AI summary · arXiv: 2209.02244v1 · math.NA
The mpEDMD Algorithm for Data-Driven Computations of Measure-Preserving Dynamical Systems
The essentials
- Study focuses on analysis and decomposition of nonlinear dynamical systems using spectral information of Koopman operators
- Koopman operators have ability to globally linearize systems, but computing their spectral information is challenging due to infinite-dimensional nature
- Authors propose new algorithm called measure-preserving extended dynamic mode decomposition (mpEDMD)
- mpEDMD is data-driven algorithm that enforces measure-preserving truncations of Koopman operators using orthogonal Procrustes problem
- Operates with general dictionary of observables, making it flexible and compatible with various types of data and pre-existing DMD-type methods
- Key contribution of mpEDMD is its ability to converge to spectral quantities of Koopman operators for general measure-preserving dynamical systems
- Theoretical proofs provided for convergence of mpEDMD in terms of projection-valued and scalar-valued spectral measures, spectra, and Koopman mode decompositions
- Investigation into convergence rates of approximating spectral measures when using delay embedding (Krylov subspaces) with increasing dictionary size
- Demonstrated effectiveness through several challenging examples, showing increased robustness to noise compared to other DMD-type methods
- Successfully captures energy conservation and cascade phenomena in experimental measurements of turbulent boundary layer flow with high Reynolds number (>6×10^4) and large state-space dimension (>10^5)
- Overall, mpEDMD offers promising approach for analyzing nonlinear dynamical systems by providing accurate approximations of Koopman operator spectral information.
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