Maria K. Cameron

University of Maryland, Department of Mathematics

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Eikonal Solvers

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.
Classification of eikonal solvers

Proposed eikonal solvers

Demo: a jet marching method (click on the figure to see a movie)

JMM movie