Hermitian Dynamic Mode Decomposition - Numerical Analysis and Software Solution

Abstract

The Dynamic Mode Decomposition (DMD) is a versatile and increasingly popular method for data driven analysis of dynamical systems that arise in a variety of applications in, e.g., computational fluid dynamics, robotics or machine learning. In the framework of numerical linear algebra, it is a data driven Rayleigh-Ritz procedure applied to a DMD matrix that is derived from the supplied data. In some applications, the physics of the underlying problem implies hermiticity of the DMD matrix, so the general DMD procedure is not computationally optimal. Furthermore, it does not guarantee important structural properties of the Hermitian eigenvalue problem and may return non-physical solutions. This paper proposes a software solution to the Hermitian (including the real symmetric) DMD matrices, accompanied with a numerical analysis that contains several fine and instructive numerical details. The eigenpairs are computed together with their residuals, and perturbation theory provides error bounds for the eigenvalues and eigenvectors. The software is developed and tested using the LAPACK package.