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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/>
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    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>--</completedDate>
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    <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>6607</id>
    <completedYear/>
    <publishedYear/>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>reportzib</type>
    <publisherName/>
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    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>--</completedDate>
    <publishedDate>2017-12-08</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">From Metastable to Coherent Sets - time-discretization schemes</title>
    <abstract language="eng">Given a time-dependent stochastic process with trajectories x(t) in a space $\Omega$, there may be sets such that the corresponding trajectories only very rarely cross the boundaries of these sets. We can analyze such a process in terms of metastability or coherence. Metastable sets M are defined in space $M\subset\Omega$, coherent sets $M(t)\subset\Omega$ are defined in space and time. Hence, if we extend the space by the time-variable t, coherent sets are metastable sets in  $\Omega\times[0,\infty]$. This relation can be exploited, because there already exist spectral algorithms for the identification of metastable sets. In this article we show that these well-established spectral algorithms (like PCCA+) 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-timediscretization scheme. The article gives an overview about different time-discretization schemes and shows their applicability in two different fields of application.</abstract>
    <identifier type="issn">1438-0064</identifier>
    <identifier type="urn">urn:nbn:de:0297-zib-66074</identifier>
    <author>Konstantin Fackeldey</author>
    <submitter>Paulina Bressel</submitter>
    <author>Péter Koltai</author>
    <author>Peter Névir</author>
    <author>Henning Rust</author>
    <author>Axel Schild</author>
    <author>Marcus Weber</author>
    <series>
      <title>ZIB-Report</title>
      <number>17-74</number>
    </series>
    <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>
    <file>https://opus4.kobv.de/opus4-zib/files/6607/ZIB-Report_17-74.pdf</file>
  </doc>
  <doc>
    <id>8537</id>
    <completedYear/>
    <publishedYear>2021</publishedYear>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>article</type>
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    <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>7014</id>
    <completedYear/>
    <publishedYear>2018</publishedYear>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst>1455</pageFirst>
    <pageLast>1485</pageLast>
    <pageNumber/>
    <edition/>
    <issue>4</issue>
    <volume>16</volume>
    <type>article</type>
    <publisherName>SIAM</publisherName>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>2018-10-02</completedDate>
    <publishedDate>--</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">A multiscale perturbation expansion approach for Markov state modeling of non-stationary molecular dynamics</title>
    <abstract language="eng">We investigate metastable dynamical systems subject to non-stationary forcing as they appear in molecular dynamics for systems driven by external fields. We show, that if the strength of the forcing is inversely proportional to the length of the slow metastable time scales of the unforced system, then the effective behavior of the forced system on slow time scales can be described by a low-dimensional reduced master equation. Our construction is explicit and uses the multiscale perturbation expansion method called two-timing, or method of multiple scales. The reduced master equation—a Markov state model—can be assembled by constructing two equilibrium Markov state models; one for the unforced system, and one for a slightly perturbed one.</abstract>
    <parentTitle language="eng">SIAM J. Multiscale Model. Simul.</parentTitle>
    <identifier type="doi">10.1137/17M1146403</identifier>
    <enrichment key="PeerReviewed">yes</enrichment>
    <enrichment key="AcceptedDate">2018-08-15</enrichment>
    <enrichment key="PreprintUrn">urn:nbn:de:0297-zib-64868</enrichment>
    <author>Péter Koltai</author>
    <submitter>Paulina Bressel</submitter>
    <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="MOL-ANALYSIS">MOL-ANALYSIS</collection>
    <collection role="projects" number="MODAL-Gesamt">MODAL-Gesamt</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>7732</id>
    <completedYear/>
    <publishedYear>2019</publishedYear>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst>4232</pageFirst>
    <pageLast>4257</pageLast>
    <pageNumber/>
    <edition/>
    <issue>11</issue>
    <volume>32</volume>
    <type>article</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>--</completedDate>
    <publishedDate>--</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Fréchet differentiable drift dependence of Perron–Frobenius and Koopman operators for non-deterministic dynamics</title>
    <abstract language="eng">We prove the Fréchet differentiability with respect to the drift of Perron–Frobenius and Koopman operators associated to time-inhomogeneous ordinary stochastic differential equations. This result relies on a similar differentiability result for pathwise expectations of path functionals of the solution of the stochastic differential equation, which we establish using Girsanov's formula. We demonstrate the significance of our result in the context of dynamical systems and operator theory, by proving continuously differentiable drift dependence of the simple eigen- and singular values and the corresponding eigen- and singular functions of the stochastic Perron–Frobenius and Koopman operators.</abstract>
    <parentTitle language="eng">Nonlinearity</parentTitle>
    <identifier type="doi">10.1088/1361-6544/ab1f2a</identifier>
    <enrichment key="PeerReviewed">yes</enrichment>
    <author>Péter Koltai</author>
    <submitter>Erlinda Körnig</submitter>
    <author>Han Cheng Lie</author>
    <author>Martin Plonka</author>
    <collection role="institutes" number="num">Numerical Mathematics</collection>
    <collection role="persons" number="lie">Lie, Han</collection>
    <collection role="projects" number="MathPlusAA1-1">MathPlusAA1-1</collection>
  </doc>
  <doc>
    <id>7969</id>
    <completedYear/>
    <publishedYear>2021</publishedYear>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue>3</issue>
    <volume>31</volume>
    <type>article</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>2021-03-02</completedDate>
    <publishedDate>--</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Transition paths of marine debris and the stability of the garbage patches</title>
    <abstract language="eng">We used transition path theory (TPT) to infer "reactive" pathways of floating marine debris trajectories. The TPT analysis was applied on a pollution-aware time-homogeneous Markov chain model constructed from trajectories produced by satellite-tracked undrogued buoys from the NOAA Global Drifter Program. The latter involved coping with the openness of the system in physical space, which further required an adaptation of the standard TPT setting. Directly connecting pollution sources along coastlines with garbage patches of varied strengths, the unveiled reactive pollution routes represent alternative targets for ocean cleanup efforts. Among our specific findings we highlight: constraining a highly probable pollution source for the Great Pacific Garbage Patch; characterizing the weakness of the Indian Ocean gyre as a trap for plastic waste; and unveiling a tendency of the subtropical gyres to export garbage toward the coastlines rather than to other gyres in the event of anomalously intense winds.</abstract>
    <parentTitle language="deu">Chaos: An Interdisciplinary Journal of Nonlinear Science</parentTitle>
    <identifier type="arxiv">2009.11234</identifier>
    <identifier type="doi">https://doi.org/10.1063/5.0030535</identifier>
    <enrichment key="PeerReviewed">yes</enrichment>
    <author>Philippe Miron</author>
    <author>Francisco Beron-Vera</author>
    <author>Luzie Helfmann</author>
    <author>Péter Koltai</author>
    <collection role="institutes" number="num">Numerical Mathematics</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>7886</id>
    <completedYear/>
    <publishedYear>2021</publishedYear>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume>31</volume>
    <type>article</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>--</completedDate>
    <publishedDate>--</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Memory-Based Reduced Modelling and Data-Based Estimation of Opinion Spreading</title>
    <abstract language="eng">We investigate opinion dynamics based on an agent-based model and are interested in predicting the evolution of the percentages of the entire agent population that share an opinion. Since these opinion percentages can be seen as an aggregated observation of the full system state, the individual opinions of each agent, we view this in the framework of the Mori–Zwanzig projection formalism. More specifically, we show how to estimate a nonlinear autoregressive model (NAR) with memory from data given by a time series of opinion percentages, and discuss its prediction capacities for various specific topologies of the agent interaction network. We demonstrate that the inclusion of memory terms significantly improves the prediction quality on examples with different network topologies.</abstract>
    <parentTitle language="eng">Journal of Nonlinear Science</parentTitle>
    <identifier type="doi">10.1007/s00332-020-09673-2</identifier>
    <enrichment key="PeerReviewed">yes</enrichment>
    <author>Niklas Wulkow</author>
    <submitter>Kalina Tsankova</submitter>
    <author>Péter 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="INNOSPREAD">INNOSPREAD</collection>
    <collection role="institutes" number="MSoCP">Modeling and Simulation of Complex Processes</collection>
  </doc>
  <doc>
    <id>6685</id>
    <completedYear/>
    <publishedYear>2018</publishedYear>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst>471</pageFirst>
    <pageLast>512</pageLast>
    <pageNumber/>
    <edition/>
    <issue>2</issue>
    <volume>28</volume>
    <type>article</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>2017-10-12</completedDate>
    <publishedDate>--</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Transition manifolds of complex metastable systems: Theory and data-driven computation of effective dynamics</title>
    <abstract language="eng">We consider complex dynamical systems showing metastable behavior but no local separation of fast and slow time scales. The article raises the question of whether such systems exhibit a low-dimensional manifold supporting its effective dynamics. For answering this question, we aim at finding nonlinear coordinates, called reaction coordinates, such that the projection of the dynamics onto these coordinates preserves the dominant time scales of the dynamics. We show that, based on a specific reducibility property, the existence of good low-dimensional reaction coordinates preserving the dominant time scales is guaranteed. Based on this theoretical framework, we develop and test a novel numerical approach for computing good reaction coordinates. The proposed algorithmic approach is fully local and thus not prone to the curse of dimension with respect to the state space of the dynamics. Hence, it is a promising method for data-based model reduction of complex dynamical systems such as molecular dynamics.</abstract>
    <parentTitle language="eng">Jounal of Nonlinear Science</parentTitle>
    <identifier type="doi">10.1007/s00332-017-9415-0</identifier>
    <enrichment key="PeerReviewed">yes</enrichment>
    <enrichment key="PreprintUrn">urn:nbn:de:0297-zib-63822</enrichment>
    <author>Andreas Bittracher</author>
    <submitter>Paulina Bressel</submitter>
    <author>Péter Koltai</author>
    <author>Stefan Klus</author>
    <author>Ralf Banisch</author>
    <author>Michael Dellnitz</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="SFB-1114-B3">SFB-1114-B3</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>
</export-example>
