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.