Proper Orthogonal Decomposition

The proper orthogonal decomposition (POD) is a method for finding a low-dimensional orthonormal basis that best approximates a collection of data. Given snapshot vectors , form the snapshot matrix

and its singular value decomposition

The first columns of are the POD modes. Their span minimizes the sum of squared orthogonal projection errors among all -dimensional subspaces, equivalently giving the best rank- approximation to in the Frobenius norm.

For a random field, the analogous modes are eigenfunctions of its covariance operator, so POD is also known as the Karhunen-Loève decomposition and is closely related to principal component analysis. In reduced-order modeling, experimental or numerical snapshots are used to compute the modes, after which the governing equations are projected onto their span, often using the Galerkin method. Sirovich's method of snapshots computes the same modes from the smaller matrix when (Sirovich 1987).