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 - 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 - 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 - GEN
AB - In backward error analysis, an approximate solution to an equation is
compared to the exact solution to a nearby "modified" equation. In numerical
ordinary differential equations, the two agree up to any power of the step
size. If the differential equation has a geometric property then the modified
equation may share it. In this way, known properties of differential equations
can be applied to the approximation. But for partial differential equations,
the known modified equations are of higher order, limiting applicability of the
theory. Therefore, we study symmetric solutions of discretized partial
differential equations that arise from a discrete variational principle. These
symmetric solutions obey infinite-dimensional functional equations. We show
that these equations admit second-order modified equations which are
Hamiltonian and also possess first-order Lagrangians in modified coordinates.
The modified equation and its associated structures are computed explicitly for
the case of rotating travelling waves in the nonlinear wave equation.
AU - McLachlan, Robert I
AU - Offen, Christian
ID - 19941
T2 - arXiv:2006.14172
TI - Backward error analysis for variational discretisations of partial differential equations
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 - 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 - 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 - CHAP
AB - Diagrammatisches Schlie{\ss}en wird im Zusammenhang mit dem Lernen von Mathmematik und ihrer Symbolsprache als wesentliche Theorie der Wissenskonstruktion diskutiert. Dabei wird h{\"{a}}ufig davon ausgegangen, dass die Wissenskonstruktion im Sinne diagrammatischen Schlie{\ss}ens erfolgt. Deskriptive Rekonstruktionen diagrammatischen Schlie{\ss}ens bei Lernenden stellen jedoch ein Desiderat der mathematikdidaktischen Forschung dar. Der vorliegende Beitrag befasst sich mit der Fragestellung, wie sich diagrammatisches Schlie{\ss}en bei Lernenden rekonstruieren l{\"{a}}sst. Als m{\"{o}}gliche Werkzeuge f{\"{u}}r eine solche Rekonstruktion werden Toulmins Argumentationsschema und Vergnauds Schema-Begriff exemplarisch angewandt, um das diagrammatische Schlie{\ss}en eines Sch{\"{u}}lerpaars beim Einstieg in die Subtraktion negativer Zahlen zu rekonstruieren. Abschlie{\ss}end wird die tats{\"{a}}chliche Eignung der beiden Ans{\"{a}}tze zur Rekonstruktion diagrammatischen Schlie{\ss}ens diskutiert.
AU - Schumacher, Jan
AU - Rezat, Sebastian
ED - Kadunz, Gert
ID - 13108
T2 - Zeichen und Sprache im Mathematikunterricht
TI - Rekonstruktion diagrammatischen Schließens beim Erlernen der Subtraktion negativer Zahlen. Vergleich zweier methodischer Zugänge
ER -
TY - JOUR
AB - In this work we present a set-oriented path following method for the computation of relative global
attractors of parameter-dependent dynamical systems. We start with an initial approximation of the
relative global attractor for a fixed parameter λ0 computed by a set-oriented subdivision method.
By using previously obtained approximations of the parameter-dependent relative global attractor
we can track it with respect to a one-dimensional parameter λ > λ0 without restarting the whole
subdivision procedure. We illustrate the feasibility of the set-oriented path following method by
exploring the dynamics in low-dimensional models for shear flows during the transition to turbulence
and of large-scale atmospheric regime changes .
AU - Gerlach, Raphael
AU - Ziessler, Adrian
AU - Eckhardt, Bruno
AU - Dellnitz, Michael
ID - 16710
JF - SIAM Journal on Applied Dynamical Systems
SN - 1536-0040
TI - A Set-Oriented Path Following Method for the Approximation of Parameter Dependent Attractors
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 -
TY - GEN
AB - In this article, we present an efficient descent method for locally Lipschitz
continuous multiobjective optimization problems (MOPs). The method is realized
by combining a theoretical result regarding the computation of descent
directions for nonsmooth MOPs with a practical method to approximate the
subdifferentials of the objective functions. We show convergence to points
which satisfy a necessary condition for Pareto optimality. Using a set of test
problems, we compare our method to the multiobjective proximal bundle method by
M\"akel\"a. The results indicate that our method is competitive while being
easier to implement. While the number of objective function evaluations is
larger, the overall number of subgradient evaluations is lower. Finally, we
show that our method can be combined with a subdivision algorithm to compute
entire Pareto sets of nonsmooth MOPs.
AU - Gebken, Bennet
AU - Peitz, Sebastian
ID - 16867
T2 - arXiv:2004.11578
TI - An efficient descent method for locally Lipschitz multiobjective optimization problems
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 - 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 - 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.
AU - Hernández Castellanos, Carlos Ignacio
AU - Ober-Blöbaum, Sina
AU - Peitz, Sebastian
ID - 16297
IS - 17
JF - International Journal of Robust and Nonlinear Control
TI - Explicit Multi-objective Model Predictive Control for Nonlinear Systems Under Uncertainty
VL - 30
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 - CHAP
AB - 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.
AU - Gerlach, Raphael
AU - Ziessler, Adrian
ED - Junge, Oliver
ED - Schütze, Oliver
ED - Ober-Blöbaum, Sina
ED - Padberg-Gehle, Kathrin
ID - 17994
SN - 2198-4182
T2 - Advances in Dynamics, Optimization and Computation
TI - The Approximation of Invariant Sets in Infinite Dimensional Dynamical Systems
VL - 304
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
JF - Foundations of Computational Mathematics
SN - 1615-3375
TI - Preservation of Bifurcations of Hamiltonian Boundary Value Problems Under Discretisation
ER -
TY - JOUR
AB - Model predictive control is a prominent approach to construct a feedback
control loop for dynamical systems. Due to real-time constraints, the major
challenge in MPC is to solve model-based optimal control problems in a very
short amount of time. For linear-quadratic problems, Bemporad et al.~have
proposed an explicit formulation where the underlying optimization problems are
solved a priori in an offline phase. In this article, we present an extension
of this concept in two significant ways. We consider nonlinear problems and --
more importantly -- problems with multiple conflicting objective functions. In
the offline phase, we build a library of Pareto optimal solutions from which we
then obtain a valid compromise solution in the online phase according to a
decision maker's preference. Since the standard multi-parametric programming
approach is no longer valid in this situation, we instead use interpolation
between different entries of the library. To reduce the number of problems that
have to be solved in the offline phase, we exploit symmetries in the dynamical
system and the corresponding multiobjective optimal control problem. The
results are verified using two different examples from autonomous driving.
AU - Ober-Blöbaum, Sina
AU - Peitz, Sebastian
ID - 16294
JF - International Journal of Robust and Nonlinear Control
TI - Explicit multiobjective model predictive control for nonlinear systems with symmetries
ER -
TY - JOUR
AB - 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.
AU - Gebken, Bennet
AU - Peitz, Sebastian
AU - Dellnitz, Michael
ID - 10595
IS - 4
JF - Journal of Global Optimization
SN - 0925-5001
TI - On the hierarchical structure of Pareto critical sets
VL - 73
ER -