Abstract

We enter a discussion as to what constitutes the ‘foundations of computational mathematics’. While not giving a definition, we give examples from image/signal processing and numerical computation where foundational issues have helped to ‘correctly’ formulate problems and guide their solution.

The question

While past chair of the organization Foundations of Computational Mathematics (FOCM), I was frequently asked what is the meaning of ‘foundations of computational mathematics’. Most people understand what computational mathematics is. So the question really centers around the meaning of ‘foundations’ in this context. Even though I have thought about this quite a while, I would not dare to try to give a precise definition of foundations – I am sure it would be picked apart. However, I would like in this presentation to give some examples where the adherence to fundamental questions has helped to shape the formulation of computational issues and more importantly contributed to their solution. The examples I choose in signal/image processing and numerical methods for PDEs are of course related to my own research. I am sure there are many other stories of the type I put forward that are waiting to be told.

The first of the three examples that I will discuss is that of image compression. This subject has grown rapidly over the last decade with an important infusion of ideas from mathematics especially the theories of wavelets and nonlinear approximation.