Naslov
Data Driven Koopman Spectral Analysis in Vandermonde-Cauchy Form via the DFT: Numerical Method and Theoretical Insights

Autori
Drmač, Zlatko ; Mezić, Igor ; Mohr, Ryan

Izvornik
SIAM journal on scientific computing (1064-8275) 41 (2019), 5; A3118-A3151

Vrsta, podvrsta i kategorija rada
Radovi u časopisima, članak, znanstveni

Ključne riječi
dynamic mode decomposition, Koopman operator, Krylov subspaces, proper orthogonal decomposition, Rayleigh--Ritz approximation, Vandermonde matrix, discrete Fourier transform, Cauchy matrix, generalized Laplace analysis

Sažetak
The goals and contributions of this paper are twofold. It provides a new numerically robust computational tool for data driven Koopman spectral analysis, based on the natural formulation via the Krylov decomposition with the Frobenius companion matrix, and by using its eigenvectors explicitly---these are the columns of 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.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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