Fast sweeping method

Fast sweeping method is a numerical method for solving boundary value problems of the Eikonal equation:

$|\nabla u(\mathbf {x} )|={\dfrac {1}{f(\mathbf {x} )}}{\text{ for }}\mathbf {x} \in \Omega$

$u(\mathbf {x} )=0{\text{ for }}\mathbf {x} \in \partial \Omega$

Fast sweeping method is an iterative method which uses upwind difference for discretization and uses Gauss-Seidel iterations with alternating sweeping ordering to solve the discretized Eikonal equation on a rectangular grid. The origins of this approach lie in control theory.