Efficient Approximation Methods for the Global Long-Term Behavior of Dynamical Systems - Theory, Algorithms and Examples

In this thesis we develop efficient numerical algorithms in order to analyze the long-term behavior of dynamical systems by transfer operator methods. Three new approaches are represented. First, a discretization via sparse grids is introduced for general systems, aiming to overcome the curse of dimension. Second, for continuous-time systems the infinitesimal generator of the associated transfer operator semigroup is treated numerically. A robust cell-to-cell approach, and a spectral method approach for smooth problems are derived. Third, focusing on the detection of conformation changes in molecules, mean field theory is used to approximate the marginal dynamics on low-dimensional subsystems. Also, conditions are given under which the Galerkin projection of a transfer operator can be related to the small random perturbation of the underlying system.

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