Proper orthogonal decomposition

The proper orthogonal decomposition is a numerical method that enables a reduction in the complexity of computer intensive simulations such as computational fluid dynamics and structural analysis (like crash simulations).

Typically in fluid dynamics and turbulences analysis, it is used to replace the Navier–Stokes equations by simpler models to solve.

The first idea behind the Proper Orthogonal Decomposition (POD), as it was originally formulated in the domain of fluid dynamics to analyze turbulences, is to decompose a random vector field u(x, t) into a set of deterministic spatial functions Φk(x) modulated by random time coefficients ak(t) so that: The first step is to sample the vector field over a period of time in what we call snapshots (as display in the image of the POD snapshots).

This snapshot method[4] is averaging the samples over the space dimension n, and correlating them with each other along the time samples p: The next step is to compute the covariance matrix C We then compute the eigenvalues and eigenvectors of C and we order them from the largest eigenvalue to the smallest.

We obtain n eigenvalues λ1,...,λn and a set of n eigenvectors arranged as columns in an n × n matrix Φ: