On the numerical approximation of the Perron-Frobenius and Koopman operator (1512.05997v3)

Abstract: Information about the behavior of dynamical systems can often be obtained by analyzing the eigenvalues and corresponding eigenfunctions of linear operators associated with a dynamical system. Examples of such operators are the Perron-Frobenius and the Koopman operator. In this paper, we will review different methods that have been developed over the last decades to compute finite-dimensional approximations of these infinite-dimensional operators - e.g. Ulam's method and Extended Dynamic Mode Decomposition (EDMD) - and highlight the similarities and differences between these approaches. The results will be illustrated using simple stochastic differential equations and molecular dynamics examples.

• The paper provides a comparative review of Ulam’s method and EDMD for approximating infinite-dimensional operators with finite-dimensional representations.
• The paper demonstrates that EDMD efficiently approximates Koopman eigenfunctions with reduced computational overhead versus traditional methods.
• The paper highlights the theoretical duality between Perron-Frobenius and Koopman operators, bridging classical schemes with data-driven techniques.

Numerical Approximation of the Perron–Frobenius and Koopman Operators

The paper under consideration provides an in-depth exploration of methods for approximating the Perron–Frobenius and Koopman operators, which are fundamental in the analysis of dynamical systems. The paper offers a comparative review of various techniques developed over recent decades, emphasizing the numerical approximation of these infinite-dimensional operators using finite-dimensional representations. The focus is particularly on Ulam's method and Extended Dynamic Mode Decomposition (EDMD), providing insightful discussions on their similarities, differences, and the intricate link between these approaches.

Summary of Objectives and Methods

The primary goal of the research is to review and elucidate methods for computing finite-dimensional approximations of the Perron–Frobenius and Koopman operators, which are important tools for understanding the global behavior of dynamical systems. These operators offer linear representations of potentially complex dynamical systems, facilitating the extraction of key dynamical features, such as invariant measures and modes, from observed data.

Key methodologies discussed include:

1. Ulam's Method: This approach provides a Galerkin discretization of the Perron–Frobenius operator using a set of disjoint measurable boxes as basis functions. By estimating transition probabilities between these boxes, it creates a finite Markov chain approximation of the global system behavior.
2. Extended Dynamic Mode Decomposition (EDMD): Distinguished for its ability to handle Koopman operator approximation, EDMD utilizes a dictionary of basis functions to project the operator into a finite-dimensional space. It accounts for the dynamics captured by Koopman modes and eigenfunctions using data-driven techniques, leading to improved approximations for dynamical behaviors that manifest in reduced dimensions.
3. Petrov–Galerkin and High-Order Methods: These approaches enhance approximation accuracy through the use of piecewise polynomial basis functions, aiming to better capture function regularities and improve the fidelity of operator approximations.

The paper systematically addresses these methods, utilizing stochastic differential equations and molecular dynamics examples to illustrate their application and efficacy.

Numerical Results and Implications

The research presents some notable numerical results, illustrating the application of the discussed methods to problems such as molecular conformations. Strong results demonstrate how EDMD, leveraging smooth basis functions, can efficiently approximate eigenfunctions with less computational overhead compared to Ulam's method. Experimental convergence of EDMD to Galerkin projections is highlighted as a significant advantage, especially in high-dimensional and stochastic contexts.

Regarding theoretical implications, the paper theorizes on the relationship and duality between Koopman and Perron–Frobenius operators, proposing that extended data-driven approaches can bridge gaps between classically distinct techniques. This realization opens avenues for further explorations into hybrid methodologies combining strengths across disciplines of dynamical systems and data science.

Practically, the insights from this paper can inform advancements in computational mechanics, atmospheric modeling, and more, where understanding complex system dynamics is crucial. As these methodologies mature, they hold promise for developing even more robust tools for the analysis and prediction of real-world dynamical systems.

Future Directions

The researchers postulate the potential for tensor-based generalizations of the discussed methods, which could further enhance applicability to high-dimensional systems. They suggest that developing efficient numerical algorithms in this space remains an open and promising area of future work.

In conclusion, the paper presents a rigorous and methodical review of methods for approximating key operators in dynamical systems analysis. It serves as a rich resource for researchers seeking to advance computational techniques for uncovering the intrinsic structures and behaviors of complex dynamical entities.