Eikonal equation

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The eikonal equation is a non-linear partial differential equation of the form

$|\nabla u(x)|=F(x), \ x\in \Omega$

subject to $u|_{\partial \Omega}=0$ , where $Ω$ is an open set in $\mathbb{R}^n$ with well-behaved boundary, $F (x)$ is a function with positive values, ∇ denotes the gradient and |·| is the Euclidean norm. Here, the right-hand side $F (x)$ is typically supplied as known input. Physically, the solution $u (x)$ is the shortest time needed to travel from the boundary $\partial \Omega$ to $x$ inside $Ω,$ with $F (x)$ being the cost (not speed) at $x$ .

A fast computational algorithm to approximate the solution to the eikonal equation is the fast marching method. In the special case when $F = 1$ , the solution gives the signed distance from $\partial \Omega$ .

The eikonal equation is encountered in problems of wave propagation, when the wave equation is approximated using the WKB theory.