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Efficient approximation methods for the global long-term behavior of dynamical systems - Theory, algorithms and examples

Péter Koltai

ISBN 978-3-8325-2752-5
168 pages, year of publication: 2010
price: 42.50 €
Efficient approximation methods for the global long-term behavior of dynamical systems - Theory, algorithms and examples
Für die Analyse des Langzeitverhaltens dynamischer Systeme werden in dieser Dissertation drei neue Ansätze, basierend auf Transferoperator-Methoden, zur Entwicklung effizienter Algorithmen vorgestellt. Zuerst leiten wir eine Diskretisierung auf dünnen Gittern her, um den Fluch der Dimension in den Griff zu bekommen. Die zweite Methode behandelt den infinitesimalen Generator der Transferoperator-Halbgruppe für zeit-kontinuierliche Systeme. Als Drittes benutzen wir mean-field-Theorie, um die Dynamik von Subsystemen zu beschreiben, wobei besonderes Augenmerk auf die Konformationsanalyse in der Moleküldynamik gerichtet wird. Des Weiteren werden Bedingungen hergeleitet, unter welchen die Galerkin-Projektion des Transferoperators mit einer kleinen zufälligen Störung des zugrunde liegenden Systems in Verbindung gebracht werden kann.

In this thesis three new transfer operator based approaches are developed for the analysis of the long-term behavior of dynamical systems. First, a sparse grid discretization is derived in order to deal with the "curse of dimension". The second method considers the infinitesimal generator of the transfer operator semigroup for continuous-time systems. The third method uses mean field theory to describe the dynamics of subsystems; here the main attention is devoted to conformation analysis in molecular dynamics. Further, conditions are derived such that the Galerkin projection of the transfer operator can be related to a small random perturbation of the underlying system.

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Keywords:
  • dynamical systems
  • transfer operator
  • infinitesimal generator
  • mean field approximation
  • numerical approximation

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