Fast marching method

The fast marching method is a numerical method created by James Sethian for solving boundary value problems of the Eikonal equation:

Typically, such a problem describes the evolution of a closed surface as a function of time ${\displaystyle u}$ with speed ${\displaystyle f(x)}$ in the normal direction at a point ${\displaystyle x}$ on the propagating surface. The speed function is specified, and the time at which the contour crosses a point ${\displaystyle x}$ is obtained by solving the equation. Alternatively, ${\displaystyle u(x)}$ can be thought of as the minimum amount of time it would take to reach ${\displaystyle \partial \Omega }$ starting from the point ${\displaystyle x}$. The fast marching method takes advantage of this optimal control interpretation of the problem in order to build a solution outwards starting from the "known information", i.e. the boundary values.

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