Data driven Koopman spectral analysis in Vandermonde-Cauchy form via the DFT: numerical method and theoretical insights

TL;DR

This paper introduces a numerically robust method for Koopman spectral analysis using Vandermonde-Cauchy matrices and DFT, providing both computational techniques and theoretical insights into snapshot reconstruction.

Contribution

It develops a new algorithm transforming Vandermonde matrices into Cauchy form via DFT for stable spectral analysis and clarifies the connection between reconstruction weights and Koopman modes.

Findings

Transforming Vandermonde matrices into Cauchy form improves numerical stability.

Explicit reconstruction formulas align with Koopman spectral theory.

The method enables accurate spectral analysis of dynamical systems.

Abstract

The goals and contributions of this paper are twofold. It provides a new computational tool for data driven Koopman spectral analysis by taking up the formidable challenge to develop a numerically robust algorithm by following the natural formulation via the Krylov decomposition with the Frobenius companion matrix, and by using its eigenvectors explicitly -- these are defined as the inverse of the notoriously ill-conditioned Vandermonde matrix. The key step to curb ill-conditioning is the discrete Fourier transform of the snapshots; in the new representation, the Vandermonde matrix is transformed into a generalized Cauchy matrix, which then allows accurate computation by specially tailored algorithms of numerical linear algebra. The second goal is to shed light on the connection between the formulas for optimal reconstruction weights when reconstructing snapshots using subsets of the computed Koopman modes. It is shown how using a certain weaker form of generalized inverses leads to explicit reconstruction formulas that match the abstract results from Koopman spectral theory, in particular the Generalized Laplace Analysis.