We develop a fast sweeping method for the factored eikonal equation. By decomposing the solution of a general eikonal equation as the product of two factors: the first factor is the solution to a simple eikonal equation (such as distance) or a previously computed solution to an approximate eikonal equation. The second factor is a necessary modification/correction. Appropriate discretization and...

The eikonal equation is instrumental in many applications in several fields ranging from computer vision to geoscience. This equation can be efficiently solved using the iterative Fast Sweeping (FS) methods and the direct Fast Marching (FM) methods. However, when used for a point source, the original eikonal equation is known to yield inaccurate numerical solutions, because of a singularity at ...

Stationary quasi-P eikonal equations, stationary Hamilton-Jacobi equations, arise from the asymptotic approximation of anisotropic wave propagation. A paraxial formulation of the quasi-P eikonal equation results in a paraxial quasi-P eikonal equation, an evolution Hamilton-Jacobi equation in a preferred direction, which provides a fast and efficient way for computing viscosity solutions of quas...

The solution for the eikonal equation with a point-source condition has an upwind singularity at the source point as the eikonal solution behaves like a distance function at and near the source. As such, the eikonal function is not differentiable at the source so that all formally high-order numerical schemes for the eikonal equation yield first-order convergence and relatively large errors. Th...

First-break traveltime tomography is based on the eikonal equation. Because the eikonal equation is solved at fixed-shot positions and only receiver positions can move along the raypath, the adjoint-state tomography relies on inversion to resolve possible contradicting information between independent shots. The double-square-root (DSR) eikonal equation allows not only the receivers but also the...

The rst arrival quasi-P wave traveltime eld in an anisotropic elastic solid solves a rst order nonlinear partial diierential equation, the qP eikonal equation. Since the qP slowness surface is convex, the rst arrival traveltime along downward propagating rays solves the paraxial qP eikonal equation, an evolution equation in depth. A second order upwind nite diierence scheme solves this paraxial...

In these notes, we present an introduction to the classical Fast Marching Method (FMM). The FMM has been first introduced to find numerical approximations of the solutions to the stationary eikonal equation. More generally we show that this method allows to find exactly the solutions of numerical schemes that satisfy a certain causality assumption. In order to motivate this method, we present t...

The complex eikonal equation in (3+1) dimensions is investigated. It is shown that this equation generates many multi-knot configurations with an arbitrary value of the Hopf index. In general, these eikonal knots do not have the toroidal symmetry. For example, a solution with topology of the trefoil knot is found. Moreover, we show that the eikonal knots provide an analytical framework in which...