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.