@unpublished{16296,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},
}
@article{16712,
author = {Dellnitz, Michael and Gebken, Bennet and Gerlach, Raphael and Klus, Stefan},
issn = {1468-9367},
journal = {Dynamical Systems},
pages = {1--19},
title = {{On the equivariance properties of self-adjoint matrices}},
doi = {10.1080/14689367.2019.1661355},
year = {2019},
}
@inproceedings{8482,
author = {Jurgelucks, Benjamin and Schulze, Veronika and Feldmann, Nadine and Claes, Leander},
booktitle = {GAMM 2019},
title = {{Arbitrary sensitivity for inverse problems in piezoelectricity}},
year = {2019},
}
@inproceedings{13106,
author = {Schumacher, Jan},
booktitle = {Beiträge zum Mathematikunterricht 2019},
location = {Regensburg},
title = {{Rekonstruktion diagrammatischen Schließens am Beispiel der Subtraktion negativer Zahlen}},
year = {2019},
}
@inbook{8577,
author = {Kuklinski, Christiane and Leis, Elena and Liebendörfer, Michael and Hochmuth, Reinhard},
booktitle = {Beiträge zum Mathematikunterricht 2019},
title = {{Erklärung von Mathematikleistung im Ingenieursstudium}},
year = {2019},
}
@article{16714,
author = {Vieluf, Solveig and Mora, Karin and Gölz, Christian and Reuter, Eva-Maria and Godde, Ben and Dellnitz, Michael and Reinsberger, Claus and Voelcker-Rehage, Claudia},
issn = {0306-4522},
journal = {Neuroscience},
pages = {203--213},
title = {{Age- and Expertise-Related Differences of Sensorimotor Network Dynamics during Force Control}},
doi = {10.1016/j.neuroscience.2018.07.025},
year = {2018},
}
@inbook{8573,
author = {Liebendörfer, Michael},
booktitle = {Beiträge zum Mathematikunterricht 2018},
editor = {Didaktik der Mathematik der Universität Paderborn, Fachgruppe},
pages = {1171--1174},
publisher = {WTM-Verlag},
title = {{Psychologische Grundbedürfnisse im frühen Mathematikstudium}},
year = {2018},
}
@article{8751,
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. The advances in algorithms and the increasing interest in Pareto-optimal solutions have led to a wide range of new applications related to optimal and feedback control, which results in new challenges such as expensive models or real-time applicability. 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 or when a solution has to be presented very quickly. This article gives an overview of recent developments in accelerating multiobjective optimal control for complex problems where either PDE constraints are present or where a feedback behavior has to be achieved. In the first case, surrogate models yield significant speed-ups. Besides classical meta-modeling techniques for multiobjective optimization, a promising alternative for control problems is to introduce a surrogate model for the system dynamics. In the case of real-time requirements, various promising model predictive control approaches have been proposed, using either fast online solvers or offline-online decomposition. We also briefly comment on dimension reduction in many-objective optimization problems as another technique for reducing the numerical effort.},
author = {Peitz, Sebastian and Dellnitz, Michael},
issn = {2297-8747},
journal = {Mathematical and Computational Applications},
number = {2},
title = {{A Survey of Recent Trends in Multiobjective Optimal Control—Surrogate Models, Feedback Control and Objective Reduction}},
doi = {10.3390/mca23020030},
volume = {23},
year = {2018},
}
@inproceedings{7766,
author = {Schumacher, Jan},
booktitle = {Beiträge zum Mathematikunterricht 2018},
publisher = {WTM-Verlag},
title = {{Semiotische Analyse von Sinnkonstruktionsprozessen bei einem innermathematischen Zugang zum Erlernen negativer Zahlen}},
year = {2018},
}
@inbook{8574,
author = {Liebendörfer, Michael and Kuklinski, Christiane and Hochmuth, Reinhard},
booktitle = {Beiträge zum Mathematikunterricht 2018},
editor = {Didaktik der Mathematik der Universität Paderborn, Fachgruppe},
pages = {1175--1178},
publisher = {WTM-Verlag},
title = {{Auswirkungen von innovativen Vorlesungen für Lehramtsstudierende in der Studieneingangsphase}},
year = {2018},
}
@inproceedings{8757,
abstract = {A framework for set‐oriented multiobjective optimal control of partial differential equations using reduced order modeling has recently been developed [1]. Following concepts from localized reduced bases methods, error estimators for the reduced cost functionals are utilized to construct a library of locally valid reduced order models. This way, a superset of the Pareto set can efficiently be computed while maintaining a prescribed error bound. In this article, this algorithm is applied to a problem with non‐smooth objective functionals. Using an academic example, we show that the extension to non‐smooth problems can be realized in a straightforward manner. We then discuss the implications on the numerical results.},
author = {Beermann, Dennis and Dellnitz, Michael and Peitz, Sebastian and Volkwein, Stefan},
booktitle = {PAMM},
issn = {1617-7061},
pages = {51--54},
title = {{POD-based multiobjective optimal control of PDEs with non-smooth objectives}},
doi = {10.1002/pamm.201710015},
year = {2018},
}
@article{10142,
author = {Steffen, Eckhard},
journal = {Australasian Journal of Combinatorics},
number = {1},
pages = {153--160},
title = {{Approximating Vizing’s independence number conjecture}},
volume = {71},
year = {2018},
}
@techreport{16967,
author = {Hochmuth, Reinhard and Biehler, Rolf and Schaper, Niclas and Kuklinski, Christiane and Lankeit, Elisa and Leis, Elena and Liebendörfer, Michael and Schürmann, Mirko},
title = {{Wirkung und Gelingensbedingungen von Unterstützungsmaßnahmen für mathmatikbezogenes Lernen in der Studieneingangsphase : Schlussbericht: Teilprojekt A der Leibniz Universität Hannover, Teilprojekte B und C der Universität Paderborn: Berichtszeitraum: 01.03.2015-31.08.2018}},
doi = {10.2314/KXP:1689534117},
year = {2018},
}