The eikonal equation is a nonlinear hyperbolic PDE whose solution τ(x) is the fastest traveltime to the point x ∈ Ω ⊂ Rn, n = 2 or 3, from the boundary ∂ Ω:
|| ∇ τ(x)|| = s(x),
τ(∂ Ω) = 0.
s(x) is the slowness function which is the reciprocal of the speed function. The eikonal equation arises in the high frequency approximation for the wave equation. In particular, it describes light propagation through an environment with possibly varying index of refraction n(x). Then the slowness function is s(x)=c-1n(x).
The eikonal equation has numerous applications in science and engineering such as computer graphics, seismic imaging, medical imaging, and many others -- see J. Sethian's webpage.
Classification of eikonal solvers, some key methods, and our contributions.
Proposed eikonal solvers
- Ordered Line Integral Methods for Solving the Eikonal Equation,
Samuel F. Potter and Maria K. Cameron,
Journal of Scientific Computing, 81(3) (2019), 2010–2050,
arXiv:1902.06825.
Software is available -- see S. Potter's webpage. - Jet Marching Methods for Solving the Eikonal Equation,
Samuel F. Potter and Maria K. Cameron,
arXiv:2009.05490.
Software is available -- see S. Potter's webpage.
Demo: a jet marching method (click on the figure to see a movie)
