POD - Proper Orthogonal Decomposition#

The Proper Orthogonal Decomposition (POD) is considered as the state-of-the-art in Reduced Order Modelling [22], especially in fluid-dynamics and in nuclear reactors applications. The algorithm is based on the Singular Value Decomposition of the snapshots matrix, which can be linked to the correlation matrix \(C\in\mathbb{R}^{N_s\times N_s}\):

\[\begin{equation*} C_{nm} = \int_\Omega u_n\cdot u_m \, d\Omega\qquad n,m = 1,\dots, N_s \end{equation*}\]

and its eigenvalue problem \(C\lambda_n = \lambda_n \boldsymbol{\eta}_n \). The POD modes are then defined with the following

\[\begin{equation*} \phi_n(\mathbf{x})= \frac{1}{\sqrt{\lambda_n}}\sum_{i=1}^{N_s} \eta_{n,i} u_i(\mathbf{x}) \qquad n = 1, \dots, N \end{equation*}\]

which provides also the orthonormality of the modes with respect to the inner product in \(L^2\).

The online phase consists in two different version of the reconstruction

\[\begin{equation*} u\simeq \sum_{n=1}^N \alpha_n\,\phi_n\qquad \qquad \alpha_n = \int_\Omega u\,\phi_n\,d\Omega \end{equation*}\]

the reduced coefficients \(\alpha_n\) are computed by projection given some test snapshots or by interpolation of the coefficients through suitable maps \(\alpha_n = \mathcal{F}(\alpha_{n,train})\), this version is known as POD-I (POD with Interpolation).

There are 6 folders containing the version of the solver for scalar and vector field, divided into offline (generation of the modes) and online (reconstruction of the field).

  • ScalarPOD_Offline

  • ScalarPOD_Online

  • VectorialPOD_Offline

  • VectorialPOD_Online

  • ScalarPODInterp_Online

  • VectorialPODInterp_Online