Journal
SIAM JOURNAL ON NUMERICAL ANALYSIS
Volume 51, Issue 1, Pages 223-247Publisher
SIAM PUBLICATIONS
DOI: 10.1137/110819986
Keywords
transfer operator; infinitesimal generator; Ulam's method; spectral method; almost-invariant set; escape rate
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Funding
- Discovery Project [DP110100068]
- Australian Research Council Centre of Excellence for Mathematics and Statistics of Complex Systems (MASCOS)
- TopMath Ph.D. program within the Elite Network of Bavaria
- TUM Graduate School
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The long-term distributions of trajectories of a flow are described by invariant densities, i.e., fixed points of an associated transfer operator. In addition, global slowly mixing structures, such as almost-invariant sets, which partition phase space into regions that are almost dynamically disconnected, can also be identified by certain eigenfunctions of this operator. Indeed, these structures are often hard to obtain by brute-force trajectory-based analyses. In a wide variety of applications, transfer operators have proven to be very efficient tools for an analysis of the global behavior of a dynamical system. The computationally most expensive step in the construction of an approximate transfer operator is the numerical integration of many short-term trajectories. In this paper, we propose to directly work with the infinitesimal generator instead of the operator, completely avoiding trajectory integration. We propose two different discretization schemes: a cell based discretization and a spectral collocation approach. Convergence can be shown in certain circumstances. We demonstrate numerically that our approach is much more efficient than the operator approach, sometimes by several orders of magnitude.
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