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

TL;DR

This paper introduces a novel numerical method that extends Ulam's approach for approximating transfer operators to higher-dimensional systems using sparse grid techniques, improving computational efficiency and applicability.

Contribution

It develops a sparse hierarchical tensor basis method for discretizing transfer operators, enabling Ulam's method to handle higher-dimensional dynamical systems.

Findings

The method is computationally efficient for high-dimensional systems.

Convergence and complexity of the approach are theoretically established.

Numerical examples demonstrate the method's effectiveness.

Abstract

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.