2. POD for Proper Orthogonal Decomposition

2.1. What is it?

The Proper Orthogonal Decomposition (POD) is a technique used to decompose a matrix and characterize it by its principal components which are called modes [AnindyaChatterjee2000]. To approximate a function z(x,t), only a finite sum of terms is required:

z(x,t) \simeq \sum_{k=1}^{m} a_k(t) \phi_k(x).

The function \phi_{k}(x) have an infinite representation. It can be chosen as a Fourier series or Chebyshev polynomials, etc. For a chosen basis of function, a set of unique time-functions a_k(t) arise. In case of the POD, the basis function are orthonormal. Meaning that:

\int_{x} \phi_{k_1} \phi_{k_2} dx &= \left\{\begin{array}{rcl} 1 & \text{if} & k_1 = k_2   \\ 0 & \text{if} & k_1 \neq k_2\end{array}\right. ,\\
a_k (t) &= \int_{x} z(x,t) \phi_k(x) dx.

The principle of the POD is to choose \phi_k(x) such that the approximation of z(x,t) is the best in a least squares sense. These orthonormal functions are called the proper orthogonal modes of the function.

When dealing with CFD simulations, the size of the domain m is usually smaller than the number of measures, snapshots, n. Hence, from the existing decomposition methods, the Singular Value Decomposition (SVD) is used. It is the snapshots methods [Cordier2006].

The Singular Value Decomposition (SVD) is a factorization operation of a matrix expressed as:

A = U \Sigma V^T,

with V diagonalizes A^TA, U diagonalizes AA^T and \Sigma is the singular value matrix which diagonal is composed by the singular values of A. Knowing that a singular value is the square root of an eigen value. u_i and v_i are eigen vectors of respectively U and V which form an orthonormal basis. Thus, the initial matrix can be rewritten:

A = \sum_{i=1}^{r} \sigma_i u_i v_i^T,

r being the rank of the matrix. If taken k < r, an approximation of the initial matrix can be constructed. This allows to compress the data as only an extract of u and v need to be stored.