TY - CONF
AB - 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.
AU - Schumacher, Jan
AU - Rezat, Sebastian
ED - Jankvist, Uffe Thomas
ED - Van den Heuvel-Panhuizen, Marja
ED - Veldhuis, Michiel
ID - 13107
KW - diagrammatic reasoning
KW - hypothetical learning trajectory
KW - induction extrapolatory method
KW - integers
KW - negative numbers
KW - permanence principle
KW - semiotics
T2 - Proceedings of the Eleventh Congress of the European Society for Research in Mathematics Education (CERME11, February 6 – 10, 2019)
TI - A Hypothetical Learning Trajectory for the Learning of the Rules for Manipulating Integers
ER -
TY - GEN
AB - It is a very challenging task to identify the objectives on which a certain
decision was based, in particular if several, potentially conflicting criteria
are equally important and a continuous set of optimal compromise decisions
exists. This task can be understood as the inverse problem of multiobjective
optimization, where the goal is to find the objective vector of a given Pareto
set. To this end, we present a method to construct the objective vector of a
multiobjective optimization problem (MOP) such that the Pareto critical set
contains a given set of data points or decision vectors. The key idea is to
consider the objective vector in the multiobjective KKT conditions as variable
and then search for the objectives that minimize the Euclidean norm of the
resulting system of equations. By expressing the objectives in a
finite-dimensional basis, we transform this problem into a homogeneous, linear
system of equations that can be solved efficiently. There are many important
potential applications of this approach. Besides the identification of
objectives (both from clean and noisy data), the method can be used for the
construction of surrogate models for expensive MOPs, which yields significant
speed-ups. Both applications are illustrated using several examples.
AU - Gebken, Bennet
AU - Peitz, Sebastian
ID - 16295
T2 - arXiv:1901.06141
TI - Inverse multiobjective optimization: Inferring decision criteria from data
ER -
TY - CHAP
AU - Kuklinski, Christiane
AU - Liebendörfer, Michael
AU - Hochmuth, Reinhard
AU - Biehler, Rolf
AU - Schaper, Niclas
AU - Lankeit, Elisa
AU - Leis, Elena
AU - Schürmann, Mirko
ID - 16966
T2 - Proceedings of {CERME} 11
TI - Features of innovative lectures that distinguish them from traditional lectures and their evaluation by attending students
ER -
TY - BOOK
ED - Rezat, Sebastian
ED - Fan, Lianghuo
ED - Hattermann, Mathias
ED - Schumacher, Jan
ED - Wuschke, Holger
ID - 13139
TI - Proceedings of the Third International Conference on Mathematics Textbook Research and Development: 16-19 September 2019 Paderborn, Germany
ER -
TY - GEN
AB - 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.
AU - Banholzer, Stefan
AU - Gebken, Bennet
AU - Dellnitz, Michael
AU - Peitz, Sebastian
AU - Volkwein, Stefan
ID - 16296
T2 - arXiv:1906.09075
TI - ROM-based multiobjective optimization of elliptic PDEs via numerical continuation
ER -
TY - JOUR
AB - 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.
AU - Ziessler, Adrian
AU - Dellnitz, Michael
AU - Gerlach, Raphael
ID - 16708
JF - SIAM Journal on Applied Dynamical Systems
SN - 1536-0040
TI - The Numerical Computation of Unstable Manifolds for Infinite Dimensional Dynamical Systems by Embedding Techniques
ER -
TY - CONF
AB - 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.
AU - Hanke, Soren
AU - Peitz, Sebastian
AU - Wallscheid, Oliver
AU - Böcker, Joachim
AU - Dellnitz, Michael
ID - 10597
SN - 9781538694145
T2 - 2019 IEEE International Symposium on Predictive Control of Electrical Drives and Power Electronics (PRECEDE)
TI - Finite-Control-Set Model Predictive Control for a Permanent Magnet Synchronous Motor Application with Online Least Squares System Identification
ER -
TY - JOUR
AU - Sahai, Tuhin
AU - Ziessler, Adrian
AU - Klus, Stefan
AU - Dellnitz, Michael
ID - 16709
JF - Nonlinear Dynamics
SN - 0924-090X
TI - Continuous relaxations for the traveling salesman problem
ER -
TY - GEN
AB - 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.
AU - Gerlach, Raphael
AU - Koltai, Péter
AU - Dellnitz, Michael
ID - 16711
T2 - arXiv:1902.08824
TI - Revealing the intrinsic geometry of finite dimensional invariant sets of infinite dimensional dynamical systems
ER -
TY - JOUR
AB - 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.
AU - Peitz, Sebastian
AU - Ober-Blöbaum, Sina
AU - Dellnitz, Michael
ID - 8753
IS - 1
JF - Acta Applicandae Mathematicae
SN - 0167-8019
TI - Multiobjective Optimal Control Methods for the Navier-Stokes Equations Using Reduced Order Modeling
VL - 161
ER -