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<export-example>
  <doc>
    <id>7223</id>
    <completedYear/>
    <publishedYear>2019</publishedYear>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst>012101</pageFirst>
    <pageLast>012101</pageLast>
    <pageNumber>11</pageNumber>
    <edition/>
    <issue/>
    <volume>29</volume>
    <type>article</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>--</completedDate>
    <publishedDate>--</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">From metastable to coherent sets - Time-discretization schemes</title>
    <abstract language="eng">In this article, we show that these well-established spectral algorithms (like PCCA+, Perron Cluster Cluster Analysis) also identify coherent sets of non-autonomous dynamical systems. For the identification of coherent sets, one has to compute a discretization (a matrix T) of the transfer operator of the process using a space-time-discretization scheme. The article gives an overview about different time-discretization schemes and shows their applicability in two different fields of application.</abstract>
    <parentTitle language="eng">Chaos: An Interdisciplinary Journal of Nonlinear Science</parentTitle>
    <identifier type="doi">10.1063/1.5058128</identifier>
    <enrichment key="PeerReviewed">yes</enrichment>
    <enrichment key="AcceptedDate">2019-01-28</enrichment>
    <enrichment key="PreprintUrn">urn:nbn:de:0297-zib-66074</enrichment>
    <author>Konstantin Fackeldey</author>
    <submitter>Marcus Weber</submitter>
    <author>Peter Koltai</author>
    <author>Peter Nevir</author>
    <author>Henning Rust</author>
    <author>Axel Schild</author>
    <author>Marcus Weber</author>
    <collection role="institutes" number="num">Numerical Mathematics</collection>
    <collection role="institutes" number="compmol">Computational Molecular Design</collection>
    <collection role="persons" number="fackeldey">Fackeldey, Konstantin</collection>
    <collection role="persons" number="weber">Weber, Marcus</collection>
    <collection role="projects" number="SFB1114-A5">SFB1114-A5</collection>
  </doc>
  <doc>
    <id>8537</id>
    <completedYear/>
    <publishedYear>2021</publishedYear>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>article</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>--</completedDate>
    <publishedDate>--</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Definition, detection and tracking of persistent structures in atmospheric flows</title>
    <abstract language="eng">Long-lived flow patterns in the atmosphere such as weather fronts, mid-latitude blockings or tropical cyclones often induce extreme weather conditions. As a consequence, their description, detection, and tracking has received increasing attention in recent years. Similar objectives also arise in diverse fields such as turbulence and combustion research, image analysis, and medical diagnostics under the headlines of "feature tracking", "coherent structure detection" or "image registration" - to name just a few. A host of different approaches to addressing the underlying, often very similar, tasks have been developed and successfully used. Here, several typical examples of such approaches are summarized, further developed and applied to meteorological data sets. Common abstract operational steps form the basis for a unifying framework for the specification of "persistent structures" involving the definition of the physical state of a system, the features of interest, and means of measuring their persistence.</abstract>
    <parentTitle language="eng">arXiv</parentTitle>
    <identifier type="arxiv">2111.13645</identifier>
    <enrichment key="PeerReviewed">no</enrichment>
    <enrichment key="opus.source">publish</enrichment>
    <enrichment key="opus.doi.autoCreate">false</enrichment>
    <enrichment key="opus.urn.autoCreate">true</enrichment>
    <author>Johannes von Lindheim</author>
    <submitter>Natalia Mikula</submitter>
    <author>Abhishek Harikrishnan</author>
    <author>Tom Dörffel</author>
    <author>Rupert Klein</author>
    <author>Peter Koltai</author>
    <author>Natalia Mikula</author>
    <author>Annette Müller</author>
    <author>Peter Névir</author>
    <author>George Pacey</author>
    <author>Robert Polzin</author>
    <author>Nikki Vercauteren</author>
    <collection role="institutes" number="vis">Visual Data Analysis</collection>
    <collection role="persons" number="ernst">Ernst, Natalia</collection>
    <collection role="institutes" number="VDcC">Visual and Data-centric Computing</collection>
    <collection role="projects" number="SFB1114-C06">SFB1114-C06</collection>
  </doc>
  <doc>
    <id>7718</id>
    <completedYear/>
    <publishedYear>2020</publishedYear>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst>3321</pageFirst>
    <pageLast>3366</pageLast>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume>30</volume>
    <type>article</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>2020-09-10</completedDate>
    <publishedDate>2020-09-10</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Extending Transition Path Theory: Periodically Driven and Finite-Time Dynamics</title>
    <parentTitle language="deu">Journal of Nonlinear Science</parentTitle>
    <identifier type="doi">https://doi.org/10.1007/s00332-020-09652-7</identifier>
    <enrichment key="PeerReviewed">yes</enrichment>
    <enrichment key="AcceptedDate">2020-08-23</enrichment>
    <author>Luzie Helfmann</author>
    <submitter>Erlinda Körnig</submitter>
    <author>Enric Ribera Borrell</author>
    <author>Christof Schütte</author>
    <author>Peter Koltai</author>
    <collection role="institutes" number="num">Numerical Mathematics</collection>
    <collection role="persons" number="schuette">Schütte, Christof</collection>
    <collection role="persons" number="ribera.borrell">Ribera Borrell, Enric</collection>
    <collection role="institutes" number="MSoCP">Modeling and Simulation of Complex Processes</collection>
    <collection role="projects" number="tipping">Stability and Tipping in Social Systems</collection>
  </doc>
  <doc>
    <id>6702</id>
    <completedYear/>
    <publishedYear>2018</publishedYear>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber>22</pageNumber>
    <edition/>
    <issue>1</issue>
    <volume>6</volume>
    <type>article</type>
    <publisherName>MDPI</publisherName>
    <publisherPlace>Basel, Switzerland</publisherPlace>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>--</completedDate>
    <publishedDate>--</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Optimal data-driven estimation of generalized Markov state models for non-equilibrium dynamics</title>
    <parentTitle language="eng">Computation</parentTitle>
    <identifier type="doi">10.3390/computation6010022</identifier>
    <enrichment key="PeerReviewed">yes</enrichment>
    <enrichment key="FulltextUrl">http://www.mdpi.com/2079-3197/6/1/22</enrichment>
    <author>Peter Koltai</author>
    <submitter>Erlinda Koernig</submitter>
    <author>Hao Wu</author>
    <author>Frank Noé</author>
    <author>Christof Schütte</author>
    <collection role="institutes" number="num">Numerical Mathematics</collection>
    <collection role="persons" number="schuette">Schütte, Christof</collection>
    <collection role="projects" number="MODAL-MedLab">MODAL-MedLab</collection>
    <collection role="projects" number="SFB1114-A5">SFB1114-A5</collection>
    <collection role="projects" number="MODAL-Gesamt">MODAL-Gesamt</collection>
  </doc>
  <doc>
    <id>6250</id>
    <completedYear/>
    <publishedYear>2016</publishedYear>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst>51</pageFirst>
    <pageLast>77</pageLast>
    <pageNumber/>
    <edition/>
    <issue>1</issue>
    <volume>3</volume>
    <type>article</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>--</completedDate>
    <publishedDate>--</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">On the numerical approximation of the Perron-Frobenius and Koopman operator</title>
    <abstract language="eng">Information about the behavior of dynamical systems can often be obtained by analyzing the eigenvalues and corresponding eigenfunctions of linear operators associated with a dynamical system. Examples of such operators are the Perron-Frobenius and the Koopman operator. In this paper, we will review di� fferent methods that have been developed over the last decades to compute � infinite-dimensional approximations of these in� finite-dimensional operators - in particular Ulam's method and Extended Dynamic Mode Decomposition (EDMD) - and highlight the similarities and di� fferences between these approaches. The results will be illustrated using simple stochastic di� fferential equations and molecular dynamics examples.</abstract>
    <parentTitle language="eng">Journal of Computational Dynamics</parentTitle>
    <identifier type="doi">10.3934/jcd.2016003</identifier>
    <enrichment key="PeerReviewed">yes</enrichment>
    <author>Stefan Klus</author>
    <submitter>Erlinda Körnig</submitter>
    <author>Peter Koltai</author>
    <author>Christof Schütte</author>
    <collection role="institutes" number="num">Numerical Mathematics</collection>
    <collection role="persons" number="schuette">Schütte, Christof</collection>
    <collection role="projects" number="ECMath-CH6">ECMath-CH6</collection>
    <collection role="projects" number="MODAL-MedLab">MODAL-MedLab</collection>
    <collection role="projects" number="SFB1114-A5">SFB1114-A5</collection>
    <collection role="projects" number="MODAL-Gesamt">MODAL-Gesamt</collection>
  </doc>
  <doc>
    <id>6267</id>
    <completedYear/>
    <publishedYear>2018</publishedYear>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst>985</pageFirst>
    <pageLast>1010</pageLast>
    <pageNumber/>
    <edition/>
    <issue>3</issue>
    <volume>28</volume>
    <type>article</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>2018-01-03</completedDate>
    <publishedDate>--</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Data-driven model reduction and transfer operator approximation</title>
    <parentTitle language="eng">Journal of Nonlinear Science</parentTitle>
    <identifier type="url">https://link.springer.com/article/10.1007/s00332-017-9437-7</identifier>
    <identifier type="doi">10.1007/s00332-017-9437-7</identifier>
    <enrichment key="PeerReviewed">yes</enrichment>
    <author>Stefan Klus</author>
    <submitter>Erlinda Koernig</submitter>
    <author>Feliks Nüske</author>
    <author>Peter Koltai</author>
    <author>Hao Wu</author>
    <author>Ioannis Kevrekidis</author>
    <author>Christof Schütte</author>
    <author>Frank Noé</author>
    <collection role="institutes" number="num">Numerical Mathematics</collection>
    <collection role="persons" number="schuette">Schütte, Christof</collection>
    <collection role="projects" number="MODAL-MedLab">MODAL-MedLab</collection>
    <collection role="projects" number="SFB1114-A5">SFB1114-A5</collection>
    <collection role="projects" number="MODAL-Gesamt">MODAL-Gesamt</collection>
  </doc>
  <doc>
    <id>6340</id>
    <completedYear/>
    <publishedYear>2016</publishedYear>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue>174103</issue>
    <volume>145</volume>
    <type>article</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>--</completedDate>
    <publishedDate>--</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">On Markov state models for non-equilibrium molecular dynamics</title>
    <parentTitle language="eng">The Journal of Chemical Physics</parentTitle>
    <identifier type="doi">10.1063/1.4966157</identifier>
    <note>2016 Editor's Choice of The Journal of Chemical Physics</note>
    <enrichment key="PeerReviewed">yes</enrichment>
    <enrichment key="PreprintUrn">urn:nbn:de:0297-zib-57869</enrichment>
    <author>Peter Koltai</author>
    <submitter>Erlinda Körnig</submitter>
    <author>Giovanni Ciccotti</author>
    <author>Christof Schütte</author>
    <collection role="institutes" number="num">Numerical Mathematics</collection>
    <collection role="persons" number="schuette">Schütte, Christof</collection>
    <collection role="projects" number="MODAL-MedLab">MODAL-MedLab</collection>
    <collection role="projects" number="MODAL-Gesamt">MODAL-Gesamt</collection>
  </doc>
  <doc>
    <id>5786</id>
    <completedYear/>
    <publishedYear/>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition>145</edition>
    <issue/>
    <volume>174103</volume>
    <type>reportzib</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>--</completedDate>
    <publishedDate>2016-03-16</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">On metastability and Markov state models for non-stationary molecular dynamics</title>
    <abstract language="eng">We utilize the theory of coherent sets to build Markov state models for non- equilibrium molecular dynamical systems. Unlike for systems in equilibrium, “meta- stable” sets in the non-equilibrium case may move as time evolves. We formalize this concept by relying on the theory of coherent sets, based on this we derive finite-time non-stationary Markov state models, and illustrate the concept and its main differences to equilibrium Markov state modeling on simple, one-dimensional examples.</abstract>
    <parentTitle language="eng">The Journal of Chemical Physics</parentTitle>
    <subTitle language="deu">2016 Editor's Choice of The Journal of Chemical Physics</subTitle>
    <identifier type="issn">1438-0064</identifier>
    <identifier type="urn">urn:nbn:de:0297-zib-57869</identifier>
    <identifier type="doi">10.1063/1.4966157</identifier>
    <enrichment key="PeerReviewed">yes</enrichment>
    <enrichment key="SourceTitle">Appeared in: The Journal of Chemical Physics 145 (2016)</enrichment>
    <author>Peter Koltai</author>
    <submitter>Erlinda Koernig</submitter>
    <author>Giovanni Ciccotti</author>
    <author>Christof Schütte</author>
    <series>
      <title>ZIB-Report</title>
      <number>16-11</number>
    </series>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>coherent set,</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Markov state model</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>non-equilibrium molecular dynamics</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>metastability</value>
    </subject>
    <collection role="msc" number="60J20">Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.) [See also 90B30, 91D10, 91D35, 91E40]</collection>
    <collection role="msc" number="60J35">Transition functions, generators and resolvents [See also 47D03, 47D07]</collection>
    <collection role="msc" number="60J60">Diffusion processes [See also 58J65]</collection>
    <collection role="institutes" number="num">Numerical Mathematics</collection>
    <collection role="persons" number="schuette">Schütte, Christof</collection>
    <collection role="projects" number="MODAL-MedLab">MODAL-MedLab</collection>
    <collection role="projects" number="no-project">no-project</collection>
    <collection role="projects" number="MODAL-Gesamt">MODAL-Gesamt</collection>
    <file>https://opus4.kobv.de/opus4-zib/files/5786/ZIB_16-11_new.pdf</file>
  </doc>
  <doc>
    <id>9659</id>
    <completedYear/>
    <publishedYear>2022</publishedYear>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>conferenceobject</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>--</completedDate>
    <publishedDate>--</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Discovering collective variable dynamics of agent-based models</title>
    <abstract language="eng">Analytical approximations of the macroscopic behavior of agent-based models (e.g.&#13;
via mean-field theory) often introduce a significant error, especially in the transient phase. For an example model called continuous-time noisy voter model, we use two data-driven approaches to learn the evolution of collective variables instead. The first approach utilizes the SINDy method to approximate the macroscopic dynamics without prior knowledge, but has proven itself to be not particularly robust. The second approach employs an informed learning strategy which includes knowledge about the agent-based model. Both approaches exhibit a considerably smaller error than the conventional analytical approximation.</abstract>
    <parentTitle language="eng">25th International Symposium on Mathematical Theory of Networks and Systems MTNS 2022</parentTitle>
    <identifier type="doi">https://doi.org/10.15495/EPub_UBT_00006809</identifier>
    <enrichment key="PeerReviewed">no</enrichment>
    <enrichment key="opus.source">publish</enrichment>
    <author>Marvin Lücke</author>
    <submitter>Stefanie Winkelmann</submitter>
    <author>Peter Koltai</author>
    <author>Stefanie Winkelmann</author>
    <author>Nora Molkethin</author>
    <author>Jobst Heitzig</author>
    <collection role="persons" number="winkelmann">Winkelmann, Stefanie</collection>
    <collection role="institutes" number="MSoCP">Modeling and Simulation of Complex Processes</collection>
    <collection role="persons" number="luecke">Lücke, Marvin</collection>
    <collection role="projects" number="MathPlusEF4-8">MathPlusEF4-8</collection>
  </doc>
  <doc>
    <id>10257</id>
    <completedYear/>
    <publishedYear>2026</publishedYear>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber>034311</pageNumber>
    <edition/>
    <issue/>
    <volume>113</volume>
    <type>article</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>2026-03-18</completedDate>
    <publishedDate>--</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Accurate mean-field equation for voter model dynamics on scale-free networks</title>
    <abstract language="eng">Understanding the emergent macroscopic behavior of dynamical systems on networks is a crucial but challenging task. One of the simplest and most effective methods to construct a reduced macroscopic model is given by mean-field theory. The resulting approximations perform well on dense and homogeneous networks but poorly on scale-free networks, which, however, are more realistic in many applications. In this paper, we introduce a modified version of the mean-field approximation for voter model dynamics on scale-free networks. The two main deviations from classical theory are that we use degree-weighted shares as coarse variables and that we introduce a correlation factor that can be interpreted as slowing down dynamics induced by interactions. We observe that the correlation factor is only a property of the network and not of the state or of parameters of the process. This approach achieves a significantly smaller approximation error than standard methods without increasing dimensionality.</abstract>
    <parentTitle language="eng">Physical Review E</parentTitle>
    <identifier type="arxiv">2509.13485</identifier>
    <identifier type="doi">10.1103/vkpx-5cvt</identifier>
    <enrichment key="opus.source">publish</enrichment>
    <enrichment key="PeerReviewed">yes</enrichment>
    <enrichment key="AcceptedDate">2026-02-24</enrichment>
    <author>Marvin Lücke</author>
    <submitter>Stefanie Winkelmann</submitter>
    <author>Stefanie Winkelmann</author>
    <author>Peter Koltai</author>
    <collection role="institutes" number="num">Numerical Mathematics</collection>
    <collection role="institutes" number="compsys">Computational Systems Biology</collection>
    <collection role="persons" number="winkelmann">Winkelmann, Stefanie</collection>
    <collection role="institutes" number="MSoCP">Modeling and Simulation of Complex Processes</collection>
    <collection role="persons" number="luecke">Lücke, Marvin</collection>
    <collection role="projects" number="DFG-CollectiveVariables">DFG-CollectiveVariables</collection>
  </doc>
</export-example>
