Discretization of transfer operators using a sparse hierarchical tensor basis - the Sparse Ulam method

The global macroscopic behaviour of a dynamical system is encoded in the eigenfunctions of a certain transfer operator associated to it. For systems with low dimensional long term dynamics, efficient techniques exist for a numerical approximation of the most important eigenfunctions, cf. DeJu99a. 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. We develop the technique, establish statements about its complexity and convergence and present two numerical examples.