TY - JOUR
AB - While 2D Gibbsian particle systems might exhibit orientational order resulting in a lattice-like structure, these particle systems do not exhibit positional order if the interaction between particles satisfies some weak assumptions. Here we investigate to which extent particles within a box of size may fluctuate from their ideal lattice position. We show that particles near the center of the box typically show a displacement at least of order . Thus we extend recent results on the hard disk model to particle systems with fairly arbitrary particle spins and interaction. Our result applies to models such as rather general continuum Potts type models, e.g. with Widom–Rowlinson or Lenard-Jones-type interaction.
AU - Richthammer, Thomas
AU - Fiedler, Michael
ID - 33481
JF - Stochastic Processes and their Applications
TI - A lower bound on the displacement of particles in 2D Gibbsian particle systems
VL - 132
ER -
TY - CONF
AU - Ridderbusch, Steffen
AU - Offen, Christian
AU - Ober-Blöbaum, Sina
AU - Goulart, Paul
ID - 21572
T2 - 2021 60th IEEE Conference on Decision and Control (CDC)
TI - Learning ODE Models with Qualitative Structure Using Gaussian Processes
ER -
TY - CONF
AU - Jiménez, F.
AU - Ober-Blöbaum, Sina
ID - 29868
T2 - Nichtlineare Sci
TI - Fractional Damping Through Restricted Calculus of Variations
VL - 31
ER -
TY - THES
AB - Ein zentraler Aspekt bei der Untersuchung dynamischer Systeme ist die Analyse ihrer invarianten Mengen wie des globalen Attraktors und (in)stabiler Mannigfaltigkeiten. Insbesondere wenn das zugrunde liegende System von einem Parameter abhängt, ist es entscheidend, sie im Bezug auf diesen Parameter effizient zu verfolgen. Für die Berechnung invarianter Mengen stützen wir uns für ihre Approximation auf numerische Algorithmen. Typischerweise können diese Methoden jedoch nur auf endlich-dimensionale dynamische Systeme angewendet werden. In dieser Arbeit präsentieren wir daher einen numerischen Rahmen für die globale dynamische Analyse unendlich-dimensionaler Systeme. Wir werden Einbettungstechniken verwenden, um das core dynamical system (CDS) zu definieren, welches ein dynamisch äquivalentes endlich-dimensionales System ist.Das CDS wird dann verwendet, um eingebettete invariante Mengen, also eins-zu-eins Bilder, mittels Mengen-orientierten numerischen Methoden zu approximieren. Bei der Konstruktion des CDS ist es entscheidend, eine geeignete Beobachtungsabbildung auszuwählen und die geeignete inverse Abbildung zu entwerfen. Dazu werden wir geeignete numerische Implementierungen des CDS für DDEs und PDEs vorstellen. Für eine nachfolgende geometrische Analyse der eingebetteten invarianten Menge betrachten wir eine Lerntechnik namens diffusion maps, die ihre intrinsische Geometrie enthüllt sowie ihre Dimension schätzt. Schließlich wenden wir unsere entwickelten numerischen Methoden an einigen bekannten unendlich-dimensionale dynamischen Systeme an, wie die Mackey-Glass-Gleichung, die Kuramoto-Sivashinsky-Gleichung und die Navier-Stokes-Gleichung.
AU - Gerlach, Raphael
ID - 32057
TI - The Computation and Analysis of Invariant Sets of Infinite-Dimensional Systems
ER -
TY - JOUR
AU - Delarue, Benjamin
AU - Ramacher, Pablo
ID - 32016
IS - 6
JF - Journal of Symplectic Geometry
TI - Asymptotic expansion of generalized Witten integrals for Hamiltonian circle actions
VL - 19
ER -
TY - JOUR
AU - Li, Jiaao
AU - Ma, Yulai
AU - Miao, Zhengke
AU - Shi, Yongtang
AU - Wang, Weifan
AU - Zhang, Cun-Quan
ID - 34042
JF - Journal of Combinatorial Theory, Series B
KW - Computational Theory and Mathematics
KW - Discrete Mathematics and Combinatorics
KW - Theoretical Computer Science
SN - 0095-8956
TI - Nowhere-zero 3-flows in toroidal graphs
VL - 153
ER -
TY - CONF
AU - Hattermann, M.
AU - Häsel-Weide, Uta
AU - Wallner, Melina
ED - Inprasitha, M.
ED - Changsri, N.
ED - Boonsena, N.
ID - 31583
T2 - Proceedings of the 44th Conference of the International Group for the Psychology of Mathematics Education
TI - Conceptualiziation processes of 6th graders for rotational symmetry
VL - 3
ER -
TY - JOUR
AB - 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.
AU - McLachlan, Robert I
AU - Offen, Christian
ID - 19938
IS - 6
JF - Foundations of Computational Mathematics
TI - Preservation of Bifurcations of Hamiltonian Boundary Value Problems Under Discretisation
VL - 20
ER -
TY - JOUR
AU - Kreusser, Lisa Maria
AU - McLachlan, Robert I
AU - Offen, Christian
ID - 19939
IS - 5
JF - Nonlinearity
SN - 0951-7715
TI - Detection of high codimensional bifurcations in variational PDEs
VL - 33
ER -
TY - CHAP
AB - 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.
AU - Flaßkamp, Kathrin
AU - Ober-Blöbaum, Sina
AU - Peitz, Sebastian
ED - Junge, Oliver
ED - Schütze, Oliver
ED - Froyland, Gary
ED - Ober-Blöbaum, Sina
ED - Padberg-Gehle, Kathrin
ID - 17411
SN - 2198-4182
T2 - Advances in Dynamics, Optimization and Computation
TI - Symmetry in Optimal Control: A Multiobjective Model Predictive Control Approach
ER -
TY - JOUR
AB - Many dimensionality and model reduction techniques rely on estimating dominant eigenfunctions of associated dynamical operators from data. Important examples include the Koopman operator and its generator, but also the Schrödinger operator. We propose a kernel-based method for the approximation of differential operators in reproducing kernel Hilbert spaces and show how eigenfunctions can be estimated by solving auxiliary matrix eigenvalue problems. The resulting algorithms are applied to molecular dynamics and quantum chemistry examples. Furthermore, we exploit that, under certain conditions, the Schrödinger operator can be transformed into a Kolmogorov backward operator corresponding to a drift-diffusion process and vice versa. This allows us to apply methods developed for the analysis of high-dimensional stochastic differential equations to quantum mechanical systems.
AU - Klus, Stefan
AU - Nüske, Feliks
AU - Hamzi, Boumediene
ID - 21819
JF - Entropy
SN - 1099-4300
TI - Kernel-Based Approximation of the Koopman Generator and Schrödinger Operator
ER -
TY - JOUR
AU - Liebendörfer, Michael
AU - Göller, Robin
AU - Biehler, Rolf
AU - Hochmuth, Reinhard
AU - Kortemeyer, Jörg
AU - Ostsieker, Laura
AU - Rode, Jana
AU - Schaper, Niclas
ID - 16961
JF - Journal für Mathematik-Didaktik
SN - 0173-5322
TI - LimSt – Ein Fragebogen zur Erhebung von Lernstrategien im mathematikhaltigen Studium
ER -
TY - CHAP
AU - Kuklinski, Christiane
AU - Leis, Elena
AU - Liebendörfer, Michael
AU - Hochmuth, Reinhard
ED - Frank, Andreas
ED - Krauss, Stefan
ED - Binder, Karin
ID - 16963
T2 - Beiträge zum {Mathematikunterricht} 2019 53. {Jahrestagung} der {Gesellschaft} für {Didaktik} der {Mathematik}.
TI - Erklärung von Mathematikleistung im Ingenieursstudium
ER -
TY - JOUR
AU - Hochmuth, Reinhard
AU - Liebendörfer, Michael
AU - Biehler, Rolf
AU - Eichler, Andreas
ID - 16964
JF - Neues Handbuch Hochschullehre
TI - Das Kompetenzzentrum Hochschuldidaktik Mathematik (khdm)
VL - 95
ER -
TY - JOUR
AU - Schürmann, Mirko
AU - Schaper, Niclas
AU - Liebendörfer, Michael
AU - Biehler, Rolf
AU - Lankeit, Elisa
AU - Hochmuth, Reinhard
AU - Ruge, Johanna
AU - Kuklinski, Christiane
ID - 16965
JF - dghd-Newsletter
TI - Ein Kurzbericht aus dem Forschungsprojekt WiGeMath-Lernzentren als Unterstützungsmaßnahme für mathematikbezogenes Lernen in der Studieneingangsphase
VL - 01/2020
ER -
TY - JOUR
AB - Multi-objective optimization is an active field of research that has many applications. Owing to its success and because decision-making processes are becoming more and more complex, there is a recent trend for incorporating many objectives into such problems. The challenge with such problems, however, is that the dimensions of the solution sets—the so-called Pareto sets and fronts—grow with the number of objectives. It is thus no longer possible to compute or to approximate the entire solution set of a given problem that contains many (e.g. more than three) objectives. On the other hand, the computation of single solutions (e.g. via scalarization methods) leads to unsatisfying results in many cases, even if user preferences are incorporated. In this article, the Pareto Explorer tool is presented—a global/local exploration tool for the treatment of many-objective optimization problems (MaOPs). In the first step, a solution of the problem is computed via a global search algorithm that ideally already includes user preferences. In the second step, a local search along the Pareto set/front of the given MaOP is performed in user specified directions. For this, several continuation-like procedures are proposed that can incorporate preferences defined in decision, objective, or in weight space. The applicability and usefulness of Pareto Explorer is demonstrated on benchmark problems as well as on an application from industrial laundry design.
AU - Schütze, Oliver
AU - Cuate, Oliver
AU - Martín, Adanay
AU - Peitz, Sebastian
AU - Dellnitz, Michael
ID - 10596
IS - 5
JF - Engineering Optimization
SN - 0305-215X
TI - Pareto Explorer: a global/local exploration tool for many-objective optimization problems
VL - 52
ER -
TY - JOUR
AB - We derive a data-driven method for the approximation of the Koopman generator called gEDMD, which can be regarded as a straightforward extension of EDMD (extended dynamic mode decomposition). This approach is applicable to deterministic and stochastic dynamical systems. It can be used for computing eigenvalues, eigenfunctions, and modes of the generator and for system identification. In addition to learning the governing equations of deterministic systems, which then reduces to SINDy (sparse identification of nonlinear dynamics), it is possible to identify the drift and diffusion terms of stochastic differential equations from data. Moreover, we apply gEDMD to derive coarse-grained models of high-dimensional systems, and also to determine efficient model predictive control strategies. We highlight relationships with other methods and demonstrate the efficacy of the proposed methods using several guiding examples and prototypical molecular dynamics problems.
AU - Klus, Stefan
AU - Nüske, Feliks
AU - Peitz, Sebastian
AU - Niemann, Jan-Hendrik
AU - Clementi, Cecilia
AU - Schütte, Christof
ID - 16288
JF - Physica D: Nonlinear Phenomena
SN - 0167-2789
TI - Data-driven approximation of the Koopman generator: Model reduction, system identification, and control
VL - 406
ER -
TY - CHAP
AB - In the development of model predictive controllers for PDE-constrained problems, the use of reduced order models is essential to enable real-time applicability. Besides local linearization approaches, proper orthogonal decomposition (POD) has been most widely used in the past in order to derive such models. Due to the huge advances concerning both theory as well as the numerical approximation, a very promising alternative based on the Koopman operator has recently emerged. In this chapter, we present two control strategies for model predictive control of nonlinear PDEs using data-efficient approximations of the Koopman operator. In the first one, the dynamic control system is replaced by a small number of autonomous systems with different yet constant inputs. The control problem is consequently transformed into a switching problem. In the second approach, a bilinear surrogate model is obtained via a convex combination of these autonomous systems. Using a recent convergence result for extended dynamic mode decomposition (EDMD), convergence of the reduced objective function can be shown. We study the properties of these two strategies with respect to solution quality, data requirements, and complexity of the resulting optimization problem using the 1-dimensional Burgers equation and the 2-dimensional Navier–Stokes equations as examples. Finally, an extension for online adaptivity is presented.
AU - Peitz, Sebastian
AU - Klus, Stefan
ID - 16289
SN - 0170-8643
T2 - Lecture Notes in Control and Information Sciences
TI - Feedback Control of Nonlinear PDEs Using Data-Efficient Reduced Order Models Based on the Koopman Operator
VL - 484
ER -
TY - JOUR
AB - The control of complex systems is of critical importance in many branches of science, engineering, and industry, many of which are governed by nonlinear partial differential equations. Controlling an unsteady fluid flow is particularly important, as flow control is a key enabler for technologies in energy (e.g., wind, tidal, and combustion), transportation (e.g., planes, trains, and automobiles), security (e.g., tracking airborne contamination), and health (e.g., artificial hearts and artificial respiration). However, the high-dimensional, nonlinear, and multi-scale dynamics make real-time feedback control infeasible. Fortunately, these high- dimensional systems exhibit dominant, low-dimensional patterns of activity that can be exploited for effective control in the sense that knowledge of the entire state of a system is not required. Advances in machine learning have the potential to revolutionize flow control given its ability to extract principled, low-rank feature spaces characterizing such complex systems.We present a novel deep learning modelpredictive control framework that exploits low-rank features of the flow in order to achieve considerable improvements to control performance. Instead of predicting the entire fluid state, we use a recurrent neural network (RNN) to accurately predict the control relevant quantities of the system, which are then embedded into an MPC framework to construct a feedback loop. In order to lower the data requirements and to improve the prediction accuracy and thus the control performance, incoming sensor data are used to update the RNN online. The results are validated using varying fluid flow examples of increasing complexity.
AU - Bieker, Katharina
AU - Peitz, Sebastian
AU - Brunton, Steven L.
AU - Kutz, J. Nathan
AU - Dellnitz, Michael
ID - 16290
JF - Theoretical and Computational Fluid Dynamics
SN - 0935-4964
TI - Deep model predictive flow control with limited sensor data and online learning
VL - 34
ER -
TY - JOUR
AB - In recent years, the success of the Koopman operator in dynamical systems
analysis has also fueled the development of Koopman operator-based control
frameworks. In order to preserve the relatively low data requirements for an
approximation via Dynamic Mode Decomposition, a quantization approach was
recently proposed in [Peitz & Klus, Automatica 106, 2019]. This way, control
of nonlinear dynamical systems can be realized by means of switched systems
techniques, using only a finite set of autonomous Koopman operator-based
reduced models. These individual systems can be approximated very efficiently
from data. The main idea is to transform a control system into a set of
autonomous systems for which the optimal switching sequence has to be computed.
In this article, we extend these results to continuous control inputs using
relaxation. This way, we combine the advantages of the data efficiency of
approximating a finite set of autonomous systems with continuous controls. We
show that when using the Koopman generator, this relaxation --- realized by
linear interpolation between two operators --- does not introduce any error for
control affine systems. This allows us to control high-dimensional nonlinear
systems using bilinear, low-dimensional surrogate models. The efficiency of the
proposed approach is demonstrated using several examples with increasing
complexity, from the Duffing oscillator to the chaotic fluidic pinball.
AU - Peitz, Sebastian
AU - Otto, Samuel E.
AU - Rowley, Clarence W.
ID - 16309
IS - 3
JF - SIAM Journal on Applied Dynamical Systems
TI - Data-Driven Model Predictive Control using Interpolated Koopman Generators
VL - 19
ER -