Colloquium

Anomalous diffusion processes are ubiquitous in physical, biological and social sciences as well as finance. They lead to non-local models, one of which is fractional diffusion. In view of its importance in basic science and applications, the efficient numerical approximation of fractional diffusion is a central theme of research. This talk surveys three numerical schemes that build on different definitions of fractional diffusion. The first method is the integral formulation and deals with singular non-integrable kernels. The second method is a PDE approach that applies to the spectral definition and exploits the extension to one higher dimension. The third method is a discretization of the Dunford-Taylor formula. We discuss pros and cons of each method, error estimates, and document their performance with a few numerical experiments.