Scientific computation has revolutionized the scientific methodology through its interplay with experiments and theory. Numerical algorithms are at the heart of this revolution. They simulate quantitative assembly of different small scale dynamics and convert it into accurate predictions of large scale phenomena. It is here that mathematics, modeling and experiments interact through scientific computation.

In this talk we will discuss few of the mathematical aspects associated with such numerical algorithms. We describe the fundamental concepts of accuracy and stability, we review applications ranging from computational fluid dynamics to audio processing, and we take a closer look into scales through two prototype algorithms: detection of edges by separation of scales and image processing by decomposition of scales.