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# Eikonal equation

## The meaning of «eikonal equation»

The eikonal equation (from Greek εἰκών, image) is a non-linear partial differential equation encountered in problems of wave propagation, when the wave equation is approximated using the WKB theory. It is derivable from Maxwell's equations of electromagnetics, and provides a link between physical (wave) optics and geometric (ray) optics.

subject to u | ∂ Ω = 0 {\displaystyle u|_{\partial \Omega }=0} , where Ω {\displaystyle \Omega } is an open set in R n {\displaystyle \mathbb {R} ^{n}} with well-behaved boundary, f ( x ) {\displaystyle f(x)} is a function with positive values, ∇ {\displaystyle \nabla } denotes the gradient and | ⋅ | {\displaystyle |\cdot |} is the Euclidean norm. Here, the right-hand side f ( x ) {\displaystyle f(x)} is typically supplied as known input. Physically, the solution u ( x ) {\displaystyle u(x)} is the shortest time needed to travel from the boundary ∂ Ω {\displaystyle \partial \Omega } to x {\displaystyle x} inside Ω , {\displaystyle \Omega ,} with f ( x ) {\displaystyle f(x)} being the speed at x {\displaystyle x} .

In the special case when f = 1 {\displaystyle f=1} , the solution gives the signed distance from ∂ Ω {\displaystyle \partial \Omega } .[citation needed]

One fast computational algorithm to approximate the solution to the eikonal equation is the fast marching method.

can be interpreted as the minimal amount of time required to travel from x {\displaystyle x} to ∂ Ω {\displaystyle \partial \Omega } , where f : Ω ¯ → ( 0 , + ∞ ) {\displaystyle f:{\bar {\Omega }}\to (0,+\infty )} is the speed of travel, and q : ∂ Ω → [ 0 , + ∞ ) {\displaystyle q:\partial \Omega \to [0,+\infty )} is an exit-time penalty. (Alternatively this can be posed as a minimal cost-to-exit by making the right-side C ( x ) / f ( x ) {\displaystyle C(x)/f(x)} and q {\displaystyle q} an exit-cost penalty.)