---
_id: '19938'
abstract:
- lang: eng
text: '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. '
article_type: original
author:
- first_name: Robert I
full_name: McLachlan, Robert I
last_name: McLachlan
- first_name: Christian
full_name: Offen, Christian
id: '85279'
last_name: Offen
orcid: https://orcid.org/0000-0002-5940-8057
citation:
ama: McLachlan RI, Offen C. Preservation of Bifurcations of Hamiltonian Boundary
Value Problems Under Discretisation. Foundations of Computational Mathematics.
2020;20(6):1363-1400. doi:10.1007/s10208-020-09454-z
apa: McLachlan, R. I., & Offen, C. (2020). Preservation of Bifurcations of Hamiltonian
Boundary Value Problems Under Discretisation. Foundations of Computational
Mathematics, 20(6), 1363–1400. https://doi.org/10.1007/s10208-020-09454-z
bibtex: '@article{McLachlan_Offen_2020, title={Preservation of Bifurcations of Hamiltonian
Boundary Value Problems Under Discretisation}, volume={20}, DOI={10.1007/s10208-020-09454-z},
number={6}, journal={Foundations of Computational Mathematics}, author={McLachlan,
Robert I and Offen, Christian}, year={2020}, pages={1363–1400} }'
chicago: 'McLachlan, Robert I, and Christian Offen. “Preservation of Bifurcations
of Hamiltonian Boundary Value Problems Under Discretisation.” Foundations of
Computational Mathematics 20, no. 6 (2020): 1363–1400. https://doi.org/10.1007/s10208-020-09454-z.'
ieee: R. I. McLachlan and C. Offen, “Preservation of Bifurcations of Hamiltonian
Boundary Value Problems Under Discretisation,” Foundations of Computational
Mathematics, vol. 20, no. 6, pp. 1363–1400, 2020.
mla: McLachlan, Robert I., and Christian Offen. “Preservation of Bifurcations of
Hamiltonian Boundary Value Problems Under Discretisation.” Foundations of Computational
Mathematics, vol. 20, no. 6, 2020, pp. 1363–400, doi:10.1007/s10208-020-09454-z.
short: R.I. McLachlan, C. Offen, Foundations of Computational Mathematics 20 (2020)
1363–1400.
date_created: 2020-10-06T16:31:46Z
date_updated: 2022-01-06T06:54:14Z
department:
- _id: '636'
doi: 10.1007/s10208-020-09454-z
extern: '1'
intvolume: ' 20'
issue: '6'
language:
- iso: eng
main_file_link:
- url: https://rdcu.be/b79aB
page: 1363-1400
publication: Foundations of Computational Mathematics
publication_status: published
status: public
title: Preservation of Bifurcations of Hamiltonian Boundary Value Problems Under Discretisation
type: journal_article
user_id: '85279'
volume: 20
year: '2020'
...
---
_id: '19939'
article_type: original
author:
- first_name: Lisa Maria
full_name: Kreusser, Lisa Maria
last_name: Kreusser
- first_name: Robert I
full_name: McLachlan, Robert I
last_name: McLachlan
- first_name: Christian
full_name: Offen, Christian
id: '85279'
last_name: Offen
orcid: https://orcid.org/0000-0002-5940-8057
citation:
ama: Kreusser LM, McLachlan RI, Offen C. Detection of high codimensional bifurcations
in variational PDEs. Nonlinearity. 2020;33(5):2335-2363. doi:10.1088/1361-6544/ab7293
apa: Kreusser, L. M., McLachlan, R. I., & Offen, C. (2020). Detection of high
codimensional bifurcations in variational PDEs. Nonlinearity, 33(5),
2335–2363. https://doi.org/10.1088/1361-6544/ab7293
bibtex: '@article{Kreusser_McLachlan_Offen_2020, title={Detection of high codimensional
bifurcations in variational PDEs}, volume={33}, DOI={10.1088/1361-6544/ab7293},
number={5}, journal={Nonlinearity}, author={Kreusser, Lisa Maria and McLachlan,
Robert I and Offen, Christian}, year={2020}, pages={2335–2363} }'
chicago: 'Kreusser, Lisa Maria, Robert I McLachlan, and Christian Offen. “Detection
of High Codimensional Bifurcations in Variational PDEs.” Nonlinearity 33,
no. 5 (2020): 2335–63. https://doi.org/10.1088/1361-6544/ab7293.'
ieee: L. M. Kreusser, R. I. McLachlan, and C. Offen, “Detection of high codimensional
bifurcations in variational PDEs,” Nonlinearity, vol. 33, no. 5, pp. 2335–2363,
2020.
mla: Kreusser, Lisa Maria, et al. “Detection of High Codimensional Bifurcations
in Variational PDEs.” Nonlinearity, vol. 33, no. 5, 2020, pp. 2335–63,
doi:10.1088/1361-6544/ab7293.
short: L.M. Kreusser, R.I. McLachlan, C. Offen, Nonlinearity 33 (2020) 2335–2363.
date_created: 2020-10-06T16:32:04Z
date_updated: 2022-01-06T06:54:14Z
department:
- _id: '636'
doi: 10.1088/1361-6544/ab7293
extern: '1'
intvolume: ' 33'
issue: '5'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://doi.org/10.1088/1361-6544/ab7293
oa: '1'
page: 2335-2363
publication: Nonlinearity
publication_identifier:
issn:
- 0951-7715
- 1361-6544
publication_status: published
status: public
title: Detection of high codimensional bifurcations in variational PDEs
type: journal_article
user_id: '85279'
volume: 33
year: '2020'
...
---
_id: '17411'
abstract:
- lang: eng
text: 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:
- first_name: Kathrin
full_name: Flaßkamp, Kathrin
last_name: Flaßkamp
- first_name: Sina
full_name: Ober-Blöbaum, Sina
last_name: Ober-Blöbaum
- first_name: Sebastian
full_name: Peitz, Sebastian
id: '47427'
last_name: Peitz
orcid: 0000-0002-3389-793X
citation:
ama: 'Flaßkamp K, Ober-Blöbaum S, Peitz S. Symmetry in Optimal Control: A Multiobjective
Model Predictive Control Approach. In: Junge O, Schütze O, Froyland G, Ober-Blöbaum
S, Padberg-Gehle K, eds. Advances in Dynamics, Optimization and Computation.
Cham: Springer; 2020. doi:10.1007/978-3-030-51264-4_9'
apa: 'Flaßkamp, K., Ober-Blöbaum, S., & Peitz, S. (2020). Symmetry in Optimal
Control: A Multiobjective Model Predictive Control Approach. In O. Junge, O. Schütze,
G. Froyland, S. Ober-Blöbaum, & K. Padberg-Gehle (Eds.), Advances in Dynamics,
Optimization and Computation. Cham: Springer. https://doi.org/10.1007/978-3-030-51264-4_9'
bibtex: '@inbook{Flaßkamp_Ober-Blöbaum_Peitz_2020, place={Cham}, title={Symmetry
in Optimal Control: A Multiobjective Model Predictive Control Approach}, DOI={10.1007/978-3-030-51264-4_9},
booktitle={Advances in Dynamics, Optimization and Computation}, publisher={Springer},
author={Flaßkamp, Kathrin and Ober-Blöbaum, Sina and Peitz, Sebastian}, editor={Junge,
Oliver and Schütze, Oliver and Froyland, Gary and Ober-Blöbaum, Sina and Padberg-Gehle,
KathrinEditors}, year={2020} }'
chicago: 'Flaßkamp, Kathrin, Sina Ober-Blöbaum, and Sebastian Peitz. “Symmetry in
Optimal Control: A Multiobjective Model Predictive Control Approach.” In Advances
in Dynamics, Optimization and Computation, edited by Oliver Junge, Oliver
Schütze, Gary Froyland, Sina Ober-Blöbaum, and Kathrin Padberg-Gehle. Cham: Springer,
2020. https://doi.org/10.1007/978-3-030-51264-4_9.'
ieee: 'K. Flaßkamp, S. Ober-Blöbaum, and S. Peitz, “Symmetry in Optimal Control:
A Multiobjective Model Predictive Control Approach,” in Advances in Dynamics,
Optimization and Computation, O. Junge, O. Schütze, G. Froyland, S. Ober-Blöbaum,
and K. Padberg-Gehle, Eds. Cham: Springer, 2020.'
mla: 'Flaßkamp, Kathrin, et al. “Symmetry in Optimal Control: A Multiobjective Model
Predictive Control Approach.” Advances in Dynamics, Optimization and Computation,
edited by Oliver Junge et al., Springer, 2020, doi:10.1007/978-3-030-51264-4_9.'
short: 'K. Flaßkamp, S. Ober-Blöbaum, S. Peitz, in: O. Junge, O. Schütze, G. Froyland,
S. Ober-Blöbaum, K. Padberg-Gehle (Eds.), Advances in Dynamics, Optimization and
Computation, Springer, Cham, 2020.'
date_created: 2020-07-27T09:50:19Z
date_updated: 2022-01-06T06:53:11Z
department:
- _id: '101'
doi: 10.1007/978-3-030-51264-4_9
editor:
- first_name: Oliver
full_name: Junge, Oliver
last_name: Junge
- first_name: Oliver
full_name: Schütze, Oliver
last_name: Schütze
- first_name: Gary
full_name: Froyland, Gary
last_name: Froyland
- first_name: Sina
full_name: Ober-Blöbaum, Sina
last_name: Ober-Blöbaum
- first_name: Kathrin
full_name: Padberg-Gehle, Kathrin
last_name: Padberg-Gehle
language:
- iso: eng
place: Cham
publication: Advances in Dynamics, Optimization and Computation
publication_identifier:
isbn:
- '9783030512637'
- '9783030512644'
issn:
- 2198-4182
- 2198-4190
publication_status: published
publisher: Springer
status: public
title: 'Symmetry in Optimal Control: A Multiobjective Model Predictive Control Approach'
type: book_chapter
user_id: '47427'
year: '2020'
...
---
_id: '21819'
abstract:
- lang: eng
text: 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.
article_number: '722'
author:
- first_name: Stefan
full_name: Klus, Stefan
last_name: Klus
- first_name: Feliks
full_name: Nüske, Feliks
id: '81513'
last_name: Nüske
orcid: 0000-0003-2444-7889
- first_name: Boumediene
full_name: Hamzi, Boumediene
last_name: Hamzi
citation:
ama: Klus S, Nüske F, Hamzi B. Kernel-Based Approximation of the Koopman Generator
and Schrödinger Operator. Entropy. 2020. doi:10.3390/e22070722
apa: Klus, S., Nüske, F., & Hamzi, B. (2020). Kernel-Based Approximation of
the Koopman Generator and Schrödinger Operator. Entropy. https://doi.org/10.3390/e22070722
bibtex: '@article{Klus_Nüske_Hamzi_2020, title={Kernel-Based Approximation of the
Koopman Generator and Schrödinger Operator}, DOI={10.3390/e22070722},
number={722}, journal={Entropy}, author={Klus, Stefan and Nüske, Feliks and Hamzi,
Boumediene}, year={2020} }'
chicago: Klus, Stefan, Feliks Nüske, and Boumediene Hamzi. “Kernel-Based Approximation
of the Koopman Generator and Schrödinger Operator.” Entropy, 2020. https://doi.org/10.3390/e22070722.
ieee: S. Klus, F. Nüske, and B. Hamzi, “Kernel-Based Approximation of the Koopman
Generator and Schrödinger Operator,” Entropy, 2020.
mla: Klus, Stefan, et al. “Kernel-Based Approximation of the Koopman Generator and
Schrödinger Operator.” Entropy, 722, 2020, doi:10.3390/e22070722.
short: S. Klus, F. Nüske, B. Hamzi, Entropy (2020).
date_created: 2021-04-28T18:06:35Z
date_updated: 2022-01-06T06:55:16Z
department:
- _id: '101'
doi: 10.3390/e22070722
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://www.mdpi.com/1099-4300/22/7/722
oa: '1'
publication: Entropy
publication_identifier:
issn:
- 1099-4300
publication_status: published
status: public
title: Kernel-Based Approximation of the Koopman Generator and Schrödinger Operator
type: journal_article
user_id: '81513'
year: '2020'
...
---
_id: '16964'
author:
- first_name: Reinhard
full_name: Hochmuth, Reinhard
last_name: Hochmuth
- first_name: Michael
full_name: Liebendörfer, Michael
last_name: Liebendörfer
- first_name: Rolf
full_name: Biehler, Rolf
last_name: Biehler
- first_name: Andreas
full_name: Eichler, Andreas
last_name: Eichler
citation:
ama: Hochmuth R, Liebendörfer M, Biehler R, Eichler A. Das Kompetenzzentrum Hochschuldidaktik
Mathematik (khdm). Neues Handbuch Hochschullehre. 2020;95:117-138.
apa: Hochmuth, R., Liebendörfer, M., Biehler, R., & Eichler, A. (2020). Das
Kompetenzzentrum Hochschuldidaktik Mathematik (khdm). Neues Handbuch Hochschullehre,
95, 117–138.
bibtex: '@article{Hochmuth_Liebendörfer_Biehler_Eichler_2020, title={Das Kompetenzzentrum
Hochschuldidaktik Mathematik (khdm)}, volume={95}, journal={Neues Handbuch Hochschullehre},
author={Hochmuth, Reinhard and Liebendörfer, Michael and Biehler, Rolf and Eichler,
Andreas}, year={2020}, pages={117–138} }'
chicago: 'Hochmuth, Reinhard, Michael Liebendörfer, Rolf Biehler, and Andreas Eichler.
“Das Kompetenzzentrum Hochschuldidaktik Mathematik (Khdm).” Neues Handbuch
Hochschullehre 95 (2020): 117–38.'
ieee: R. Hochmuth, M. Liebendörfer, R. Biehler, and A. Eichler, “Das Kompetenzzentrum
Hochschuldidaktik Mathematik (khdm),” Neues Handbuch Hochschullehre, vol.
95, pp. 117–138, 2020.
mla: Hochmuth, Reinhard, et al. “Das Kompetenzzentrum Hochschuldidaktik Mathematik
(Khdm).” Neues Handbuch Hochschullehre, vol. 95, 2020, pp. 117–38.
short: R. Hochmuth, M. Liebendörfer, R. Biehler, A. Eichler, Neues Handbuch Hochschullehre
95 (2020) 117–138.
date_created: 2020-05-16T07:32:23Z
date_updated: 2022-01-06T06:53:00Z
department:
- _id: '10'
intvolume: ' 95'
page: 117-138
publication: Neues Handbuch Hochschullehre
status: public
title: Das Kompetenzzentrum Hochschuldidaktik Mathematik (khdm)
type: journal_article
user_id: '30933'
volume: 95
year: '2020'
...
---
_id: '10596'
abstract:
- lang: eng
text: 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.
article_type: original
author:
- first_name: Oliver
full_name: Schütze, Oliver
last_name: Schütze
- first_name: Oliver
full_name: Cuate, Oliver
last_name: Cuate
- first_name: Adanay
full_name: Martín, Adanay
last_name: Martín
- first_name: Sebastian
full_name: Peitz, Sebastian
id: '47427'
last_name: Peitz
orcid: https://orcid.org/0000-0002-3389-793X
- first_name: Michael
full_name: Dellnitz, Michael
last_name: Dellnitz
citation:
ama: 'Schütze O, Cuate O, Martín A, Peitz S, Dellnitz M. Pareto Explorer: a global/local
exploration tool for many-objective optimization problems. Engineering Optimization.
2020;52(5):832-855. doi:10.1080/0305215x.2019.1617286'
apa: 'Schütze, O., Cuate, O., Martín, A., Peitz, S., & Dellnitz, M. (2020).
Pareto Explorer: a global/local exploration tool for many-objective optimization
problems. Engineering Optimization, 52(5), 832–855. https://doi.org/10.1080/0305215x.2019.1617286'
bibtex: '@article{Schütze_Cuate_Martín_Peitz_Dellnitz_2020, title={Pareto Explorer:
a global/local exploration tool for many-objective optimization problems}, volume={52},
DOI={10.1080/0305215x.2019.1617286},
number={5}, journal={Engineering Optimization}, author={Schütze, Oliver and Cuate,
Oliver and Martín, Adanay and Peitz, Sebastian and Dellnitz, Michael}, year={2020},
pages={832–855} }'
chicago: 'Schütze, Oliver, Oliver Cuate, Adanay Martín, Sebastian Peitz, and Michael
Dellnitz. “Pareto Explorer: A Global/Local Exploration Tool for Many-Objective
Optimization Problems.” Engineering Optimization 52, no. 5 (2020): 832–55.
https://doi.org/10.1080/0305215x.2019.1617286.'
ieee: 'O. Schütze, O. Cuate, A. Martín, S. Peitz, and M. Dellnitz, “Pareto Explorer:
a global/local exploration tool for many-objective optimization problems,” Engineering
Optimization, vol. 52, no. 5, pp. 832–855, 2020.'
mla: 'Schütze, Oliver, et al. “Pareto Explorer: A Global/Local Exploration Tool
for Many-Objective Optimization Problems.” Engineering Optimization, vol.
52, no. 5, 2020, pp. 832–55, doi:10.1080/0305215x.2019.1617286.'
short: O. Schütze, O. Cuate, A. Martín, S. Peitz, M. Dellnitz, Engineering Optimization
52 (2020) 832–855.
date_created: 2019-07-10T08:14:39Z
date_updated: 2022-01-06T06:50:46Z
department:
- _id: '101'
doi: 10.1080/0305215x.2019.1617286
intvolume: ' 52'
issue: '5'
language:
- iso: eng
page: 832-855
publication: Engineering Optimization
publication_identifier:
issn:
- 0305-215X
- 1029-0273
publication_status: published
status: public
title: 'Pareto Explorer: a global/local exploration tool for many-objective optimization
problems'
type: journal_article
user_id: '47427'
volume: 52
year: '2020'
...
---
_id: '16288'
abstract:
- lang: eng
text: 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.
article_number: '132416'
author:
- first_name: Stefan
full_name: Klus, Stefan
last_name: Klus
- first_name: Feliks
full_name: Nüske, Feliks
id: '81513'
last_name: Nüske
orcid: 0000-0003-2444-7889
- first_name: Sebastian
full_name: Peitz, Sebastian
id: '47427'
last_name: Peitz
orcid: https://orcid.org/0000-0002-3389-793X
- first_name: Jan-Hendrik
full_name: Niemann, Jan-Hendrik
last_name: Niemann
- first_name: Cecilia
full_name: Clementi, Cecilia
last_name: Clementi
- first_name: Christof
full_name: Schütte, Christof
last_name: Schütte
citation:
ama: 'Klus S, Nüske F, Peitz S, Niemann J-H, Clementi C, Schütte C. Data-driven
approximation of the Koopman generator: Model reduction, system identification,
and control. Physica D: Nonlinear Phenomena. 2020;406. doi:10.1016/j.physd.2020.132416'
apa: 'Klus, S., Nüske, F., Peitz, S., Niemann, J.-H., Clementi, C., & Schütte,
C. (2020). Data-driven approximation of the Koopman generator: Model reduction,
system identification, and control. Physica D: Nonlinear Phenomena, 406.
https://doi.org/10.1016/j.physd.2020.132416'
bibtex: '@article{Klus_Nüske_Peitz_Niemann_Clementi_Schütte_2020, title={Data-driven
approximation of the Koopman generator: Model reduction, system identification,
and control}, volume={406}, DOI={10.1016/j.physd.2020.132416},
number={132416}, journal={Physica D: Nonlinear Phenomena}, author={Klus, Stefan
and Nüske, Feliks and Peitz, Sebastian and Niemann, Jan-Hendrik and Clementi,
Cecilia and Schütte, Christof}, year={2020} }'
chicago: 'Klus, Stefan, Feliks Nüske, Sebastian Peitz, Jan-Hendrik Niemann, Cecilia
Clementi, and Christof Schütte. “Data-Driven Approximation of the Koopman Generator:
Model Reduction, System Identification, and Control.” Physica D: Nonlinear
Phenomena 406 (2020). https://doi.org/10.1016/j.physd.2020.132416.'
ieee: 'S. Klus, F. Nüske, S. Peitz, J.-H. Niemann, C. Clementi, and C. Schütte,
“Data-driven approximation of the Koopman generator: Model reduction, system identification,
and control,” Physica D: Nonlinear Phenomena, vol. 406, 2020.'
mla: 'Klus, Stefan, et al. “Data-Driven Approximation of the Koopman Generator:
Model Reduction, System Identification, and Control.” Physica D: Nonlinear
Phenomena, vol. 406, 132416, 2020, doi:10.1016/j.physd.2020.132416.'
short: 'S. Klus, F. Nüske, S. Peitz, J.-H. Niemann, C. Clementi, C. Schütte, Physica
D: Nonlinear Phenomena 406 (2020).'
date_created: 2020-03-13T12:35:40Z
date_updated: 2022-01-06T06:52:48Z
department:
- _id: '101'
doi: 10.1016/j.physd.2020.132416
intvolume: ' 406'
language:
- iso: eng
publication: 'Physica D: Nonlinear Phenomena'
publication_identifier:
issn:
- 0167-2789
publication_status: published
status: public
title: 'Data-driven approximation of the Koopman generator: Model reduction, system
identification, and control'
type: journal_article
user_id: '47427'
volume: 406
year: '2020'
...
---
_id: '16289'
abstract:
- lang: eng
text: 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.
author:
- first_name: Sebastian
full_name: Peitz, Sebastian
id: '47427'
last_name: Peitz
orcid: https://orcid.org/0000-0002-3389-793X
- first_name: Stefan
full_name: Klus, Stefan
last_name: Klus
citation:
ama: 'Peitz S, Klus S. Feedback Control of Nonlinear PDEs Using Data-Efficient Reduced
Order Models Based on the Koopman Operator. In: Lecture Notes in Control and
Information Sciences. Vol 484. Lecture Notes in Control and Information Sciences.
Cham: Springer; 2020:257-282. doi:10.1007/978-3-030-35713-9_10'
apa: 'Peitz, S., & Klus, S. (2020). Feedback Control of Nonlinear PDEs Using
Data-Efficient Reduced Order Models Based on the Koopman Operator. In Lecture
Notes in Control and Information Sciences (Vol. 484, pp. 257–282). Cham: Springer.
https://doi.org/10.1007/978-3-030-35713-9_10'
bibtex: '@inbook{Peitz_Klus_2020, place={Cham}, series={Lecture Notes in Control
and Information Sciences}, title={Feedback Control of Nonlinear PDEs Using Data-Efficient
Reduced Order Models Based on the Koopman Operator}, volume={484}, DOI={10.1007/978-3-030-35713-9_10},
booktitle={Lecture Notes in Control and Information Sciences}, publisher={Springer},
author={Peitz, Sebastian and Klus, Stefan}, year={2020}, pages={257–282}, collection={Lecture
Notes in Control and Information Sciences} }'
chicago: 'Peitz, Sebastian, and Stefan Klus. “Feedback Control of Nonlinear PDEs
Using Data-Efficient Reduced Order Models Based on the Koopman Operator.” In Lecture
Notes in Control and Information Sciences, 484:257–82. Lecture Notes in Control
and Information Sciences. Cham: Springer, 2020. https://doi.org/10.1007/978-3-030-35713-9_10.'
ieee: 'S. Peitz and S. Klus, “Feedback Control of Nonlinear PDEs Using Data-Efficient
Reduced Order Models Based on the Koopman Operator,” in Lecture Notes in Control
and Information Sciences, vol. 484, Cham: Springer, 2020, pp. 257–282.'
mla: Peitz, Sebastian, and Stefan Klus. “Feedback Control of Nonlinear PDEs Using
Data-Efficient Reduced Order Models Based on the Koopman Operator.” Lecture
Notes in Control and Information Sciences, vol. 484, Springer, 2020, pp. 257–82,
doi:10.1007/978-3-030-35713-9_10.
short: 'S. Peitz, S. Klus, in: Lecture Notes in Control and Information Sciences,
Springer, Cham, 2020, pp. 257–282.'
date_created: 2020-03-13T12:38:52Z
date_updated: 2022-01-06T06:52:48Z
department:
- _id: '101'
doi: 10.1007/978-3-030-35713-9_10
intvolume: ' 484'
language:
- iso: eng
page: 257-282
place: Cham
publication: Lecture Notes in Control and Information Sciences
publication_identifier:
isbn:
- '9783030357122'
- '9783030357139'
issn:
- 0170-8643
- 1610-7411
publication_status: published
publisher: Springer
series_title: Lecture Notes in Control and Information Sciences
status: public
title: Feedback Control of Nonlinear PDEs Using Data-Efficient Reduced Order Models
Based on the Koopman Operator
type: book_chapter
user_id: '47427'
volume: 484
year: '2020'
...
---
_id: '16290'
abstract:
- lang: eng
text: 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.
article_type: original
author:
- first_name: Katharina
full_name: Bieker, Katharina
id: '32829'
last_name: Bieker
- first_name: Sebastian
full_name: Peitz, Sebastian
id: '47427'
last_name: Peitz
orcid: https://orcid.org/0000-0002-3389-793X
- first_name: Steven L.
full_name: Brunton, Steven L.
last_name: Brunton
- first_name: J. Nathan
full_name: Kutz, J. Nathan
last_name: Kutz
- first_name: Michael
full_name: Dellnitz, Michael
last_name: Dellnitz
citation:
ama: Bieker K, Peitz S, Brunton SL, Kutz JN, Dellnitz M. Deep model predictive flow
control with limited sensor data and online learning. Theoretical and Computational
Fluid Dynamics. 2020;34:577–591. doi:10.1007/s00162-020-00520-4
apa: Bieker, K., Peitz, S., Brunton, S. L., Kutz, J. N., & Dellnitz, M. (2020).
Deep model predictive flow control with limited sensor data and online learning.
Theoretical and Computational Fluid Dynamics, 34, 577–591. https://doi.org/10.1007/s00162-020-00520-4
bibtex: '@article{Bieker_Peitz_Brunton_Kutz_Dellnitz_2020, title={Deep model predictive
flow control with limited sensor data and online learning}, volume={34}, DOI={10.1007/s00162-020-00520-4},
journal={Theoretical and Computational Fluid Dynamics}, author={Bieker, Katharina
and Peitz, Sebastian and Brunton, Steven L. and Kutz, J. Nathan and Dellnitz,
Michael}, year={2020}, pages={577–591} }'
chicago: 'Bieker, Katharina, Sebastian Peitz, Steven L. Brunton, J. Nathan Kutz,
and Michael Dellnitz. “Deep Model Predictive Flow Control with Limited Sensor
Data and Online Learning.” Theoretical and Computational Fluid Dynamics
34 (2020): 577–591. https://doi.org/10.1007/s00162-020-00520-4.'
ieee: K. Bieker, S. Peitz, S. L. Brunton, J. N. Kutz, and M. Dellnitz, “Deep model
predictive flow control with limited sensor data and online learning,” Theoretical
and Computational Fluid Dynamics, vol. 34, pp. 577–591, 2020.
mla: Bieker, Katharina, et al. “Deep Model Predictive Flow Control with Limited
Sensor Data and Online Learning.” Theoretical and Computational Fluid Dynamics,
vol. 34, 2020, pp. 577–591, doi:10.1007/s00162-020-00520-4.
short: K. Bieker, S. Peitz, S.L. Brunton, J.N. Kutz, M. Dellnitz, Theoretical and
Computational Fluid Dynamics 34 (2020) 577–591.
date_created: 2020-03-13T12:40:09Z
date_updated: 2022-01-06T06:52:48Z
department:
- _id: '101'
doi: 10.1007/s00162-020-00520-4
intvolume: ' 34'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://link.springer.com/content/pdf/10.1007/s00162-020-00520-4.pdf
oa: '1'
page: 577–591
project:
- _id: '52'
name: Computing Resources Provided by the Paderborn Center for Parallel Computing
publication: Theoretical and Computational Fluid Dynamics
publication_identifier:
issn:
- 0935-4964
- 1432-2250
publication_status: published
status: public
title: Deep model predictive flow control with limited sensor data and online learning
type: journal_article
user_id: '47427'
volume: 34
year: '2020'
...
---
_id: '16309'
abstract:
- lang: eng
text: "In recent years, the success of the Koopman operator in dynamical systems\r\nanalysis
has also fueled the development of Koopman operator-based control\r\nframeworks.
In order to preserve the relatively low data requirements for an\r\napproximation
via Dynamic Mode Decomposition, a quantization approach was\r\nrecently proposed
in [Peitz & Klus, Automatica 106, 2019]. This way, control\r\nof nonlinear dynamical
systems can be realized by means of switched systems\r\ntechniques, using only
a finite set of autonomous Koopman operator-based\r\nreduced models. These individual
systems can be approximated very efficiently\r\nfrom data. The main idea is to
transform a control system into a set of\r\nautonomous systems for which the optimal
switching sequence has to be computed.\r\nIn this article, we extend these results
to continuous control inputs using\r\nrelaxation. This way, we combine the advantages
of the data efficiency of\r\napproximating a finite set of autonomous systems
with continuous controls. We\r\nshow that when using the Koopman generator, this
relaxation --- realized by\r\nlinear interpolation between two operators --- does
not introduce any error for\r\ncontrol affine systems. This allows us to control
high-dimensional nonlinear\r\nsystems using bilinear, low-dimensional surrogate
models. The efficiency of the\r\nproposed approach is demonstrated using several
examples with increasing\r\ncomplexity, from the Duffing oscillator to the chaotic
fluidic pinball."
author:
- first_name: Sebastian
full_name: Peitz, Sebastian
id: '47427'
last_name: Peitz
orcid: 0000-0002-3389-793X
- first_name: Samuel E.
full_name: Otto, Samuel E.
last_name: Otto
- first_name: Clarence W.
full_name: Rowley, Clarence W.
last_name: Rowley
citation:
ama: Peitz S, Otto SE, Rowley CW. Data-Driven Model Predictive Control using Interpolated
Koopman Generators. SIAM Journal on Applied Dynamical Systems. 2020;19(3):2162-2193.
doi:10.1137/20M1325678
apa: Peitz, S., Otto, S. E., & Rowley, C. W. (2020). Data-Driven Model Predictive
Control using Interpolated Koopman Generators. SIAM Journal on Applied Dynamical
Systems, 19(3), 2162–2193. https://doi.org/10.1137/20M1325678
bibtex: '@article{Peitz_Otto_Rowley_2020, title={Data-Driven Model Predictive Control
using Interpolated Koopman Generators}, volume={19}, DOI={10.1137/20M1325678},
number={3}, journal={SIAM Journal on Applied Dynamical Systems}, author={Peitz,
Sebastian and Otto, Samuel E. and Rowley, Clarence W.}, year={2020}, pages={2162–2193}
}'
chicago: 'Peitz, Sebastian, Samuel E. Otto, and Clarence W. Rowley. “Data-Driven
Model Predictive Control Using Interpolated Koopman Generators.” SIAM Journal
on Applied Dynamical Systems 19, no. 3 (2020): 2162–93. https://doi.org/10.1137/20M1325678.'
ieee: S. Peitz, S. E. Otto, and C. W. Rowley, “Data-Driven Model Predictive Control
using Interpolated Koopman Generators,” SIAM Journal on Applied Dynamical
Systems, vol. 19, no. 3, pp. 2162–2193, 2020.
mla: Peitz, Sebastian, et al. “Data-Driven Model Predictive Control Using Interpolated
Koopman Generators.” SIAM Journal on Applied Dynamical Systems, vol. 19,
no. 3, 2020, pp. 2162–93, doi:10.1137/20M1325678.
short: S. Peitz, S.E. Otto, C.W. Rowley, SIAM Journal on Applied Dynamical Systems
19 (2020) 2162–2193.
date_created: 2020-03-17T09:53:01Z
date_updated: 2022-01-06T06:52:48Z
department:
- _id: '101'
doi: 10.1137/20M1325678
intvolume: ' 19'
issue: '3'
language:
- iso: eng
main_file_link:
- url: https://epubs.siam.org/doi/pdf/10.1137/20M1325678
page: 2162-2193
publication: SIAM Journal on Applied Dynamical Systems
status: public
title: Data-Driven Model Predictive Control using Interpolated Koopman Generators
type: journal_article
user_id: '47427'
volume: 19
year: '2020'
...
---
_id: '13108'
abstract:
- lang: ger
text: 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.
author:
- first_name: Jan
full_name: Schumacher, Jan
id: '26809'
last_name: Schumacher
orcid: 0000-0002-6676-9774
- first_name: Sebastian
full_name: Rezat, Sebastian
last_name: Rezat
citation:
ama: 'Schumacher J, Rezat S. Rekonstruktion diagrammatischen Schließens beim Erlernen
der Subtraktion negativer Zahlen. Vergleich zweier methodischer Zugänge. In: Kadunz
G, ed. Zeichen und Sprache im Mathematikunterricht. Springer; 2020. doi:10.1007/978-3-662-61194-4_5'
apa: Schumacher, J., & Rezat, S. (2020). Rekonstruktion diagrammatischen Schließens
beim Erlernen der Subtraktion negativer Zahlen. Vergleich zweier methodischer
Zugänge. In G. Kadunz (Ed.), Zeichen und Sprache im Mathematikunterricht.
Springer. https://doi.org/10.1007/978-3-662-61194-4_5
bibtex: '@inbook{Schumacher_Rezat_2020, title={Rekonstruktion diagrammatischen Schließens
beim Erlernen der Subtraktion negativer Zahlen. Vergleich zweier methodischer
Zugänge}, DOI={10.1007/978-3-662-61194-4_5},
booktitle={Zeichen und Sprache im Mathematikunterricht}, publisher={Springer},
author={Schumacher, Jan and Rezat, Sebastian}, editor={Kadunz, GertEditor}, year={2020}
}'
chicago: Schumacher, Jan, and Sebastian Rezat. “Rekonstruktion diagrammatischen
Schließens beim Erlernen der Subtraktion negativer Zahlen. Vergleich zweier methodischer
Zugänge.” In Zeichen und Sprache im Mathematikunterricht, edited by Gert
Kadunz. Springer, 2020. https://doi.org/10.1007/978-3-662-61194-4_5.
ieee: J. Schumacher and S. Rezat, “Rekonstruktion diagrammatischen Schließens beim
Erlernen der Subtraktion negativer Zahlen. Vergleich zweier methodischer Zugänge,”
in Zeichen und Sprache im Mathematikunterricht, G. Kadunz, Ed. Springer,
2020.
mla: Schumacher, Jan, and Sebastian Rezat. “Rekonstruktion diagrammatischen Schließens
beim Erlernen der Subtraktion negativer Zahlen. Vergleich zweier methodischer
Zugänge.” Zeichen und Sprache im Mathematikunterricht, edited by Gert Kadunz,
Springer, 2020, doi:10.1007/978-3-662-61194-4_5.
short: 'J. Schumacher, S. Rezat, in: G. Kadunz (Ed.), Zeichen und Sprache im Mathematikunterricht,
Springer, 2020.'
date_created: 2019-08-30T12:00:51Z
date_updated: 2022-01-06T06:51:28Z
department:
- _id: '360'
- _id: '98'
doi: 10.1007/978-3-662-61194-4_5
editor:
- first_name: Gert
full_name: Kadunz, Gert
last_name: Kadunz
language:
- iso: ger
publication: Zeichen und Sprache im Mathematikunterricht
publication_status: published
publisher: Springer
quality_controlled: '1'
status: public
title: Rekonstruktion diagrammatischen Schließens beim Erlernen der Subtraktion negativer
Zahlen. Vergleich zweier methodischer Zugänge
type: book_chapter
user_id: '26809'
year: '2020'
...
---
_id: '31873'
author:
- first_name: Jan
full_name: Schumacher, Jan
id: '26809'
last_name: Schumacher
orcid: 0000-0002-6676-9774
citation:
ama: 'Schumacher J. Deduktion und Abduktion beim diagrammatischen Schließen – das
didaktische Potential der Peirceschen Semiotik. In: LibreCat University; 2020.
doi:10.17877/DE290R-21555'
apa: Schumacher, J. (2020). Deduktion und Abduktion beim diagrammatischen Schließen
– das didaktische Potential der Peirceschen Semiotik. https://doi.org/10.17877/DE290R-21555
bibtex: '@inproceedings{Schumacher_2020, title={Deduktion und Abduktion beim diagrammatischen
Schließen – das didaktische Potential der Peirceschen Semiotik}, DOI={10.17877/DE290R-21555},
publisher={LibreCat University}, author={Schumacher, Jan}, year={2020} }'
chicago: Schumacher, Jan. “Deduktion Und Abduktion Beim Diagrammatischen Schließen
– Das Didaktische Potential Der Peirceschen Semiotik.” LibreCat University, 2020.
https://doi.org/10.17877/DE290R-21555.
ieee: 'J. Schumacher, “Deduktion und Abduktion beim diagrammatischen Schließen –
das didaktische Potential der Peirceschen Semiotik,” 2020, doi: 10.17877/DE290R-21555.'
mla: Schumacher, Jan. Deduktion Und Abduktion Beim Diagrammatischen Schließen
– Das Didaktische Potential Der Peirceschen Semiotik. LibreCat University,
2020, doi:10.17877/DE290R-21555.
short: 'J. Schumacher, in: LibreCat University, 2020.'
date_created: 2022-06-14T07:10:29Z
date_updated: 2022-06-14T07:15:34Z
department:
- _id: '360'
- _id: '98'
doi: 10.17877/DE290R-21555
language:
- iso: eng
publication_status: published
publisher: LibreCat University
status: public
title: Deduktion und Abduktion beim diagrammatischen Schließen – das didaktische Potential
der Peirceschen Semiotik
type: conference
user_id: '26809'
year: '2020'
...
---
_id: '34789'
article_type: original
author:
- first_name: Habib
full_name: Amiri, Habib
last_name: Amiri
- first_name: Helge
full_name: Glöckner, Helge
id: '178'
last_name: Glöckner
- first_name: Alexander
full_name: Schmeding, Alexander
last_name: Schmeding
citation:
ama: Amiri H, Glöckner H, Schmeding A. Lie groupoids of mappings taking values in
a Lie groupoid. Archivum Mathematicum. 2020;56(5):307–356. doi:10.5817/AM2020-5-307
apa: Amiri, H., Glöckner, H., & Schmeding, A. (2020). Lie groupoids of mappings
taking values in a Lie groupoid. Archivum Mathematicum, 56(5), 307–356.
https://doi.org/10.5817/AM2020-5-307
bibtex: '@article{Amiri_Glöckner_Schmeding_2020, title={Lie groupoids of mappings
taking values in a Lie groupoid}, volume={56}, DOI={10.5817/AM2020-5-307},
number={5}, journal={Archivum Mathematicum}, author={Amiri, Habib and Glöckner,
Helge and Schmeding, Alexander}, year={2020}, pages={307–356} }'
chicago: 'Amiri, Habib, Helge Glöckner, and Alexander Schmeding. “Lie Groupoids
of Mappings Taking Values in a Lie Groupoid.” Archivum Mathematicum 56,
no. 5 (2020): 307–356. https://doi.org/10.5817/AM2020-5-307.'
ieee: 'H. Amiri, H. Glöckner, and A. Schmeding, “Lie groupoids of mappings taking
values in a Lie groupoid,” Archivum Mathematicum, vol. 56, no. 5, pp. 307–356,
2020, doi: 10.5817/AM2020-5-307.'
mla: Amiri, Habib, et al. “Lie Groupoids of Mappings Taking Values in a Lie Groupoid.”
Archivum Mathematicum, vol. 56, no. 5, 2020, pp. 307–356, doi:10.5817/AM2020-5-307.
short: H. Amiri, H. Glöckner, A. Schmeding, Archivum Mathematicum 56 (2020) 307–356.
date_created: 2022-12-21T19:13:24Z
date_updated: 2022-12-21T19:15:59Z
department:
- _id: '10'
- _id: '87'
- _id: '93'
doi: 10.5817/AM2020-5-307
intvolume: ' 56'
issue: '5'
keyword:
- 22A22
- '22E65'
- '22E67'
- 46T10
- 47H30
- 58D15
- 58H05
language:
- iso: eng
page: 307–356
publication: Archivum Mathematicum
publication_identifier:
issn:
- 0044-8753
quality_controlled: '1'
status: public
title: Lie groupoids of mappings taking values in a Lie groupoid
type: journal_article
user_id: '178'
volume: 56
year: '2020'
...
---
_id: '34787'
article_type: original
author:
- first_name: Helge
full_name: Glöckner, Helge
id: '178'
last_name: Glöckner
- first_name: Niku
full_name: Masbough, Niku
last_name: Masbough
citation:
ama: Glöckner H, Masbough N. Products of regular locally compact spaces are k_R-spaces.
Topology Proceedings. 2020;55:35–38.
apa: Glöckner, H., & Masbough, N. (2020). Products of regular locally compact
spaces are k_R-spaces. Topology Proceedings, 55, 35–38.
bibtex: '@article{Glöckner_Masbough_2020, title={Products of regular locally compact
spaces are k_R-spaces}, volume={55}, journal={Topology Proceedings}, author={Glöckner,
Helge and Masbough, Niku}, year={2020}, pages={35–38} }'
chicago: 'Glöckner, Helge, and Niku Masbough. “Products of Regular Locally Compact
Spaces Are K_R-Spaces.” Topology Proceedings 55 (2020): 35–38.'
ieee: H. Glöckner and N. Masbough, “Products of regular locally compact spaces are
k_R-spaces,” Topology Proceedings, vol. 55, pp. 35–38, 2020.
mla: Glöckner, Helge, and Niku Masbough. “Products of Regular Locally Compact Spaces
Are K_R-Spaces.” Topology Proceedings, vol. 55, 2020, pp. 35–38.
short: H. Glöckner, N. Masbough, Topology Proceedings 55 (2020) 35–38.
date_created: 2022-12-21T19:06:45Z
date_updated: 2022-12-21T20:06:44Z
department:
- _id: '10'
- _id: '87'
- _id: '93'
intvolume: ' 55'
keyword:
- 54B10
- 54D45
- 54D50
language:
- iso: eng
page: 35–38
publication: Topology Proceedings
publication_identifier:
issn:
- 0146-4124
quality_controlled: '1'
status: public
title: Products of regular locally compact spaces are k_R-spaces
type: journal_article
user_id: '178'
volume: 55
year: '2020'
...
---
_id: '34808'
abstract:
- lang: eng
text: "For suitable finite-dimensional smooth manifolds M (possibly with various\r\nkinds
of boundary or corners), locally convex topological vector spaces F and\r\nnon-negative
integers k, we construct continuous linear operators S_n from the\r\nspace of
F-valued k times continuously differentiable functions on M to the\r\ncorresponding
space of smooth functions such that S_n(f) converges to f in\r\nC^k(M,F) as n
tends to infinity, uniformly for f in compact subsets of\r\nC^k(M,F). We also
study the existence of continuous linear right inverses for\r\nrestriction maps
from C^k(M,F) to C^k(L,F) if L is a closed subset of M,\r\nendowed with a C^k-manifold
structure turning the inclusion map from L to M\r\ninto a C^k-map. Moreover, we
construct continuous linear right inverses for\r\nrestriction operators between
spaces of sections in vector bundles in many\r\nsituations, and smooth local right
inverses for restriction operators between\r\nmanifolds of mappings. We also obtain
smoothing results for sections in fibre\r\nbundles."
author:
- first_name: Helge
full_name: Glöckner, Helge
id: '178'
last_name: Glöckner
citation:
ama: Glöckner H. Smoothing operators for vector-valued functions and extension operators.
arXiv:200600254. Published online 2020.
apa: Glöckner, H. (2020). Smoothing operators for vector-valued functions and extension
operators. In arXiv:2006.00254.
bibtex: '@article{Glöckner_2020, title={Smoothing operators for vector-valued functions
and extension operators}, journal={arXiv:2006.00254}, author={Glöckner, Helge},
year={2020} }'
chicago: Glöckner, Helge. “Smoothing Operators for Vector-Valued Functions and Extension
Operators.” ArXiv:2006.00254, 2020.
ieee: H. Glöckner, “Smoothing operators for vector-valued functions and extension
operators,” arXiv:2006.00254. 2020.
mla: Glöckner, Helge. “Smoothing Operators for Vector-Valued Functions and Extension
Operators.” ArXiv:2006.00254, 2020.
short: H. Glöckner, ArXiv:2006.00254 (2020).
date_created: 2022-12-22T07:51:53Z
date_updated: 2022-12-22T07:52:42Z
department:
- _id: '10'
- _id: '87'
- _id: '93'
external_id:
arxiv:
- '2006.00254'
language:
- iso: eng
publication: arXiv:2006.00254
status: public
title: Smoothing operators for vector-valued functions and extension operators
type: preprint
user_id: '178'
year: '2020'
...
---
_id: '29399'
author:
- first_name: D. J. N.
full_name: Limebeer, D. J. N.
last_name: Limebeer
- first_name: Sina
full_name: Ober-Blöbaum, Sina
id: '16494'
last_name: Ober-Blöbaum
- first_name: F. H.
full_name: Farshi, F. H.
last_name: Farshi
citation:
ama: Limebeer DJN, Ober-Blöbaum S, Farshi FH. Variational integrators for dissipative
systems. IEEE Transactions on Automatic Control. 2020;65(4):1381-1396.
apa: Limebeer, D. J. N., Ober-Blöbaum, S., & Farshi, F. H. (2020). Variational
integrators for dissipative systems. IEEE Transactions on Automatic Control,
65(4), 1381–1396.
bibtex: '@article{Limebeer_Ober-Blöbaum_Farshi_2020, title={Variational integrators
for dissipative systems}, volume={65(4)}, journal={IEEE Transactions on Automatic
Control}, author={Limebeer, D. J. N. and Ober-Blöbaum, Sina and Farshi, F. H.},
year={2020}, pages={1381–1396} }'
chicago: 'Limebeer, D. J. N., Sina Ober-Blöbaum, and F. H. Farshi. “Variational
Integrators for Dissipative Systems.” IEEE Transactions on Automatic Control
65(4) (2020): 1381–96.'
ieee: D. J. N. Limebeer, S. Ober-Blöbaum, and F. H. Farshi, “Variational integrators
for dissipative systems,” IEEE Transactions on Automatic Control, vol.
65(4), pp. 1381–1396, 2020.
mla: Limebeer, D. J. N., et al. “Variational Integrators for Dissipative Systems.”
IEEE Transactions on Automatic Control, vol. 65(4), 2020, pp. 1381–96.
short: D.J.N. Limebeer, S. Ober-Blöbaum, F.H. Farshi, IEEE Transactions on Automatic
Control 65(4) (2020) 1381–1396.
date_created: 2022-01-18T11:17:25Z
date_updated: 2022-01-21T08:26:46Z
department:
- _id: '636'
language:
- iso: eng
page: 1381-1396
publication: IEEE Transactions on Automatic Control
status: public
title: Variational integrators for dissipative systems
type: journal_article
user_id: '15694'
volume: 65(4)
year: '2020'
...
---
_id: '16297'
abstract:
- lang: eng
text: "In real-world problems, uncertainties (e.g., errors in the measurement,\r\nprecision
errors) often lead to poor performance of numerical algorithms when\r\nnot explicitly
taken into account. This is also the case for control problems,\r\nwhere optimal
solutions can degrade in quality or even become infeasible. Thus,\r\nthere is
the need to design methods that can handle uncertainty. In this work,\r\nwe consider
nonlinear multi-objective optimal control problems with uncertainty\r\non the
initial conditions, and in particular their incorporation into a\r\nfeedback loop
via model predictive control (MPC). In multi-objective optimal\r\ncontrol, an
optimal compromise between multiple conflicting criteria has to be\r\nfound. For
such problems, not much has been reported in terms of uncertainties.\r\nTo address
this problem class, we design an offline/online framework to compute\r\nan approximation
of efficient control strategies. This approach is closely\r\nrelated to explicit
MPC for nonlinear systems, where the potentially expensive\r\noptimization problem
is solved in an offline phase in order to enable fast\r\nsolutions in the online
phase. In order to reduce the numerical cost of the\r\noffline phase, we exploit
symmetries in the control problems. Furthermore, in\r\norder to ensure optimality
of the solutions, we include an additional online\r\noptimization step, which
is considerably cheaper than the original\r\nmulti-objective optimization problem.
We test our framework on a car\r\nmaneuvering problem where safety and speed are
the objectives. The\r\nmulti-objective framework allows for online adaptations
of the desired\r\nobjective. Alternatively, an automatic scalarizing procedure
yields very\r\nefficient feedback controls. Our results show that the method is
capable of\r\ndesigning driving strategies that deal better with uncertainties
in the initial\r\nconditions, which translates into potentially safer and faster
driving\r\nstrategies."
author:
- first_name: Carlos Ignacio
full_name: Hernández Castellanos, Carlos Ignacio
last_name: Hernández Castellanos
- first_name: Sina
full_name: Ober-Blöbaum, Sina
id: '16494'
last_name: Ober-Blöbaum
- first_name: Sebastian
full_name: Peitz, Sebastian
id: '47427'
last_name: Peitz
orcid: https://orcid.org/0000-0002-3389-793X
citation:
ama: Hernández Castellanos CI, Ober-Blöbaum S, Peitz S. Explicit Multi-objective
Model Predictive Control for Nonlinear Systems Under Uncertainty. International
Journal of Robust and Nonlinear Control. 2020;30(17):7593-7618. doi:10.1002/rnc.5197
apa: Hernández Castellanos, C. I., Ober-Blöbaum, S., & Peitz, S. (2020). Explicit
Multi-objective Model Predictive Control for Nonlinear Systems Under Uncertainty.
International Journal of Robust and Nonlinear Control, 30(17), 7593–7618.
https://doi.org/10.1002/rnc.5197
bibtex: '@article{Hernández Castellanos_Ober-Blöbaum_Peitz_2020, title={Explicit
Multi-objective Model Predictive Control for Nonlinear Systems Under Uncertainty},
volume={30(17)}, DOI={10.1002/rnc.5197},
journal={International Journal of Robust and Nonlinear Control}, author={Hernández
Castellanos, Carlos Ignacio and Ober-Blöbaum, Sina and Peitz, Sebastian}, year={2020},
pages={7593–7618} }'
chicago: 'Hernández Castellanos, Carlos Ignacio, Sina Ober-Blöbaum, and Sebastian
Peitz. “Explicit Multi-Objective Model Predictive Control for Nonlinear Systems
Under Uncertainty.” International Journal of Robust and Nonlinear Control
30(17) (2020): 7593–7618. https://doi.org/10.1002/rnc.5197.'
ieee: 'C. I. Hernández Castellanos, S. Ober-Blöbaum, and S. Peitz, “Explicit Multi-objective
Model Predictive Control for Nonlinear Systems Under Uncertainty,” International
Journal of Robust and Nonlinear Control, vol. 30(17), pp. 7593–7618, 2020,
doi: 10.1002/rnc.5197.'
mla: Hernández Castellanos, Carlos Ignacio, et al. “Explicit Multi-Objective Model
Predictive Control for Nonlinear Systems Under Uncertainty.” International
Journal of Robust and Nonlinear Control, vol. 30(17), 2020, pp. 7593–618,
doi:10.1002/rnc.5197.
short: C.I. Hernández Castellanos, S. Ober-Blöbaum, S. Peitz, International Journal
of Robust and Nonlinear Control 30(17) (2020) 7593–7618.
date_created: 2020-03-13T12:45:56Z
date_updated: 2022-01-21T09:55:39Z
department:
- _id: '101'
doi: 10.1002/rnc.5197
language:
- iso: eng
page: 7593-7618
publication: International Journal of Robust and Nonlinear Control
status: public
title: Explicit Multi-objective Model Predictive Control for Nonlinear Systems Under
Uncertainty
type: journal_article
user_id: '15694'
volume: 30(17)
year: '2020'
...
---
_id: '29398'
author:
- first_name: C. I. O.
full_name: Hernández Castellanos, C. I. O.
last_name: Hernández Castellanos
- first_name: G.
full_name: Schütze, G.
last_name: Schütze
- first_name: J.-Q.
full_name: Sun, J.-Q.
last_name: Sun
- first_name: Sina
full_name: Ober-Blöbaum, Sina
id: '16494'
last_name: Ober-Blöbaum
- first_name: G.
full_name: Morales-Luna, G.
last_name: Morales-Luna
citation:
ama: Hernández Castellanos CIO, Schütze G, Sun J-Q, Ober-Blöbaum S, Morales-Luna
G. Numerical computation of lightly multi-objective robust optimal solutions by
means of generalized cell mapping. Mathematics. 2020;8(11):1959.
apa: Hernández Castellanos, C. I. O., Schütze, G., Sun, J.-Q., Ober-Blöbaum, S.,
& Morales-Luna, G. (2020). Numerical computation of lightly multi-objective
robust optimal solutions by means of generalized cell mapping. Mathematics,
8(11):1959.
bibtex: '@article{Hernández Castellanos_Schütze_Sun_Ober-Blöbaum_Morales-Luna_2020,
title={Numerical computation of lightly multi-objective robust optimal solutions
by means of generalized cell mapping}, volume={8(11):1959}, journal={Mathematics},
author={Hernández Castellanos, C. I. O. and Schütze, G. and Sun, J.-Q. and Ober-Blöbaum,
Sina and Morales-Luna, G.}, year={2020} }'
chicago: Hernández Castellanos, C. I. O., G. Schütze, J.-Q. Sun, Sina Ober-Blöbaum,
and G. Morales-Luna. “Numerical Computation of Lightly Multi-Objective Robust
Optimal Solutions by Means of Generalized Cell Mapping.” Mathematics 8(11):1959
(2020).
ieee: C. I. O. Hernández Castellanos, G. Schütze, J.-Q. Sun, S. Ober-Blöbaum, and
G. Morales-Luna, “Numerical computation of lightly multi-objective robust optimal
solutions by means of generalized cell mapping,” Mathematics, vol. 8(11):1959,
2020.
mla: Hernández Castellanos, C. I. O., et al. “Numerical Computation of Lightly Multi-Objective
Robust Optimal Solutions by Means of Generalized Cell Mapping.” Mathematics,
vol. 8(11):1959, 2020.
short: C.I.O. Hernández Castellanos, G. Schütze, J.-Q. Sun, S. Ober-Blöbaum, G.
Morales-Luna, Mathematics 8(11):1959 (2020).
date_created: 2022-01-18T11:08:28Z
date_updated: 2022-01-21T09:57:35Z
department:
- _id: '636'
language:
- iso: eng
publication: Mathematics
status: public
title: Numerical computation of lightly multi-objective robust optimal solutions by
means of generalized cell mapping
type: journal_article
user_id: '15694'
volume: 8(11):1959
year: '2020'
...
---
_id: '29422'
author:
- first_name: Y.
full_name: Lishkova, Y.
last_name: Lishkova
- first_name: Sina
full_name: Ober-Blöbaum, Sina
id: '16494'
last_name: Ober-Blöbaum
- first_name: M.
full_name: Cannon, M.
last_name: Cannon
- first_name: S.
full_name: Leyendecker, S.
last_name: Leyendecker
citation:
ama: 'Lishkova Y, Ober-Blöbaum S, Cannon M, Leyendecker S. A multirate variational
approach to simulation and optimal control for flexible spacecraft. In: Accepted
for Publication in Proceedings of 2020 AAS/AIAA Astrodynamics Specialist Conference
- Lake Tahoe. ; 2020.'
apa: Lishkova, Y., Ober-Blöbaum, S., Cannon, M., & Leyendecker, S. (2020). A
multirate variational approach to simulation and optimal control for flexible
spacecraft. Accepted for Publication in Proceedings of 2020 AAS/AIAA Astrodynamics
Specialist Conference - Lake Tahoe.
bibtex: '@inproceedings{Lishkova_Ober-Blöbaum_Cannon_Leyendecker_2020, title={A
multirate variational approach to simulation and optimal control for flexible
spacecraft}, booktitle={Accepted for publication in Proceedings of 2020 AAS/AIAA
Astrodynamics Specialist Conference - Lake Tahoe}, author={Lishkova, Y. and Ober-Blöbaum,
Sina and Cannon, M. and Leyendecker, S.}, year={2020} }'
chicago: Lishkova, Y., Sina Ober-Blöbaum, M. Cannon, and S. Leyendecker. “A Multirate
Variational Approach to Simulation and Optimal Control for Flexible Spacecraft.”
In Accepted for Publication in Proceedings of 2020 AAS/AIAA Astrodynamics Specialist
Conference - Lake Tahoe, 2020.
ieee: Y. Lishkova, S. Ober-Blöbaum, M. Cannon, and S. Leyendecker, “A multirate
variational approach to simulation and optimal control for flexible spacecraft,”
2020.
mla: Lishkova, Y., et al. “A Multirate Variational Approach to Simulation and Optimal
Control for Flexible Spacecraft.” Accepted for Publication in Proceedings of
2020 AAS/AIAA Astrodynamics Specialist Conference - Lake Tahoe, 2020.
short: 'Y. Lishkova, S. Ober-Blöbaum, M. Cannon, S. Leyendecker, in: Accepted for
Publication in Proceedings of 2020 AAS/AIAA Astrodynamics Specialist Conference
- Lake Tahoe, 2020.'
date_created: 2022-01-18T14:37:53Z
date_updated: 2022-01-21T13:45:40Z
department:
- _id: '636'
language:
- iso: eng
publication: Accepted for publication in Proceedings of 2020 AAS/AIAA Astrodynamics
Specialist Conference - Lake Tahoe
status: public
title: A multirate variational approach to simulation and optimal control for flexible
spacecraft
type: conference
user_id: '15694'
year: '2020'
...
---
_id: '29423'
author:
- first_name: T.
full_name: Faulwasser, T.
last_name: Faulwasser
- first_name: K.
full_name: Flaßkamp, K.
last_name: Flaßkamp
- first_name: Sina
full_name: Ober-Blöbaum, Sina
id: '16494'
last_name: Ober-Blöbaum
- first_name: 'K. '
full_name: 'Worthmann, K. '
last_name: Worthmann
citation:
ama: 'Faulwasser T, Flaßkamp K, Ober-Blöbaum S, Worthmann K. A dissipativity characterization
of velocity turnpikes in optimal control problems for mechanical systems. In:
24th International Symposium on Mathematical Theory of Networks and Systems.
; 2020.'
apa: Faulwasser, T., Flaßkamp, K., Ober-Blöbaum, S., & Worthmann, K. (2020).
A dissipativity characterization of velocity turnpikes in optimal control problems
for mechanical systems. 24th International Symposium on Mathematical Theory
of Networks and Systems.
bibtex: '@inproceedings{Faulwasser_Flaßkamp_Ober-Blöbaum_Worthmann_2020, title={A
dissipativity characterization of velocity turnpikes in optimal control problems
for mechanical systems}, booktitle={24th International Symposium on Mathematical
Theory of Networks and Systems}, author={Faulwasser, T. and Flaßkamp, K. and Ober-Blöbaum,
Sina and Worthmann, K. }, year={2020} }'
chicago: Faulwasser, T., K. Flaßkamp, Sina Ober-Blöbaum, and K. Worthmann. “A Dissipativity
Characterization of Velocity Turnpikes in Optimal Control Problems for Mechanical
Systems.” In 24th International Symposium on Mathematical Theory of Networks
and Systems, 2020.
ieee: T. Faulwasser, K. Flaßkamp, S. Ober-Blöbaum, and K. Worthmann, “A dissipativity
characterization of velocity turnpikes in optimal control problems for mechanical
systems,” 2020.
mla: Faulwasser, T., et al. “A Dissipativity Characterization of Velocity Turnpikes
in Optimal Control Problems for Mechanical Systems.” 24th International Symposium
on Mathematical Theory of Networks and Systems, 2020.
short: 'T. Faulwasser, K. Flaßkamp, S. Ober-Blöbaum, K. Worthmann, in: 24th International
Symposium on Mathematical Theory of Networks and Systems, 2020.'
date_created: 2022-01-18T14:42:07Z
date_updated: 2022-01-21T13:46:57Z
department:
- _id: '636'
language:
- iso: eng
publication: 24th International Symposium on Mathematical Theory of Networks and Systems
status: public
title: A dissipativity characterization of velocity turnpikes in optimal control problems
for mechanical systems
type: conference
user_id: '15694'
year: '2020'
...