Latent Modes of Nonlinear Flows
A Koopman Theory Analysis

Investigates new models of nonlinear flows, how they can be approximated and linearized, based on Koopman theory analysis.

Ido Cohen (Author), Guy Gilboa (Author)

9781009323857, Cambridge University Press

Paperback / softback, published 29 June 2023

75 pages
22.9 x 15.2 x 0.3 cm, 0.108 kg

Extracting the latent underlying structures of complex nonlinear local and nonlocal flows is essential for their analysis and modeling. In this Element the authors attempt to provide a consistent framework through Koopman theory and its related popular discrete approximation - dynamic mode decomposition (DMD). They investigate the conditions to perform appropriate linearization, dimensionality reduction and representation of flows in a highly general setting. The essential elements of this framework are Koopman eigenfunctions (KEFs) for which existence conditions are formulated. This is done by viewing the dynamic as a curve in state-space. These conditions lay the foundations for system reconstruction, global controllability, and observability for nonlinear dynamics. They examine the limitations of DMD through the analysis of Koopman theory and propose a new mode decomposition technique based on the typical time profile of the dynamics.

1. Introduction
2. Preliminaries
3. Motivation for This Work
4. Koopman Eigenfunctions and Modes
5. Koopman Theory for Partial Differential Equation
6. Mode Decomposition Based on Time State-Space Mapping
7. Examples
8. Conclusion
List of Symbols
List of Abbreviations
Appendix A Extended Dynamic Mode Decomposition Induced from Inverse Mapping
Appendix B Sparse Representation
References.

Subject Areas: Numerical analysis [PBKS]