Eaton abstract a new finitedifference technique is presented for solving the eikonal equation for inhomogeneous, transversely isotropic media. Derivation of the eikonal equation physics stack exchange. Although many authors have discussed various numerical methods for. A highly scalable massively parallel fast marching method. It is shown that the ansatz is a form of bateman potential, and the derivation explicitly defines rays, phase fronts, and the field polarization. Finitedifferencesolution of the eikonal equation for transversely isotropic media david w. Eikonal tomography complements traditional surface wave tomography in several ways. The relation between both methods is studyed in section iv. Lemma applied to eikonal equation eikonal equation.
Let us now derive the wave equation assuming that the wavelength is small. Since the eikonal equation does not uniquely specify a weak solution, we need to look for a specific solution a viscosity solution or entropy solution. There are currently, however, few algorithms that can e. The eikonal approximation in quantum mechanics works for processes involving the scattering of particles with large incoming momentum and when the scattering angle is very small. Because such an equation appears in a variety of applications, it is essential to develop fast, e cient numerical methods to solve such an equation.
Shajesh department of physics and astronomy, university of oklahoma, 440 w. Abstract for the bgk equation in the large scale hyperbolic limit, the density of particles can be transformed as the. Raytracing and eikonal solutions for lowfrequency wave. Eikonal equation, a nonlinear partial differential equation encountered in problems of wave propagation eikonal approximation, a method of approximation useful in wave scattering equations operation eikonal, an operation conducted by the national security agency in collaboration. While the solution of the scalar wave equation represents the wavefield px, y, z. This paper presents a novel solver for the eikonal equation that is designed to run e. By using nonconjugate subalgebras of ranks 1, 2, and 3, we performed the symmetry reduction of the equation 1 to differential equations with less a number of independent variables. Paraxial eikonal equations and applications william w. The scheme relies on an upwind finite difference approximation to the gradient and a resulting causality relationship that lends itself to a dijkstralike programming approach. An accurate and efficient eikonal equation solver with variable grid spacing wenjing wu, geox systems ltd. Harmonic plane waves and the eikonal approximation clearly, a plane wave of the form.
The theory of viscosity solutions for eikonal equations or more general hamiltonjacobi equations is welldeveloped when x is a euclidean space. It is stressed that both algorithms should be considered. Fast semilagrangian schemes for the eikonal equation and. The eikonal equation from german eikonal, which is from greek, image is a nonlinear partial differential equation encountered in problems of wave propagation, when the wave equation is approximated using the wkb theory. Falcone, the minimum time problem and its applications to front propagation, in motion by mean curvature and related topics, a.
The proposed method manages a list of active nodes and iteratively updates the solutions on those grid points until they. An accurate and efficient eikonal equation solver with. As plancks constant treated as a free parameter tends. The mathematical theory of geometrical optics can be regarded as the theory of the eikonal equation. We introduce and analyze a fast version of the semilagrangian algorithm for front propagation originally proposed in m.
The new algorithm is obtained using the local definition of the. On the eikonal function cornbleet 1996 radio science. The fast marching method is a widely used numerical method for solving the eikonal static hamiltonjacobi equation, which is a firstorder hyperbolic partial differential equation arising from a variety of applications, such as computational geometry, computational fluid dynamics, computer vision, materials science, optimal control, etc. In there, the authors use the approximation of low wavelengths high wavenumbers, to neglect some terms in the wave equation and arrive at the eikonal equation. Redistancing by flow of time dependent eikonal equation litien cheng. It describes simd, shared memory, and distributed memory machine models. Redistancing by flow of time dependent eikonal equation.
Propertiespreserving high order numerical methods for a kinetic eikonal equation songting luoa, nicholas paynea adepartment of mathematics, iowa state university, ames, ia 50011. In image analysis, for example, shortest paths defined by imagedriven metrics have been proposed for segmentation and the tracking of whitematter pathways in the diffusion weighted images of the brain. We present a family of fast and accurate dijkstralike solvers for the eikonal equation and factored eikonal equation which compute solutions on a regular grid by solving local variational minimization problems. The method is an extension of other recently developed, isotropic finitedifference algorithms. Fast methods for the eikonal and related hamilton jacobi. Eikonal evolution and gluon radiation alex kovner1,2 and urs achim wiedemann1. A computational study of the fast marching and the fast sweeping methods for the eikonal equation is given. An ecient solution to the eikonal equation on parametric. In the papers 16,17,18,19,20,21,22, we studied the eikonal equation using the subgroup structure of the proper orthochronous group p 1, 4. In this paper a fast sweeping method for computing the numerical solution of eikonal equations on a rectangular grid is presented. Margrave abstract highfrequency approximations to the wave. In 3d, we present two different families of algorithms which. When used as part of a more general computational pipeline, it is essential that effort be expended to minimize the computational cost of this. While solutions of the eikonal equation are used in their own right in many physical problems, such solutions are also used as building blocks in more general computational schemes such as in remeshing and in imagevolume analysis e.
Fast sweeping methods for eikonal equations on triangular. On classification of symmetry reductions for the eikonal. This equation, like the eikonal equation, is an equation in the first partial derivatives and is of second degree. Nowack department of earth and atmospheric sciences, purdue university, west lafayette, in 47907, usa accepted 1992 january 15. This is a study report on eikonal approximations which was undertaken under the guidance. To formulate a wellposed initialvalue problem on equation, it is sufficient to specify at some closed surface and to choose one of the two branches. Eikonal physics underlies recent approaches towards calculating the produced. The solution of the eikonal equation may have singularities. Eikonal is the german form of the greek word, meaning likeness, icon or image. We would like to devise a similar algorithm for equation 2. The derivation is classic and can be found in many popular textbooks. I think i came up with a way but i wonder if there is a simpler way to argue this. The focusing problem for the eikonal equation article pdf available in journal of evolution equations 31 june 2002 with 36 reads how we measure reads.
In principles of optics by max born and emil wolf, the eikonal equation is derived from maxwells equations. In the language of differential equations, the main advantage the eikonal approximation offers is that the equations reduce to a differential equation in a single. Ordered line integral methods for solving the eikonal equation. Application to image analysis anand rangarajan and karthik s. Department of computer and information science and engineering university of florida, gainesville, fl, usa abstract. The algorithm takes advantage of the upwind nature of the eikonal equation in order to update the value of each grid point by a number of times bounded by the number of its neighbors.
Finitedifference solution to the eikonal equation for. First, there is no explicit regularization and, hence, the method is largely free from ad hoc choices. The eikonal equation has a wide range of applications. However, finding solutions of maxwells equations is an extremely difficult task.
In section iii we modify the method to solve the eikonal equation satisfying a given characteristic data at in nity. From the form of the eikonal equation there results a remarkable analogy between geometrical optics and the mechanics of material particles. Fo r a n m dimensional riemannian sp ace of c onstant curvature k with the metric ds 2. The method as we implement it does, however, involve smoothing in. All forms of the eikonal equation are firstorder partial differential equations. Raytracing and eikonal solutions for lowfrequency wavefields. The fast marching method is a numerical algorithm for solving the eikonal equation on a rectangular orthogonal mesh in o m log m steps, where m is the total number of grid points. In seismology the eikonal equation is used to calculate the travel time of the optimal trajectories of seismic. In this appendix, i remind the reader how the eikonal equation is derived from the wave equation.
Propertiespreserving high order numerical methods for a. A simple derivation of the eikonal and transport equations. It is derivable from maxwells equations of electromagnetics, and provides a link between physical wave optics and geometric ray optics. Our methods converge linearly but compute significantly more accurate solutions than competing first order methods. The eikonal equation in its simplest form says that the magnitude of the gradient of the eikonal is constant. The eikonal approximation was not born in the study of quantum mechanics. The motion of a material particle is determined by the hamiltonjacobi equation 16. There was proli c activity in the application of eikonal approximation in high energy physics, especially in qcd. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. The crucial idea is that each sweeping ordering follows a family of. It is a raytheoretical approximation to the scalar wave equation.
Eikonal equations are fundamental to describe propagation of a wave front or interfaces in various disciplines of sciences and technology. The method is an iterative method which uses upwind difference for discretization and uses gaussseidel iterations with alternating sweeping ordering to solve the discretized system. Eikonal equation article about eikonal equation by the. Eikonal approximation university of oklahoma physics. We study the focusing problem for the eikonal equation.
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