<?xml version="1.0" encoding="utf-8"?>
<export-example>
  <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>6382</id>
    <completedYear/>
    <publishedYear/>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>reportzib</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>--</completedDate>
    <publishedDate>2017-05-03</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&#13;
separation of fast and slow time scales. The article raises the question of whether&#13;
such systems exhibit a low-dimensional manifold supporting its effective dynamics.&#13;
For answering this question, we aim at finding nonlinear coordinates, called reaction&#13;
coordinates, such that the projection of the dynamics onto these coordinates preserves&#13;
the dominant time scales of the dynamics. We show that, based on a specific&#13;
reducibility property, the existence of good low-dimensional reaction coordinates&#13;
preserving the dominant time scales is guaranteed. Based on this theoretical framework,&#13;
we develop and test a novel numerical approach for computing good reaction&#13;
coordinates. The proposed algorithmic approach is fully local and thus not prone to&#13;
the curse of dimension with respect to the state space of the dynamics. Hence, it is&#13;
a promising method for data-based model reduction of complex dynamical systems&#13;
such as molecular dynamics.</abstract>
    <identifier type="issn">1438-0064</identifier>
    <identifier type="urn">urn:nbn:de:0297-zib-63822</identifier>
    <enrichment key="PeerReviewed">yes</enrichment>
    <enrichment key="SubmissionStatus">accepted for publication</enrichment>
    <enrichment key="AcceptedDate">2017-09-23</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>
    <series>
      <title>ZIB-Report</title>
      <number>17-22</number>
    </series>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>metastability</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>slow dynamics</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>effective dynamics</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>transition manifold</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>embedding</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>transfer operator</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>reaction coordinate</value>
    </subject>
    <collection role="msc" number="47B38">Operators on function spaces (general)</collection>
    <collection role="msc" number="60H35">Computational methods for stochastic equations [See also 65C30]</collection>
    <collection role="msc" number="82C31">Stochastic methods (Fokker-Planck, Langevin, etc.) [See also 60H10]</collection>
    <collection role="institutes" number="num">Numerical Mathematics</collection>
    <collection role="persons" number="schuette">Schütte, Christof</collection>
    <collection role="projects" number="SFB1114-C3">SFB1114-C3</collection>
    <file>https://opus4.kobv.de/opus4-zib/files/6382/ZIB-Report_17-22.pdf</file>
  </doc>
  <doc>
    <id>8436</id>
    <completedYear/>
    <publishedYear>2023</publishedYear>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst>449</pageFirst>
    <pageLast>488</pageLast>
    <pageNumber/>
    <edition/>
    <issue>2</issue>
    <volume>21</volume>
    <type>article</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>--</completedDate>
    <publishedDate>--</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Optimal Reaction Coordinates: Variational Characterization and Sparse Computation</title>
    <abstract language="eng">Reaction coordinates (RCs) are indicators of hidden, low-dimensional mechanisms that govern the long-term behavior of high-dimensional stochastic processes. We present a novel and general variational characterization of optimal RCs and provide conditions for their existence. Optimal RCs are minimizers of a certain loss function, and reduced models based on them guarantee a good approximation of the statistical long-term properties of the original high-dimensional process. We show that for slow-fast systems, metastable systems, and other systems with known good RCs, the novel theory reproduces previous insight. Remarkably, for reversible systems, the numerical effort required to evaluate the loss function scales only with the variability of the underlying, low-dimensional mechanism, and not with that of the full system. The theory provided lays the foundation for an efficient and data-sparse computation of RCs via modern machine learning techniques.</abstract>
    <parentTitle language="eng">Multiscale Modelling &amp; Simulation</parentTitle>
    <identifier type="arxiv">2107.10158</identifier>
    <identifier type="doi">10.1137/21M1448367</identifier>
    <enrichment key="PeerReviewed">yes</enrichment>
    <enrichment key="opus.source">publish</enrichment>
    <enrichment key="AcceptedDate">30.10.2022</enrichment>
    <author>Andreas Bittracher</author>
    <submitter>Ekaterina Engel</submitter>
    <author>Mattes Mollenhauer</author>
    <author>Péter Koltai</author>
    <author>Christof Schütte</author>
    <collection role="persons" number="schuette">Schütte, Christof</collection>
    <collection role="projects" number="no-project">no-project</collection>
    <collection role="institutes" number="MSoCP">Modeling and Simulation of Complex Processes</collection>
  </doc>
</export-example>
