Dimensionality reduction for studying physical phenomena

Abstract

We explore dimensionality reduction in the context of model based and model free approaches. In the model based approach there is a governing equation or a set of rules relating quantities of interest, whereas in a model free setting there are no rules or equations, only data is available. As an example consider the problem of squealing noise in a brake. The model based approach relates quantities of interest like mass distribution within the brake, damping, stiffness and other properties of a brake material, speed of rotation etc with a dynamical equation, the steady state behaviour can be obtained by converting it to an eigenvalue problem and finding eigenvalues and eigenvectors (which are related to squeal frequency and mode shapes of a disc brake). The model reduction problem could be posed as projecting the eigenvalue problem to a lower dimensional space, while preserving important eigenvalues and eigenvectors. In contrast, the model free approach starts with data, i.e., a set of parameter values which correspond to squeal and the values which correspond to no-squeal. If the number of parameters responsible for squeal is very large, then dimensionality reduction is concerned with reducing the number of these parameters or ranking these parameters in order of importance. We illustrate pros and cons of model based and model free dimensionality reduction with some numerical examples.