The method of Proper Orthogonal Decomposition

The Proper Orthogonal Decomposition (POD) is a post-processing technique. It is a multi-variate statistical method that aims at obtaining a compact representation of the data. This method may serve two purposes, namely order reduction by projecting high-dimensional data into a lower-dimensional space and feature extraction by revealing  relevant, but unexpected, structure hidden in the data.

It takes a given set of data and extracts basis functions, that contain as much “energy” as possible. By taking first modes of POD, we can modelize almost all the problem because they have all the energy. 

Let $ q_{k}(x), k = 1 ... N_t $ be a set of observations (called snapshots) at point x of the domain that could be obtained by a PIV experimental measurements or by numerical simulation. The goal of POD is to find functions $\Phi$ such that :
$$ \frac{<\mid(q,\Phi)\mid^{2}>}{\mid\mid\Phi\mid\mid^{2}} $$
is maximized. $<.>$ is an average, $\mid\mid.\mid\mid$ the induced norm. Solving the optimization problem leads to an eigenvalue problem, where the functions $\Phi$ are the eigenfunctions.

In our problem, the resolution gives us the eigenfunctions and by taking first modes, we can modelize the velocity : 

First mode for velocity corresponds to the mean (numerical results)-UFirst mode for velocity corresponds to the mean (numerical results)-V

Figure 3.11 - First mode for velocity corresponds to the mean (numerical results)

Figure 3.12 – POD first modes

Figure 3.13 – POD reconstruction