Rigorous data-driven computation of spectral properties of Koopman operators for dynamical systems

Abstract

Koopman operators are infinite-dimensional operators that globally linearize nonlinear dynamical systems, making their spectral information useful for understanding dynamics. However, Koopman operators can have continuous spectra and infinite-dimensional invariant subspaces, making computing their spectral information a considerable challenge. This paper describes data-driven algorithms with rigorous convergence guarantees for computing spectral information of Koopman operators from trajectory data. We introduce residual dynamic mode decomposition (ResDMD), which provides the first scheme for computing the spectra and pseudospectra of general Koopman operators from snapshot data without spectral pollution. Using the resolvent operator and ResDMD, we also compute smoothed approximations of spectral measures associated with measure-preserving dynamical systems. We prove explicit convergence theorems for our algorithms, which can achieve high-order convergence even for chaotic systems, when computing the density of the continuous spectrum and the discrete spectrum. We demonstrate our algorithms on the tent map, Gauss iterated map, nonlinear pendulum, double pendulum, Lorenz system, and an 11-dimensional extended Lorenz system. Finally, we provide kernelized variants of our algorithms for dynamical systems with a high-dimensional state-space. This allows us to compute the spectral measure associated with the dynamics of a protein molecule that has a 20,046-dimensional state-space, and compute nonlinear Koopman modes with error bounds for turbulent flow past aerofoils with Reynolds number > 10 5 that has a 295,122dimensional state-space.

Modal Analysis of Spatiotemporal Data via Multi-fidelity Multi-variate Gaussian Processes

“…Although DMD-type algorithms appeal to Koopman spectral density of the system, they are known to be sensitive to sensor noise. To overcome this issue, other variants of DMD such as total-least-squares DMD [11], residual DMD [12], and sparsity-promoting DMD [13] were developed and shown to be superior to the regular DMD method in robustness to the noise and spectral pollution. However, it is still a deterministic model and cannot consider the statistics of the data, which motivated the development of probabilistic models [14,15].…”

“…In spite of their prevalent usage, current modal analysis techniques encounter some challenges. Some of them include (1) Spectral pollution caused by discretization of the dynamics as well as the measurement noise [12], (2) Inconvenience to utilize temporally non-uniformly distributed data, due to the usual underlying assumption of uniform time step size in conventional DMD methods [12]. This work primarily focuses on addressing the latter challenge.…”

On Numerical Approximations of the Koopman Operator

“…This can potentially be hard to do because of issues such as spectral pollution-see remarks at the end of Section 3; also note the general long-standing problems of spectral pollution and computing the full spectrum of Schrödinger operators on a lattice were recently solved in [15]. The recent work [16] enables computation of full spectral measures using the combination of resolvent operator techniques (used for the first time in the Koopman operator context in [17]) and ResDMD-an extension of Dynamic Mode Decomposition (introduced next) technique that incorporates computation of residues from data snapshots (computation of residues was considered earlier in [18]).…”

“…Computation of residues was considered in [18] to address the problem of spectral pollution, where discretization introduces spurious eigenvalues. As mentioned before, the recent work [16] provides another method to resolve the spectral pollution problem, introducing ResDMD-an extension of Dynamic Mode Decomposition that incorporates computation of residues from data snapshots. The relationship between GLA and finite section methods was studied in [40].…”