Eikonal equation

The eikonal equation (from German Eikonal, which is 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.

The eikonal equation is of the form

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

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