An introduction to the proper orthogonal decomposition

A tutorial is presented on the Proper Orthogonal Decomposition (POD), which finds applications in computationally processing large amounts of high-dimensional data with the aim of obtaining low-dimensional descriptions that capture much of the phenomena of interest. The discrete version of the POD, which is the singular value decomposition (SVD) of matrices, is described in some detail. The continuous version of the POD is outlined. Low-rank approximations to data using the SVD are discussed. The SVD and the eigenvalue decomposition are compared. Two geometric interpretations of the SVD/POD are given. Computational strategies (using standard software) are mentioned. Two numerical examples are provided: one shows low-rank approximations of a surface, and the other demonstrates simple a posteriori analysis of data from a simulated vibroimpact system. Some relevant computer code is supplied.