Multiscale problems are challenging for computational processing due to the large number of unknowns required to just represent a multiscale function. This is mathematically related to the Nyquist – Shannon theorem for spectrally band-limited functions. We will review different methodologies for handling this challenge from computational and data science. We will then focus on recent developments in computational analogues to the analytical techniques for averaging dynamical systems and homogenization of elliptic operators. We will study a class of algorithms, which exploits scale separation in the solution and an alternative methodology, which compresses the spectrum in the solution. The latter is sometimes labeled as seamless methods.