3 - Proper orthogonal decomposition

Summary

The proper orthogonal decomposition (POD) provides a basis for the modal decomposition of an ensemble of functions, such as data obtained in the course of experiments. Its properties suggest that it is the preferred basis to use in various applications. The most striking of these is optimality: it provides the most efficient way of capturing the dominant components of an infinite-dimensional process with only finitely many, and often surprisingly few, “modes.”

The POD was introduced in the context of turbulence by Lumley in [220]. In other disciplines the same procedure goes by the names: Karhunen–Loève decomposition, principal components analysis, singular systems analysis, and singular value decomposition. The basis functions it yields are variously called: empirical eigenfunctions, empirical basis functions, and empirical orthogonal functions. According to Yaglom (see [221]), the POD was introduced independently by numerous people at different times, including Kosambi [197], Loève [215], Karhunen [183], Pougachev [285], and Obukhov [272]. Lorenz [216], whose name we have already met in another context, suggested its use in weather prediction. The procedure has been used in various disciplines other than fluid mechanics, including random variables [275], image processing [313], signal analysis [5], data compression [7], process identification and control in chemical engineering [118,119], and oceanography [286]. Computational packages based on the POD are now readily available (an early example appeared in [11]).

In the bulk of these applications, the POD is used to analyze experimental data with a view to extracting dominant features and trends – in particular coherent structures. In the context of turbulence and other complex spatio-temporal fields, these will typically be patterns in space and time. However, our goal is somewhat different.