On the convergence of Hermitian Dynamic Mode Decomposition
File(s)
Published version
Author(s)
Boullé, Nicolas
Colbrook, Matthew J
Type
Journal Article
Abstract
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 that approximates the Koopman operator associated with an unknown nonlinear dynamical system, using discrete-time snapshots. This approach preserves the self-adjointness of the operator in its finite-dimensional approximations. We prove that, under suitably broad conditions, the spectral measures corresponding to the eigenvalues and eigenfunctions computed by Hermitian DMD converge to those of the underlying Koopman operator. This result also applies to skew-Hermitian systems (after multiplication by i), applicable to generators of continuous-time measure-preserving systems. Along the way, we establish a general theorem on the convergence of spectral measures for finite sections of self-adjoint operators, including those that are unbounded, which is of independent interest to the wider spectral community. We numerically demonstrate our results by applying them to two-dimensional Schrödinger equations.
Date Issued
2025-02-01
Date Acceptance
2024-10-10
Citation
Physica D: Nonlinear Phenomena, 2025, 472
ISSN
0167-2789
Publisher
Elsevier
Journal / Book Title
Physica D: Nonlinear Phenomena
Volume
472
Copyright Statement
© 2024 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
License URL
Identifier
10.1016/j.physd.2024.134405
Publication Status
Published
Article Number
134405
Date Publish Online
2024-12-12