---
_id: '58811'
abstract:
- lang: eng
  text: |-
    <jats:title>Abstract</jats:title>
              <jats:p>Fractional dissipation is a powerful tool to study nonlocal physical phenomena such as damping models. The design of geometric, in particular, variational integrators for the numerical simulation of such systems relies on a variational formulation of the model. In Jiménez and Ober-Blöbaum (J Nonlinear Sci 31:46, 2021), a new approach is proposed to deal with dissipative systems including fractionally damped systems in a variational way for both, the continuous and discrete setting. It is based on the doubling of variables and their fractional derivatives. The aim of this work is to derive higher-order fractional variational integrators by means of convolution quadrature (CQ) based on backward difference formulas. We then provide numerical methods that are of order 2 improving a previous result in Jiménez and Ober-Blöbaum (J Nonlinear Sci 31:46, 2021). The convergence properties of the fractional variational integrators and saturation effects due to the approximation of the fractional derivatives by CQ are studied numerically.</jats:p>
article_number: '38'
author:
- first_name: Khaled
  full_name: Hariz Belgacem, Khaled
  last_name: Hariz Belgacem
- first_name: Fernando
  full_name: Jiménez, Fernando
  last_name: Jiménez
- first_name: Sina
  full_name: Ober-Blöbaum, Sina
  last_name: Ober-Blöbaum
citation:
  ama: Hariz Belgacem K, Jiménez F, Ober-Blöbaum S. Fractional Variational Integrators
    Based on Convolution Quadrature. <i>Journal of Nonlinear Science</i>. 2025;35(2).
    doi:<a href="https://doi.org/10.1007/s00332-025-10131-0">10.1007/s00332-025-10131-0</a>
  apa: Hariz Belgacem, K., Jiménez, F., &#38; Ober-Blöbaum, S. (2025). Fractional
    Variational Integrators Based on Convolution Quadrature. <i>Journal of Nonlinear
    Science</i>, <i>35</i>(2), Article 38. <a href="https://doi.org/10.1007/s00332-025-10131-0">https://doi.org/10.1007/s00332-025-10131-0</a>
  bibtex: '@article{Hariz Belgacem_Jiménez_Ober-Blöbaum_2025, title={Fractional Variational
    Integrators Based on Convolution Quadrature}, volume={35}, DOI={<a href="https://doi.org/10.1007/s00332-025-10131-0">10.1007/s00332-025-10131-0</a>},
    number={238}, journal={Journal of Nonlinear Science}, publisher={Springer Science
    and Business Media LLC}, author={Hariz Belgacem, Khaled and Jiménez, Fernando
    and Ober-Blöbaum, Sina}, year={2025} }'
  chicago: Hariz Belgacem, Khaled, Fernando Jiménez, and Sina Ober-Blöbaum. “Fractional
    Variational Integrators Based on Convolution Quadrature.” <i>Journal of Nonlinear
    Science</i> 35, no. 2 (2025). <a href="https://doi.org/10.1007/s00332-025-10131-0">https://doi.org/10.1007/s00332-025-10131-0</a>.
  ieee: 'K. Hariz Belgacem, F. Jiménez, and S. Ober-Blöbaum, “Fractional Variational
    Integrators Based on Convolution Quadrature,” <i>Journal of Nonlinear Science</i>,
    vol. 35, no. 2, Art. no. 38, 2025, doi: <a href="https://doi.org/10.1007/s00332-025-10131-0">10.1007/s00332-025-10131-0</a>.'
  mla: Hariz Belgacem, Khaled, et al. “Fractional Variational Integrators Based on
    Convolution Quadrature.” <i>Journal of Nonlinear Science</i>, vol. 35, no. 2,
    38, Springer Science and Business Media LLC, 2025, doi:<a href="https://doi.org/10.1007/s00332-025-10131-0">10.1007/s00332-025-10131-0</a>.
  short: K. Hariz Belgacem, F. Jiménez, S. Ober-Blöbaum, Journal of Nonlinear Science
    35 (2025).
date_created: 2025-02-24T11:52:16Z
date_updated: 2025-02-24T11:55:58Z
doi: 10.1007/s00332-025-10131-0
intvolume: '        35'
issue: '2'
language:
- iso: eng
publication: Journal of Nonlinear Science
publication_identifier:
  issn:
  - 0938-8974
  - 1432-1467
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: Fractional Variational Integrators Based on Convolution Quadrature
type: journal_article
user_id: '98857'
volume: 35
year: '2025'
...
---
_id: '63329'
abstract:
- lang: eng
  text: "<jats:title>Abstract</jats:title><jats:p>Recent experimental work has revealed
    that interstitial fluid flow can mobilize two types of tumor cell migration mechanisms.
    One is a chemotactic-driven mechanism where chemokine (chemical component) bounded
    to the extracellular matrix (ECM) is released and skewed in the flow direction.
    This leads to higher chemical concentrations downstream which the tumor cells
    can sense and migrate toward. The other is a mechanism where the flowing fluid
    imposes a stress on the tumor cells which triggers them to go in the upstream
    direction. Researchers have suggested that these two migration modes possibly
    can play a role in metastatic behavior, i.e., the process where tumor cells are
    able to break loose from the primary tumor and move to nearby lymphatic vessels.
    In Waldeland and Evje (J Biomech 81:22–35, 2018), a mathematical cell–fluid model
    was put forward based on a mixture theory formulation. It was demonstrated that
    the model was able to capture the main characteristics of the two competing migration
    mechanisms. The objective of the current work is to seek deeper insight into certain
    qualitative aspects of these competing mechanisms by means of mathematical methods.
    For that purpose, we propose a simpler version of the cell–fluid model mentioned
    above but such that the two competing migration mechanisms are retained. An initial
    cell distribution in a one-dimensional slab is exposed to a constant fluid flow
    from one end to the other, consistent with the experimental setup. Then, we explore
    by means of analytical estimates the long-time behavior of the two competing migration
    mechanisms for two different scenarios: (i) when the initial cell volume fraction
    is low and (ii) when the initial cell volume fraction is high. In particular,
    it is demonstrated in a strict mathematical sense that for a sufficiently low
    initial cell volume fraction, the downstream migration dominates in the sense
    that the solution converges to a downstream-dominated steady state as time elapses.
    On the other hand, with a sufficiently high initial cell volume fraction, the
    upstream migration mechanism is the stronger in the sense that the solution converges
    to an upstream-dominated steady state.\r\n</jats:p>"
author:
- first_name: Steinar
  full_name: Evje, Steinar
  last_name: Evje
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Evje S, Winkler M. Mathematical Analysis of Two Competing Cancer Cell Migration
    Mechanisms Driven by Interstitial Fluid Flow. <i>Journal of Nonlinear Science</i>.
    2020;30(4):1809-1847. doi:<a href="https://doi.org/10.1007/s00332-020-09625-w">10.1007/s00332-020-09625-w</a>
  apa: Evje, S., &#38; Winkler, M. (2020). Mathematical Analysis of Two Competing
    Cancer Cell Migration Mechanisms Driven by Interstitial Fluid Flow. <i>Journal
    of Nonlinear Science</i>, <i>30</i>(4), 1809–1847. <a href="https://doi.org/10.1007/s00332-020-09625-w">https://doi.org/10.1007/s00332-020-09625-w</a>
  bibtex: '@article{Evje_Winkler_2020, title={Mathematical Analysis of Two Competing
    Cancer Cell Migration Mechanisms Driven by Interstitial Fluid Flow}, volume={30},
    DOI={<a href="https://doi.org/10.1007/s00332-020-09625-w">10.1007/s00332-020-09625-w</a>},
    number={4}, journal={Journal of Nonlinear Science}, publisher={Springer Science
    and Business Media LLC}, author={Evje, Steinar and Winkler, Michael}, year={2020},
    pages={1809–1847} }'
  chicago: 'Evje, Steinar, and Michael Winkler. “Mathematical Analysis of Two Competing
    Cancer Cell Migration Mechanisms Driven by Interstitial Fluid Flow.” <i>Journal
    of Nonlinear Science</i> 30, no. 4 (2020): 1809–47. <a href="https://doi.org/10.1007/s00332-020-09625-w">https://doi.org/10.1007/s00332-020-09625-w</a>.'
  ieee: 'S. Evje and M. Winkler, “Mathematical Analysis of Two Competing Cancer Cell
    Migration Mechanisms Driven by Interstitial Fluid Flow,” <i>Journal of Nonlinear
    Science</i>, vol. 30, no. 4, pp. 1809–1847, 2020, doi: <a href="https://doi.org/10.1007/s00332-020-09625-w">10.1007/s00332-020-09625-w</a>.'
  mla: Evje, Steinar, and Michael Winkler. “Mathematical Analysis of Two Competing
    Cancer Cell Migration Mechanisms Driven by Interstitial Fluid Flow.” <i>Journal
    of Nonlinear Science</i>, vol. 30, no. 4, Springer Science and Business Media
    LLC, 2020, pp. 1809–47, doi:<a href="https://doi.org/10.1007/s00332-020-09625-w">10.1007/s00332-020-09625-w</a>.
  short: S. Evje, M. Winkler, Journal of Nonlinear Science 30 (2020) 1809–1847.
date_created: 2025-12-18T19:38:02Z
date_updated: 2025-12-18T20:00:26Z
doi: 10.1007/s00332-020-09625-w
intvolume: '        30'
issue: '4'
language:
- iso: eng
page: 1809-1847
publication: Journal of Nonlinear Science
publication_identifier:
  issn:
  - 0938-8974
  - 1432-1467
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: Mathematical Analysis of Two Competing Cancer Cell Migration Mechanisms Driven
  by Interstitial Fluid Flow
type: journal_article
user_id: '31496'
volume: 30
year: '2020'
...
---
_id: '21941'
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: Péter
  full_name: Koltai, Péter
  last_name: Koltai
- first_name: Hao
  full_name: Wu, Hao
  last_name: Wu
- first_name: Ioannis
  full_name: Kevrekidis, Ioannis
  last_name: Kevrekidis
- first_name: Christof
  full_name: Schütte, Christof
  last_name: Schütte
- first_name: Frank
  full_name: Noé, Frank
  last_name: Noé
citation:
  ama: Klus S, Nüske F, Koltai P, et al. Data-Driven Model Reduction and Transfer
    Operator Approximation. <i>Journal of Nonlinear Science</i>. 2018:985-1010. doi:<a
    href="https://doi.org/10.1007/s00332-017-9437-7">10.1007/s00332-017-9437-7</a>
  apa: Klus, S., Nüske, F., Koltai, P., Wu, H., Kevrekidis, I., Schütte, C., &#38;
    Noé, F. (2018). Data-Driven Model Reduction and Transfer Operator Approximation.
    <i>Journal of Nonlinear Science</i>, 985–1010. <a href="https://doi.org/10.1007/s00332-017-9437-7">https://doi.org/10.1007/s00332-017-9437-7</a>
  bibtex: '@article{Klus_Nüske_Koltai_Wu_Kevrekidis_Schütte_Noé_2018, title={Data-Driven
    Model Reduction and Transfer Operator Approximation}, DOI={<a href="https://doi.org/10.1007/s00332-017-9437-7">10.1007/s00332-017-9437-7</a>},
    journal={Journal of Nonlinear Science}, author={Klus, Stefan and Nüske, Feliks
    and Koltai, Péter and Wu, Hao and Kevrekidis, Ioannis and Schütte, Christof and
    Noé, Frank}, year={2018}, pages={985–1010} }'
  chicago: Klus, Stefan, Feliks Nüske, Péter Koltai, Hao Wu, Ioannis Kevrekidis, Christof
    Schütte, and Frank Noé. “Data-Driven Model Reduction and Transfer Operator Approximation.”
    <i>Journal of Nonlinear Science</i>, 2018, 985–1010. <a href="https://doi.org/10.1007/s00332-017-9437-7">https://doi.org/10.1007/s00332-017-9437-7</a>.
  ieee: S. Klus <i>et al.</i>, “Data-Driven Model Reduction and Transfer Operator
    Approximation,” <i>Journal of Nonlinear Science</i>, pp. 985–1010, 2018.
  mla: Klus, Stefan, et al. “Data-Driven Model Reduction and Transfer Operator Approximation.”
    <i>Journal of Nonlinear Science</i>, 2018, pp. 985–1010, doi:<a href="https://doi.org/10.1007/s00332-017-9437-7">10.1007/s00332-017-9437-7</a>.
  short: S. Klus, F. Nüske, P. Koltai, H. Wu, I. Kevrekidis, C. Schütte, F. Noé, Journal
    of Nonlinear Science (2018) 985–1010.
date_created: 2021-04-30T16:59:03Z
date_updated: 2022-01-06T06:55:20Z
department:
- _id: '101'
doi: 10.1007/s00332-017-9437-7
extern: '1'
language:
- iso: eng
page: 985-1010
publication: Journal of Nonlinear Science
publication_identifier:
  issn:
  - 0938-8974
  - 1432-1467
publication_status: published
status: public
title: Data-Driven Model Reduction and Transfer Operator Approximation
type: journal_article
user_id: '81513'
year: '2018'
...
---
_id: '16715'
author:
- first_name: Andreas
  full_name: Bittracher, Andreas
  last_name: Bittracher
- first_name: Péter
  full_name: Koltai, Péter
  last_name: Koltai
- first_name: Stefan
  full_name: Klus, Stefan
  last_name: Klus
- first_name: Ralf
  full_name: Banisch, Ralf
  last_name: Banisch
- first_name: Michael
  full_name: Dellnitz, Michael
  last_name: Dellnitz
- first_name: Christof
  full_name: Schütte, Christof
  last_name: Schütte
citation:
  ama: Bittracher A, Koltai P, Klus S, Banisch R, Dellnitz M, Schütte C. Transition
    Manifolds of Complex Metastable Systems. <i>Journal of Nonlinear Science</i>.
    2018;28:471-512. doi:<a href="https://doi.org/10.1007/s00332-017-9415-0">10.1007/s00332-017-9415-0</a>
  apa: Bittracher, A., Koltai, P., Klus, S., Banisch, R., Dellnitz, M., &#38; Schütte,
    C. (2018). Transition Manifolds of Complex Metastable Systems. <i>Journal of Nonlinear
    Science</i>, <i>28</i>, 471–512. <a href="https://doi.org/10.1007/s00332-017-9415-0">https://doi.org/10.1007/s00332-017-9415-0</a>
  bibtex: '@article{Bittracher_Koltai_Klus_Banisch_Dellnitz_Schütte_2018, title={Transition
    Manifolds of Complex Metastable Systems}, volume={28}, DOI={<a href="https://doi.org/10.1007/s00332-017-9415-0">10.1007/s00332-017-9415-0</a>},
    journal={Journal of Nonlinear Science}, author={Bittracher, Andreas and Koltai,
    Péter and Klus, Stefan and Banisch, Ralf and Dellnitz, Michael and Schütte, Christof},
    year={2018}, pages={471–512} }'
  chicago: 'Bittracher, Andreas, Péter Koltai, Stefan Klus, Ralf Banisch, Michael
    Dellnitz, and Christof Schütte. “Transition Manifolds of Complex Metastable Systems.”
    <i>Journal of Nonlinear Science</i> 28 (2018): 471–512. <a href="https://doi.org/10.1007/s00332-017-9415-0">https://doi.org/10.1007/s00332-017-9415-0</a>.'
  ieee: A. Bittracher, P. Koltai, S. Klus, R. Banisch, M. Dellnitz, and C. Schütte,
    “Transition Manifolds of Complex Metastable Systems,” <i>Journal of Nonlinear
    Science</i>, vol. 28, pp. 471–512, 2018.
  mla: Bittracher, Andreas, et al. “Transition Manifolds of Complex Metastable Systems.”
    <i>Journal of Nonlinear Science</i>, vol. 28, 2018, pp. 471–512, doi:<a href="https://doi.org/10.1007/s00332-017-9415-0">10.1007/s00332-017-9415-0</a>.
  short: A. Bittracher, P. Koltai, S. Klus, R. Banisch, M. Dellnitz, C. Schütte, Journal
    of Nonlinear Science 28 (2018) 471–512.
date_created: 2020-04-16T14:09:31Z
date_updated: 2022-01-06T06:52:55Z
department:
- _id: '101'
doi: 10.1007/s00332-017-9415-0
intvolume: '        28'
language:
- iso: eng
page: 471-512
publication: Journal of Nonlinear Science
publication_identifier:
  issn:
  - 0938-8974
  - 1432-1467
publication_status: published
status: public
title: Transition Manifolds of Complex Metastable Systems
type: journal_article
user_id: '47427'
volume: 28
year: '2018'
...
---
_id: '16592'
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: Marin
  full_name: Kobilarov, Marin
  last_name: Kobilarov
citation:
  ama: Flaßkamp K, Ober-Blöbaum S, Kobilarov M. Solving Optimal Control Problems by
    Exploiting Inherent Dynamical Systems Structures. <i>Journal of Nonlinear Science</i>.
    2012:599-629. doi:<a href="https://doi.org/10.1007/s00332-012-9140-7">10.1007/s00332-012-9140-7</a>
  apa: Flaßkamp, K., Ober-Blöbaum, S., &#38; Kobilarov, M. (2012). Solving Optimal
    Control Problems by Exploiting Inherent Dynamical Systems Structures. <i>Journal
    of Nonlinear Science</i>, 599–629. <a href="https://doi.org/10.1007/s00332-012-9140-7">https://doi.org/10.1007/s00332-012-9140-7</a>
  bibtex: '@article{Flaßkamp_Ober-Blöbaum_Kobilarov_2012, title={Solving Optimal Control
    Problems by Exploiting Inherent Dynamical Systems Structures}, DOI={<a href="https://doi.org/10.1007/s00332-012-9140-7">10.1007/s00332-012-9140-7</a>},
    journal={Journal of Nonlinear Science}, author={Flaßkamp, Kathrin and Ober-Blöbaum,
    Sina and Kobilarov, Marin}, year={2012}, pages={599–629} }'
  chicago: Flaßkamp, Kathrin, Sina Ober-Blöbaum, and Marin Kobilarov. “Solving Optimal
    Control Problems by Exploiting Inherent Dynamical Systems Structures.” <i>Journal
    of Nonlinear Science</i>, 2012, 599–629. <a href="https://doi.org/10.1007/s00332-012-9140-7">https://doi.org/10.1007/s00332-012-9140-7</a>.
  ieee: K. Flaßkamp, S. Ober-Blöbaum, and M. Kobilarov, “Solving Optimal Control Problems
    by Exploiting Inherent Dynamical Systems Structures,” <i>Journal of Nonlinear
    Science</i>, pp. 599–629, 2012.
  mla: Flaßkamp, Kathrin, et al. “Solving Optimal Control Problems by Exploiting Inherent
    Dynamical Systems Structures.” <i>Journal of Nonlinear Science</i>, 2012, pp.
    599–629, doi:<a href="https://doi.org/10.1007/s00332-012-9140-7">10.1007/s00332-012-9140-7</a>.
  short: K. Flaßkamp, S. Ober-Blöbaum, M. Kobilarov, Journal of Nonlinear Science
    (2012) 599–629.
date_created: 2020-04-16T05:53:07Z
date_updated: 2022-01-06T06:52:52Z
department:
- _id: '101'
doi: 10.1007/s00332-012-9140-7
language:
- iso: eng
page: 599-629
publication: Journal of Nonlinear Science
publication_identifier:
  issn:
  - 0938-8974
  - 1432-1467
publication_status: published
status: public
title: Solving Optimal Control Problems by Exploiting Inherent Dynamical Systems Structures
type: journal_article
user_id: '15701'
year: '2012'
...