Estimating long term behavior of flows without trajectory integration: the infinitesimal generator approach

TL;DR

This paper introduces a novel method for analyzing the long-term behavior of flows by directly discretizing the infinitesimal generator, eliminating the need for trajectory integration and significantly improving computational efficiency.

Contribution

The paper presents two discretization schemes for the infinitesimal generator, demonstrating convergence and superior efficiency over traditional transfer operator methods.

Findings

The infinitesimal generator approach is more efficient than the transfer operator method.

The proposed schemes show convergence under certain conditions.

Numerical experiments confirm significant speed-ups, sometimes by orders of magnitude.

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.