On the Convergence of Hermitian Dynamic Mode Decomposition
Nicolas Boullé Matthew J. Colbrook
In this work, we study the convergence of Hermitian Dynamic Mode
Decomposition (DMD) to the spectral properties of self-adjoint Koopman
operators. Hermitian DMD is a data-driven method for approximating the Koopman
operator associated with an unknown nonlinear dynamical system from
discrete-time snapshots, while preserving the self-adjointness of the operator
on its finite-dimensional approximations. We show that, under suitable
conditions, the eigenvalues and eigenfunctions of HDMD converge to the spectral
properties of the underlying Koopman operator. Along the way, we establish a
general theorem on the convergence of spectral measures, and demonstrate our
results numerically on the two-dimensional Schr\"odinger equation.