Kernel Proper Orthogonal Decomposition (KPOD)

This document offers a brief reference guide to the use of the Kernel Proper Orthogonal Decomposition (KPOD).

Theory

The Kernel POD is a nonlinear variant of the POD. It seeks to capture nonlinear correlations between snapshots via their embedding into a higher dimensional feature space (that we shall denote as \(xi\)), defined by a kernel function, and performs the POD there [2].

Thus, the main step is the computation of this kernel function. To this end, we compute the euclidean distance between snapshots \(\mathbf{d}_i \in \mathbb{R}^{n_S}\) as

\[M_{ij} = \| \mathbf{d}_i - \mathbf{d}_j \|^2 \;.\]

In the following, we use Radial Basis Function (RBFs) as kernel function. We compute the inverse length-scale of the RBF as:

\[\gamma = - \frac{\log{k_m}}{M_{ij}}\]

where \(k_m\) is a minimum value for the kernelized correlation. This leads to the kernelized expression of the correlation matrix as:

\[\mathbf{K}_\xi = \exp{(-\gamma \| \mathbf{d}_i - \mathbf{d}_j \|^2)} \;.\]

From this point onward, the procedure is the one of a standard POD: the temporal modes \(\psi^\xi_r\) are eigenvectors of \(\mathbf{K}^\xi\) (that now is a nonlinear manifold), the amplitudes are \(\sigma_r^\xi = \sqrt{\lambda^xi}\), and the spatial modes \(\phi^\xi_r\) are obtained via projection.

Example with MODULO

The kPOD is called in MODULO as:

from modulo_vki import ModuloVKI

# --- Initialize MODULO object
m = ModuloVKI(data=D,svd_solver='svd_scipy_sparse')
M_DIST=[1,19]
Phi_kPOD, Psi_kPOD, Sigma_kPOD,K_zeta = m.kPOD(M_DIST=M_DIST,k_m=k_m,
cent=True, K_out=True)

where M_DIST defines the snapshot indexes of which to compute the distance, k_m is the minimum kernelized correlation threshold described above, cent=True performs centering of \(\mathbf{K}^\xi\) and K_out=True returns the kernelized correlation matrix.

Note

The hyper parameter \(k_m\) heavily influences the outcomes of the kPOD. For large \(k_m\) then \(\mathbf{K}^\xi \rightarrow \mathbf{K}\), whilst smaller \(k_m\) will identify correlation happening at a specific frequency. The matrix is Toeplitz, and its eigenvectors tends naturally towards the Fourier basis. Outcome of this step would remind the filtering as in the diagonally filtered SPOD (Sieber et al.), but with more pronounced focus on a specific frequency range.