Data driven Koopman spectral analysis in Vandermonde-Cauchy form via the DFT: numerical method and theoretical insights
For researchers in dynamical systems and data-driven modeling, this work provides a practical solution to the long-standing numerical instability in Koopman analysis using the Krylov decomposition approach.
This paper introduces a numerically robust algorithm for data-driven Koopman spectral analysis by transforming the ill-conditioned Vandermonde matrix into a generalized Cauchy matrix via the discrete Fourier transform, enabling accurate computation. It also reveals connections between optimal reconstruction weights and Koopman spectral theory, specifically Generalized Laplace Analysis.
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.