Although fast sweeping methods have existed in control theory, it was first proposed for Eikonal equations by Hongkai Zhao, an applied mathematician at the University of California, Irvine.
12.
The fast sweeping method is an iterative method which uses upwind difference for discretization and uses Gauss Seidel iterations with alternating sweeping ordering to solve the discretized Eikonal equation on a rectangular grid.
13.
Sweeping algorithms such as the "'fast sweeping method ( FSM ) "'are highly efficient for solving Eikonal equations when the corresponding characteristic curves do not change direction very often.
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The more technical eikonal equation can be solved using standard differential equations solvers offered by a few numerical software packages mentioned in the Wikipedia List of numerical analysis software and it offers a higher precision order.
15.
Under reasonable hypothesis on the " initial " data, the eikonal equation admits a local solution, but a global smooth solution ( e . g . a solution for all time in the geometrical optics case ) is not possible.
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If " ? " is a subset of the Euclidean space "'R " "'n " with piecewise smooth boundary, then the signed distance function is differentiable almost everywhere, and its gradient satisfies the eikonal equation
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In the case of an Eikonal equation, the first method to do so was developed by Jon N . Tsitsiklis using a control-theoretic approach : followed shortly by Sethian's work on high order finite difference Dijkstra-like Fast Marching Methods.
18.
Given that a smooth data fitting curve n ( altitude ) is provided by the ITU for the refraction index n, and that the values of n differs from 1 only by something of the order 10 & minus; 4, a numerical solution of the eikonal equation can be considered.
19.
With the definition k : = \ nabla _ x \ varphi, \ omega : =-\ varphi _ t, the eikonal equation is precisely the dispersion relation that results by plugging the plane wave solution e ^ { i ( k \ cdot x-\ omega t ) } into the wave equation.
20.
Ray optics and electromagnetism are related by the fact that the eikonal equation gives a second electromagnetic formula of the same form as the potential equation above where the line of constant potential has been replaced by a line of constant phase, and the force lines have been replaced by normal vectors coming out of the constant phase line at right angles.
