Model Reduction via the Karhunen-Loeve Expansion Part I: An Exposition

Abstract

In formulating mathematical models for dynamical systems, obtaining a high degree of qualitative correctness (i.e. predictive capability) may not be the only objective. The model must be useful for its intended application, and models of reduced complexity are attractive in many cases. In Part I of this paper we provide an exposition of some techniques that are useful in finding models of reduced complexity for dynamical systems involving flows. The material presented here is not new. The techniques we discuss are based on classical theory such as the Karhunen-Loeve expansion and the method of Galerkin, and the more recent concept of "coherent structures". They have been heavily exploited in a wide range of areas in science and engineering. The attempt here is to present this collection of important methods and ideas together, at a high level of detail, in coherent form, and in the context of model reduction for simulation and control. In this manner we lead in to Part II which illustr...