Dynamic Mode Decomposition: Theory and Data Reconstruction

Abstract

Dynamic Mode Decomposition (DMD) is a data-driven decomposition technique extracting spatio-temporal patterns of time-dependent phenomena. In this paper, we perform a comprehensive theoretical analysis of various variants of DMD. We provide a systematic advancement of these and examine the interrelations. In addition, several results of each variant are proven. Our main result is the exact reconstruction property. To this end, a new modification of scaling factors is presented and a new concept of an error scaling is introduced to guarantee an error-free reconstruction of the data.

Constrained Dynamic Mode Decomposition

“…Hence, we can assume the relation n > m + 1 and the linear independence of the vectors. Before we explain the algorithm of DMD shown in the first part of Algorithm 1 (lines 1-7), we describe the principles of DMD [36,37], which help understand our approach in the next section.…”

“…is an eigenvector of the matrix A with eigenvalue λ j [36]) and are arranged column-wise in the matrix Θ (line 6). Finally, the DMD amplitudes a j ∈ C are computed by solving min∥ΘΛa − x 1 ∥ via a least-squares solver (line 7).…”

Visualization and Selection of Dynamic Mode Decomposition Components for Unsteady Flow

“…Before formulating the algorithm of DMD, we summarize the concepts of DMD [13,14] and compare them with DFT. These aspects help understand the visualization techniques and their principles.…”

“…However, it differs in the calculation of amplitudes from the standard literature. The new definition of amplitudes, which uses the second snapshot instead of the first one for the reconstruction, was introduced by Krake et al [13]. We make use of this new formulation by improving the representation of DMD components and integrating them to arrive at novel and more adequate DMD visualizations.…”

“…Finally, the amplitudes a j and the error scaling c 0 are calculated (lines [9][10][11][12]. By this choice of amplitudes, Krake et al [13] proved the following reconstruction property:…”