# 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:

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:

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:

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:

\(r\) 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.