Eikonal equation

An eikonal equation (from Greek εἰκών, image^[1][2]) is a non-linear first-order partial differential equation that is encountered in problems of wave propagation.

The classical eikonal equation in geometric optics is a differential equation of the form

${\displaystyle |\nabla u(x)|=n(x)}$

(1)

where ${\displaystyle x}$ lies in an open subset of ${\displaystyle \mathbb {R} ^{n}}$, ${\displaystyle n(x)}$ is a positive function, ${\displaystyle \nabla }$ denotes the gradient, and ${\displaystyle |\cdot |}$ is the Euclidean norm. The function ${\displaystyle n}$ is given and one seeks solutions ${\displaystyle u}$. In the context of geometric optics, the function ${\displaystyle n}$ is the refractive index of the medium.

More generally, an eikonal equation is an equation of the form

${\displaystyle H(x,\nabla u(x))=0}$

(2)

where ${\displaystyle H}$ is a function of ${\displaystyle 2n}$ variables. Here the function ${\displaystyle H}$ is given, and ${\displaystyle u}$ is the solution. If ${\displaystyle H(x,y)=|y|-n(x)}$, then equation (2) becomes (1).

Eikonal equations naturally arise in the WKB method^[3] and the study of Maxwell's equations.^[4] Eikonal equations provide a link between physical (wave) optics and geometric (ray) optics.

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