On Alternative Algorithms for Computing Dynamic Mode Decomposition

Abstract

Dynamic mode decomposition (DMD) is a data-driven, modal decomposition technique that describes spatiotemporal features of high-dimensional dynamic data. The method is equation-free in the sense that it does not require knowledge of the underlying governing equations. The main purpose of this article is to introduce new alternatives to the currently accepted algorithm for calculating the dynamic mode decomposition. We present two new algorithms which are more economical from a computational point of view, which is an advantage when working with large data. With a few illustrative examples, we demonstrate the applicability of the introduced algorithms.

Delay-Embedding Spatio-Temporal Dynamic Mode Decomposition

“…This approach to implementing the DMD method is called exact DMD, since Tu et al [16] proves that DMD modes computed by ( 21) are the exact eigenvectors of A. DMD modes computed by (17) are known as projected eigenvectors of A. See [38,39] for some other results. In this case, the projected matrix of D, in (3), has the following presentation:…”

“…However, the introduction of the new variables is made at the expense of reducing the number of samples in the training data set. Hence, the number of these new variables (number of rows in the Hankel matrix), in (39), has to be a balance between the ability to detect dominant modes and the accuracy of the estimated model. The following algorithm (Algorithm 3) provides a step-by-step implementation of parallel delay-embedding DMD:…”

An Improved Approach for Implementing Dynamic Mode Decomposition with Control

“…This subsection introduces a novel approach to the DMDc method, which enhances efficiency over the standard approach. It was partly introduced in our recent conference report [29]. For completeness of the exposition, we will describe this novel methodology below.…”