduce the computational cost, in particular in a multi query context such as uncertainty quantification, parameter estimation or optimization. In this article, we show how to combine reduced order modeling and multiobjective optimal control techniques in order to efficiently solve multiobjective optimal control problems constrained by PDEs. We consider a global, derivative free optimization method as well as a local, gradient-based approach for which the optimality system is derived in two different ways. The methods are compared with regard to the solution quality as well as the computational effort and they are illustrated using the example of the flow around a cylinder and a backward-facing-step channel flow.},
author = {Peitz, Sebastian and Ober-Blöbaum, Sina and Dellnitz, Michael},
issn = {0167-8019},
journal = {Acta Applicandae Mathematicae},
number = {1},
pages = {171–199},
title = {{Multiobjective Optimal Control Methods for the Navier-Stokes Equations Using Reduced Order Modeling}},
doi = {10.1007/s10440-018-0209-7},
volume = {161},
year = {2019},
}
@article{10593,
abstract = {We present a new framework for optimal and feedback control of PDEs using Koopman operator-based reduced order models (K-ROMs). The Koopman operator is a linear but infinite-dimensional operator which describes the dynamics of observables. A numerical approximation of the Koopman operator therefore yields a linear system for the observation of an autonomous dynamical system. In our approach, by introducing a finite number of constant controls, the dynamic control system is transformed into a set of autonomous systems and the corresponding optimal control problem into a switching time optimization problem. This allows us to replace each of these systems by a K-ROM which can be solved orders of magnitude faster. By this approach, a nonlinear infinite-dimensional control problem is transformed into a low-dimensional linear problem. Using a recent convergence result for the numerical approximation via Extended Dynamic Mode Decomposition (EDMD), we show that the value of the K-ROM based objective function converges in measure to the value of the full objective function. To illustrate the results, we consider the 1D Burgers equation and the 2D Navier–Stokes equations. The numerical experiments show remarkable performance concerning both solution times and accuracy.},
author = {Peitz, Sebastian and Klus, Stefan},
issn = {0005-1098},
journal = {Automatica},
pages = {184--191},
title = {{Koopman operator-based model reduction for switched-system control of PDEs}},
doi = {10.1016/j.automatica.2019.05.016},
volume = {106},
year = {2019},
}
@article{21,
abstract = {We address the general mathematical problem of computing the inverse p-th
root of a given matrix in an efficient way. A new method to construct iteration
functions that allow calculating arbitrary p-th roots and their inverses of
symmetric positive definite matrices is presented. We show that the order of
convergence is at least quadratic and that adaptively adjusting a parameter q
always leads to an even faster convergence. In this way, a better performance
than with previously known iteration schemes is achieved. The efficiency of the
iterative functions is demonstrated for various matrices with different
densities, condition numbers and spectral radii.},
author = {Richters, Dorothee and Lass, Michael and Walther, Andrea and Plessl, Christian and Kühne, Thomas},
journal = {Communications in Computational Physics},
number = {2},
pages = {564--585},
publisher = {Global Science Press},
title = {{A General Algorithm to Calculate the Inverse Principal p-th Root of Symmetric Positive Definite Matrices}},
doi = {10.4208/cicp.OA-2018-0053},
volume = {25},
year = {2019},
}