Paper Info
Title
Estimating Long-Term Behavior of Flows without Trajectory Integration: The Infinitesimal Generator Approach.
Abstract
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.
Year
DOI
Venue
2013
10.1137/110819986
SIAM JOURNAL ON NUMERICAL ANALYSIS
Keywords
Field
DocType
transfer operator,infinitesimal generator,Ulam's method,spectral method,almost-invariant set,escape rate
Discretization,Mathematical optimization,Eigenfunction,Shift operator,Multiplication operator,Mathematical analysis,Operator (computer programming),Infinitesimal generator,Mathematics,Dynamical system,Transfer operator
Journal
Volume
Issue
ISSN
51
1
0036-1429
Citations 
PageRank 
References 
9
0.93
2
Authors
3
Name
Order
Citations
PageRank
Gary Froyland113017.19
Oliver Junge212821.57
Péter Koltai3193.87