Towards tensor-based methods for the numerical approximation of the Perron-Frobenius and Koopman operator

TL;DR

This paper introduces a tensor-based reformulation of numerical methods for approximating the Perron-Frobenius and Koopman operators, enabling efficient analysis of high-dimensional dynamical systems by exploiting low-rank tensor structures.

Contribution

It proposes a novel tensor formulation of Ulam's method and EDMD, reducing computational complexity for high-dimensional systems through low-rank tensor approximations.

Findings

Tensor reformulation reduces computational time and memory usage.

Low-rank tensor approximations effectively capture system dynamics.

Method demonstrated on simple stochastic differential equations.

Abstract

The global behavior of dynamical systems can be studied by analyzing the eigenvalues and corresponding eigenfunctions of linear operators associated with the system. Two important operators which are frequently used to gain insight into the system's behavior are the Perron-Frobenius operator and the Koopman operator. Due to the curse of dimensionality, computing the eigenfunctions of high-dimensional systems is in general infeasible. We will propose a tensor-based reformulation of two numerical methods for computing finite-dimensional approximations of the aforementioned infinite-dimensional operators, namely Ulam's method and Extended Dynamic Mode Decomposition (EDMD). The aim of the tensor formulation is to approximate the eigenfunctions by low-rank tensors, potentially resulting in a significant reduction of the time and memory required to solve the resulting eigenvalue problems, provided that such a low-rank tensor decomposition exists. Typically, not all variables of a high-dimensional dynamical system contribute equally to the system's behavior, often the dynamics can be decomposed into slow and fast processes, which is also reflected in the eigenfunctions. Thus, the weak coupling between different variables might be approximated by low-rank tensor cores. We will illustrate the efficiency of the tensor-based formulation of Ulam's method and EDMD using simple stochastic differential equations.