Dynamic Mode Decomposition (DMD)

This document offers a brief reference guide to the use of the Dynamic Mode Decomposition (DMD).

Theory

The DMD seeks to overcome the spectral-mixing induced by the POD compression, while simultaneously avoid _windowing_ problems caused by the dependency of the DFT on the fundamental tone of the data to be decomposed. The main idea introduced by Schmid (2010) [4], is to fit a linear system onto a reduced POD projection of the dataset.

The algorithm we implement in MODULO is the one proposed by Tu et al. (2013) [5], and a more robust theoretical formulation can also be found in Mendez (2021) [6].

This method creates two portions of the snapshot matrix \(\mathbf{D}\) that differ of one step in time, that is:

\[\mathbf{D}_1 = [\mathbf{d}_1, \dots, \mathbf{d}_{n_t - 1}] \in \mathbb{R}^{n_S \times n_t - 1} \;,\]

and

\[\mathbf{D}_2 = [\mathbf{d}_2, \dots, \mathbf{d}_{n_t}] \in \mathbb{R}^{n_S \times n_t - 1} .\]

Then, the linear system propagator \(\mathbf{P}: \mathbf{d}_{t} \rightarrow \mathbf{d}_{t+\Delta t} \in \mathbb{R}^{n_S \times n_S}\) can be solved by least square. This can be computed via POD:

\[\mathbf{P} \approx \mathbf{D}_2 \mathbf{\Psi}_P \mathbf{\Sigma}_P^{-1} \mathbf{\Phi}_P^\top \;.\]

Yet, this can be ill-conditioned and computationally prohibitive to compute and it is further projected onto a reduced POD space spanned by the basis \(\tilde{\mathbf{\Phi}} = [\phi^\bullet, \dots, \phi^\bullet_{R}]\), where \(n_r \ll n_t\). This leads to a system in \(\mathbb{R}^{n_r \times n_r}\) wich reads:

\[\mathbf{D}_2 \approx \mathbf{P}{\mathbf{D}}_1 \rightarrow \tilde{\mathbf{\Phi}}_P^\top \mathbf{D}_2 \approx \tilde{\mathbf{\Phi}}_P^\top \mathbf{P} \tilde{\mathbf{\Phi}}_P \tilde{\mathbf{\Phi}}_P^\top \mathbf{D}_1 \;,\]

that we can compact writing:

\[\tilde{\mathbf{V}}_2 \approx \tilde{\mathbf{S}} \tilde{\mathbf{V}}_1 \;.\]

The reduced-order propagator is then:

\[\tilde{\mathbf{S}} = \tilde{\mathbf{\Phi}}_P^\top \mathbf{P} \tilde{\mathbf{\Phi}}_P = \tilde{\mathbf{\Phi}}_P^\top \mathbf{D}_2 \tilde{\mathbf{\Psi}} \tilde{\mathbf{\Sigma}}^{-1} \;.\]

The eigenvalue decomposition of this propagator governs the evolution of the reduced system, and the eigenvalues dictate the stability of each mode: \(\|\lambda_r\|<1\) implies vanishing modes, \(\|\lambda_r\|>1\) exploding ones and \(\|\lambda_r\|=1\) are purely harmonic (like DFT). Note that, in this latter case, the frequency is not constrained to be a multiple of the fundamental tone \(f_0 = 1/T = 1/n_t \Delta t\).

At last, the complex spatial structures of the DMD can be computed by mapping to the original space using the POD basis:

\[\mathbf{\Phi}_D = \tilde{\Phi}_P \mathbf{Q}\]

where \(\mathbf{Q}\) are the eigenvectors of the propagator \(\tilde{\mathbf{S}}\).

Example in MODULO

# --- Initialize MODULO object
m = ModuloVKI(data=np.nan_to_num(D))
# Compute the POD using PIP's method
Phi_D, Lambda, freqs, a0s = m.DMD(save_dmd, F_S=F_S)

The DMD module returns the spatial structures \(\mathbf{\Phi}_D\), the complex eigenvalues \(\lambda_r\) of the reduced-order propagator \(\tilde{\mathbf{S}}\), the frequency associated with each DMD mode, and their amplitudes \(\mathbf{a}_0\).