Introduction to the Fast Marching Method

In these notes, we present an introduction to the classical Fast Marching Method (FMM). The FMM has been first introduced to find numerical approximations of the solutions to the stationary eikonal equation. More generally we show that this method allows to find exactly the solutions of numerical schemes that satisfy a certain causality assumption. In order to motivate this method, we present the shape from shading problem, introduce the notion of viscosity solutions to the stationary eikonal equation and consider finite differences schemes. We also provide various comparison principles for the eikonal equation and for the schemes. We give an error estimate between the numerical solution and the viscosity solution. We finally indicate some extensions of the FMM to more general equations than the stationary eikonal equation.