Iterative method for solving the eikonal equation

Abstract

The paper present principles and derivation of the iterative method for solving the eikonal equation. The eikonal equation, which defines the relationship between the phase of the optical wave Φ (r) and the refractive index n(r) , i.e. |grad Φ (r)| 2 = n 2 ( r ), represents the fundamental equation in geometrical optics. It describes the evolution of the wavefront, which is given by the equation Φ (r) = C , of the electromagnetic wave in the limit of infinite frequency or zero wavelength. The eikonal equation is the nonlinear partial differential equation (PDE) of the first order. This classification makes the eikonal equation of rather diffcult to solve, both analytically and numerically. Several algorithms have been developed to solve the eikonal equation: Dijkstra's algorithm, fast marching method, fast sweeping method, label-correcting methods, etc. Major disadvantage of these methods is that their convergence puts rather high requirements on the density of the computing grid. It is known that finite element method (FEM) offers much more memory and time efficient approach to solve PDEs. Unfortunately, FEM cannot be applied to solve eikonal equation directly due to its first order. In order to provide the fast and memory efficient solution of the eikonal equation, it is suggested to solve a generalized version of the eikonal equation, which is of the second order and which can be solved using FEM. Then, iterative procedure for computing the corrections of the obtained numerical solution is developed. It is shown that the computed series converges to the solution of the original eikonal equation.