Ergodic Theory, Dynamic Mode Decomposition, and Computation of Spectral Properties of the Koopman Operator

By Hassan Arbabi, Igor Mezi'c. SIAM Journal on Applied Dynamical Systems, 2016

538 citations recorded by the source corpus.

We establish the convergence of a class of numerical algorithms, known as dynamic mode decomposition (DMD), for computation of the eigenvalues and eigenfunctions of the infinite-dimensional Koopman operator. The algorithms act on data coming from observables on a state space, arranged in Hankel-type matrices. The proofs utilize the assumption that the underlying dynamical system is ergodic. This includes the classical measure-preserving systems, as well as systems whose attractors support a physical measure. Our approach relies on the observation that vector projections in DMD can be used to approximate the function projections by the virtue of Birkhoff's ergodic theorem. Using this fact, we show that applying DMD to Hankel data matrices in the limit of infinite-time observations yields the true Koopman eigenfunctions and eigenvalues. We also show that the singular value decomposition, which is the central part of most DMD algorithms, converges to the proper orthogonal decomposition of observables. We use...

Subfields: Artificial Intelligence, Civil and Structural Engineering, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/84606a365a1172474997df474d5fd4dc4b217ff8

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Citations in the corpus

On Numerical Approximations of the Koopman Operator (2020, methodology): “The idea is that one might start with a single observable, and due to its evolution span an invariant subspace of the Koopman operator (note the connection of such methods with the Takens embedding theorem ideas [4, 20]).”
Data-driven discovery of Koopman eigenfunctions for control (2017, methodology): “For chaotic systems, the use of delay coordinates provides a promising embedding [67]–[69], but delay embedded models have not yet been extended to incorporate control.”
Data-driven discovery of coordinates and governing equations (2019, methodology): “Methods such as dynamic mode decomposition (DMD) [4, 5, 10, 74], extended DMD [75], kernel DMD [76], and time-delay DMD [77, 78] build linear models for dynamics, but these methods rely on a proper set of coordinates for linearization.”
Data-driven spectral analysis of the Koopman operator (2017, methodology): “ theorem describes how to exploit the CD kernel for recovering the density ˆ of the AC part: Theorem 2 (Density of AC part) If = at + ac + sc with d ac = ˆdis a positive measure supported on [0;1] and (m k)N k=0 its Fourier coecients (14), then for Lebesgue almost all 2[0;1] lim N!1 h N+ 1 K~ N(e”
Learning Koopman Invariant Subspaces for Dynamic Mode Decomposition (2017, methodology): “Figure 4 shows the eigenvalues estimated by LKIS-DMD (n = 16), linear Hankel DMD [41], [42] (delay 8), and DMDs with”