# POD for Proper Orthogonal Decomposition¶

## 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:

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

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.