Paper Info
Title
Discretization of the Frobenius-Perron Operator Using a Sparse Haar Tensor Basis: The Sparse Ulam Method
Abstract
The global macroscopic behavior of a dynamical system is encoded in the eigenfunctions of the associated Frobenius-Perron operator. For systems with low dimensional long term dynamics, efficient techniques exist for a numerical approximation of the most important eigenfunctions; cf. [M. Dellnitz and O. Junge, SIAM J. Numer. Anal., 36 (1999), pp. 491-515]. They are based on a projection of the operator onto a space of piecewise constant functions supported on a neighborhood of the attractor—Ulam's method. In this paper we develop a numerical technique which makes Ulam's approach applicable to systems with higher dimensional long term dynamics. It is based on ideas for the treatment of higher dimensional partial differential equations using sparse grids [C. Zenger, Sparse grids, in Parallel Algorithms for Partial Differential Equations (Kiel, 1990), Vieweg, Braunschweig, 1991, pp. 241-251; H.-J. Bungartz and M. Griebel, Acta Numer., 13 (2004), pp. 147-269]. Here, we use a sparse Haar tensor basis as the underlying approximation space. We develop the technique, establish statements about its complexity and convergence, and present two numerical examples.
Year
DOI
Venue
2009
10.1137/080716864
SIAM J. Numerical Analysis
Keywords
Field
DocType
higher dimensional long term,sparse grid,higher dimensional partial differential,numerical technique,sparse ulam method,sparse haar tensor basis,numerical approximation,frobenius-perron operator,acta numer,numerical example,low dimensional long term,sparse grids
Discretization,Tensor,Mathematical analysis,Sparse approximation,Projection (linear algebra),Numerical analysis,Sparse grid,Partial differential equation,Transfer operator,Mathematics
Journal
Volume
Issue
ISSN
47
5
0036-1429
Citations 
PageRank 
References 
8
0.89
4
Authors
2
Name
Order
Citations
PageRank
Oliver Junge112821.57
Péter Koltai2193.87