Abstract
Information about the behavior of dynamical systems can often be obtained by analyzing the eigenvalues and corresponding eigenfunctions of linear operators associated with a dynamical system. Examples of such operators are the Perron-Frobenius and the Koopman operator. In this paper, we will review different methods that have been developed over the last decades to compute finite-dimensional approximations of these infinite-dimensional operators - in particular Ulam's method and Extended Dynamic Mode Decomposition (EDMD) - and highlight the similarities and differences between these approaches. The results will be illustrated using simple stochastic differential equations and molecular dynamics examples.
Original language | English |
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Pages (from-to) | 51-79 |
Number of pages | 29 |
Journal | Journal of Computational Dynamics |
Volume | 3 |
Issue number | 1 |
DOIs | |
Publication status | Published - Jan 2016 |
Keywords
- Extended dynamic mode decomposition
- Galerkin methods
- Koopman operator
- Perron-Frobenius operator
- Transfer operator
- Ulam's method
ASJC Scopus subject areas
- Computational Mechanics
- Computational Mathematics