The Proper Orthogonal Decomposition (Prof. Scott T.M. Dawson)

This lecture was given by Prof. Scott T.M. Dawson, Illinois Institute of Technology, USA in the framework of the von Karman Lecture Series on Machine Learning for Fluid Mechanics organized by the von Karman Institute and the Université libre de Bruxelles in February 2020. The proper orthogonal decomposition (POD) is one of the most ubiquitous data analysis and modeling technique in fluid mechanics. Since many of the properties of the PODare inherited from the singular value decomposition (SVD), we start with a discussion of the SVD, and describe those of its properties that are particularly useful for understanding the POD. Our discussion of the POD starts by characterizing the POD as a property of a given dataset, before discussing how the same concept arises when considering continuous (in space and time) dynamical systems. We will describe various variants of the POD that have emerged over last half a century, and how they are related (such as spectral and space-only POD). As well as giving a broad overview, we will discuss some of the technical details often omitted and/or taken for granted when POD is applied in practice, such as how to correctly incorporate nonstandard inner product weights. We finish by using a simple example to demonstrate properties and methods of implementation for the POD.