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  <doc>
    <id>1131</id>
    <completedYear/>
    <publishedYear/>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>doctoralthesis</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>2009-06-03</completedDate>
    <publishedDate>2009-06-03</publishedDate>
    <thesisDateAccepted>2008-12-17</thesisDateAccepted>
    <title language="eng">Statistical Error Estimation and Grid-free Hierarchical Refinement in Conformation Dynamics</title>
    <abstract language="eng">The understanding of geometric structures and dynamical properties of molecular conformations gives insight into molecular long-term behavior. The identification of metastable conformations together with their life times and transition patterns is the intention of conformation dynamics. Conformation dynamics is a multi-scale approach that leads to a reduced description of the dynamical system in terms of a stochastic transition probability matrix. The present thesis deals with the error analysis of computed matrices and the resulting matrix functions. Since conformational membership vectors, as they are computed by the Robust Perron Cluster Analysis (PCCA+), form an invariant subspace of the transition matrix, subspace-based error estimators are of particular interest. The decomposition of the state space into basis functions and the approximation of integrals by Monte-Carlo quadrature give rise to row-wise correlated random matrices, for which stochastic norms are computed. Together with an appropriate statistical model for the distribution of matrix rows, this allows for the calculation of error bounds and error distributions of the invariant subspace and other variables of interest. Equilibration of errors among the basis functions can be achieved by enhanced sampling in regions where the trajectories are mixing slowly. Hierarchical refinement of such basis functions systematically improves the clustering into metastable conformations by reducing the error in the corresponding invariant subspace. These techniques allow for an evaluation of simulation results and pave the way for the analysis of larger molecules. Moreover, the extension of PCCA+ to non-reversible Markov chains, verified by the corresponding perturbation theory, and the modification of the objective function for the case of soft membership vectors represent a further generalization of the clustering method, thus continuing the development from PCCA over PCCA+ to PCCA++. The methods developed in this thesis are useful for but not limited to conformation dynamics. In fact, they are applicable to a broader class of problems which combine domain decomposition with Monte-Carlo quadrature. Possible application areas may include the chemical master equation or quantum dynamical systems.</abstract>
    <abstract language="deu">Das Verständnis von geometrischen Strukturen und dynamischen Eigenschaften molekularer Konformationen ist essentiell für die Vorhersage des Langzeitverhaltens von Molekülen. Die Identifikation metastabiler Konformationen sowie die Bestimmung von Übergangswahrscheinlichkeiten und Haltezeiten sind Bestandteil der Konformationdynamik. Dabei handelt es sich um eine Mehrskalenmethode, die auf eine reduzierte Beschreibung des Systems mittels einer stochastischen Übergangsmatrix führt. In der vorliegenden Dissertation wurde untersucht, wie man die Genauigkeit der Matrizen sowie der daraus berechneten Größen quantifizieren kann. Im Mittelpunkt stehen dabei Fehlerschätzer für den invarianten Unterraum, da die rechten Eigenvektoren als Grundlage der Robusten Perron Cluster Analyse (PCCA+) zur Identifizierung der metastabilen Konformationen dienen. Die Zerlegung des Zustandsraumes in Basisfunktionen sowie die Approximation der Matrixeinträge mittels Monte-Carlo-Quadratur führen zu zeilenweise korrelierten Zufallsmatrizen. Mit Hilfe einer stochastischen Norm sowie einem geeigneten statistischen Modell für die Verteilung der Matrixzeilen können u.a. Fehlerschranken und -verteilungen für den invarianten Unterraum brechnet werden. Eine Equilibrierung des Fehlers zwischen den Basisfunktionen kann durch erweitertes Sampling in solchen Regionen erreicht werden, in denen die Trajektorien nur langsam mischen.Eine hierarchische Zerlegung dieser Basisfunktionen verbessert systematisch die Zerlegung in metastabile Konformationen, indem sie den Fehler im invarianten Unterraum reduziert. Diese Techniken gestatten eine Evaluierung der Simulationsergebnisse und ebnen den Weg zur Behandlung komplexerer Moleküle. Desweiteren wurden Verallgemeinerungen der PCCA+ untersucht. Die Erweiterung der PCCA+ auf nicht-reversible Markov-Ketten sowie die Modifizierung der Zielfunktion für den Fall der weichen Clusterung setzen die Entwicklung von der PCCA über PCCA+ zu PCCA++ fort. Somit können neue Anwendungsfelder für dieses Cluster-Verfahren erschlossen werden. Die Methoden wurden zwar in Rahmen der Konformationsdynamik entwickelt, jedoch lassen sie sich auf eine weite Problemklasse anwenden, in der Gebietszerlegungsverfahren mit Monte-Carlo-Quadratur kombiniert werden. Mögliche Anwendungsgebiete umfassen die chemische Master-Gleichung oder quantenchemische Systeme.</abstract>
    <identifier type="opus3-id">1180</identifier>
    <identifier type="urn">urn:nbn:de:kobv:188-fudissthesis000000008079-9</identifier>
    <author>Susanna Röblitz</author>
    <submitter>unknown unknown</submitter>
    <advisor>Peter Deuflhard</advisor>
    <advisor>Wilhelm Huisinga</advisor>
    <subject>
      <language>deu</language>
      <type>uncontrolled</type>
      <value>Konformationsdynamik</value>
    </subject>
    <subject>
      <language>deu</language>
      <type>uncontrolled</type>
      <value>Metastabilität</value>
    </subject>
    <subject>
      <language>deu</language>
      <type>uncontrolled</type>
      <value>gitterfreie Methoden</value>
    </subject>
    <subject>
      <language>deu</language>
      <type>uncontrolled</type>
      <value>Monte-Carlo-Quadratur</value>
    </subject>
    <subject>
      <language>deu</language>
      <type>uncontrolled</type>
      <value>Perron-Cluster-Analyse</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>conformation dynamics</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>metastability</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>mesh-free methods</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Monte Carlo quadrature</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Perron cluster analysis</value>
    </subject>
    <collection role="ddc" number="510">Mathematik</collection>
    <collection role="msc" number="37M05">Simulation</collection>
    <collection role="msc" number="37M25">Computational methods for ergodic theory (approximation of invariant measures, computation of Lyapunov exponents, entropy)</collection>
    <collection role="msc" number="62M05">Markov processes: estimation</collection>
    <collection role="msc" number="65C05">Monte Carlo methods</collection>
    <collection role="msc" number="92E10">Molecular structure (graph-theoretic methods, methods of differential topology, etc.)</collection>
    <collection role="collections" number="">Dissertationen</collection>
    <collection role="institutes" number="num">Numerical Mathematics</collection>
    <collection role="persons" number="susanna.roeblitz">Röblitz, Susanna</collection>
    <collection role="projects" number="Matheon-A19">Matheon-A19</collection>
    <thesisGrantor>Freie Universität Berlin</thesisGrantor>
    <file>https://opus4.kobv.de/opus4-zib/files/1131/thesis_final_el.pdf</file>
  </doc>
  <doc>
    <id>4583</id>
    <completedYear/>
    <publishedYear>2013</publishedYear>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst>147</pageFirst>
    <pageLast>179</pageLast>
    <pageNumber/>
    <edition/>
    <issue>2</issue>
    <volume>7</volume>
    <type>article</type>
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    <belongsToBibliography>0</belongsToBibliography>
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    <title language="eng">Fuzzy spectral clustering by PCCA+: application to Markov state models and data classification</title>
    <parentTitle language="eng">Advances in Data Analysis and Classification</parentTitle>
    <identifier type="doi">10.1007/s11634-013-0134-6</identifier>
    <enrichment key="PeerReviewed">yes</enrichment>
    <author>Susanna Röblitz</author>
    <submitter> Stötzel</submitter>
    <author>Marcus Weber</author>
    <collection role="institutes" number="num">Numerical Mathematics</collection>
    <collection role="institutes" number="compmol">Computational Molecular Design</collection>
    <collection role="institutes" number="compsys">Computational Systems Biology</collection>
    <collection role="persons" number="susanna.roeblitz">Röblitz, Susanna</collection>
    <collection role="persons" number="weber">Weber, Marcus</collection>
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    <id>4640</id>
    <completedYear/>
    <publishedYear>2013</publishedYear>
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    <language>eng</language>
    <pageFirst>110</pageFirst>
    <pageLast>194</pageLast>
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    <edition/>
    <issue/>
    <volume>139</volume>
    <type>article</type>
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    <title language="eng">Adaptive spectral clustering with application to tripeptide conformation analysis</title>
    <parentTitle language="eng">The Journal of Chemical Physics</parentTitle>
    <identifier type="doi">10.1063/1.4830409</identifier>
    <enrichment key="PeerReviewed">yes</enrichment>
    <author>Fiete Haack</author>
    <submitter>Adam Nielsen</submitter>
    <author>Konstantin Fackeldey</author>
    <author>Susanna Röblitz</author>
    <author>Olga Scharkoi</author>
    <author>Marcus Weber</author>
    <author>Burkhard Schmidt</author>
    <collection role="institutes" number="num">Numerical Mathematics</collection>
    <collection role="institutes" number="compmol">Computational Molecular Design</collection>
    <collection role="institutes" number="compsys">Computational Systems Biology</collection>
    <collection role="persons" number="fackeldey">Fackeldey, Konstantin</collection>
    <collection role="persons" number="susanna.roeblitz">Röblitz, Susanna</collection>
    <collection role="persons" number="weber">Weber, Marcus</collection>
    <collection role="projects" number="BMS-Nielsen">BMS-Nielsen</collection>
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