Topic
Eikonal equation
About: Eikonal equation is a research topic. Over the lifetime, 3449 publications have been published within this topic receiving 66112 citations.
Papers published on a yearly basis
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Papers
TL;DR: A fast marching level set method is presented for monotonically advancing fronts, which leads to an extremely fast scheme for solving the Eikonal equation.
Abstract: A fast marching level set method is presented for monotonically advancing fronts, which leads to an extremely fast scheme for solving the Eikonal equation. Level set methods are numerical techniques for computing the position of propagating fronts. They rely on an initial value partial differential equation for a propagating level set function and use techniques borrowed from hyperbolic conservation laws. Topological changes, corner and cusp development, and accurate determination of geometric properties such as curvature and normal direction are naturally obtained in this setting. This paper describes a particular case of such methods for interfaces whose speed depends only on local position. The technique works by coupling work on entropy conditions for interface motion, the theory of viscosity solutions for Hamilton-Jacobi equations, and fast adaptive narrow band level set methods. The technique is applicable to a variety of problems, including shape-from-shading problems, lithographic development calculations in microchip manufacturing, and arrival time problems in control theory.
3,663 citations
TL;DR: The development of Fast Marching Methods is reviewed, including the theoretical and numerical underpinnings; details of the computational schemes, including higher order versions; and examples of the techniques in a collection of different areas are demonstrated.
Abstract: Fast Marching Methods are numerical schemes for computing solutions to the nonlinear Eikonal equation and related static Hamilton--Jacobi equations Based on entropy-satisfying upwind schemes and fast sorting techniques, they yield consistent, accurate, and highly efficient algorithms They are optimal in the sense that the computational complexity of the algorithms is O(N log N), where N is the total number of points in the domain The schemes are of use in a variety of applications, including problems in shape offsetting, computing distances from complex curves and surfaces, shape-from-shading, photolithographic development, computing first arrivals in seismic travel times, construction of shortest geodesics on surfaces, optimal path planning around obstacles, and visibility and reflection calculations In this paper, we review the development of these techniques, including the theoretical and numerical underpinnings; provide details of the computational schemes, including higher order versions; and demonstrate the techniques in a collection of different areas
1,565 citations
TL;DR: The Fast Marching Method is extended to triangulated domains with the same computational complexity and is provided as an optimal time algorithm for computing the geodesic distances and thereby extracting shortest paths on triangulate manifolds.
Abstract: 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 this paper we extend the Fast Marching Method to triangulated domains with the same computational complexity. As an application, we provide an optimal time algorithm for computing the geodesic distances and thereby extracting shortest paths on triangulated manifolds.
1,211 citations
TL;DR: Monotonicity and stability properties of the fast sweeping algorithm are proven and it is shown that 2 n Gauss-Seidel iterations is enough for the distance function in n dimensions.
Abstract: In this paper a fast sweeping method for computing the numerical solution of Eikonal equations on a rectangular grid is presented. The method is an iterative method which uses upwind difference for discretization and uses Gauss-Seidel iterations with alternating sweeping ordering to solve the discretized system. The crucial idea is that each sweeping ordering follows a family of characteristics of the corresponding Eikonal equation in a certain direction simultaneously. The method has an optimal complexity of O(N) for N grid points and is extremely simple to implement in any number of dimensions. Monotonicity and stability properties of the fast sweeping algorithm are proven. Convergence and error estimates of the algorithm for computing the distance function is studied in detail. It is shown that 2 n Gauss-Seidel iterations is enough for the distance function in n dimensions. An estimation of the number of iterations for general Eikonal equations is also studied. Numerical examples are used to verify the analysis.
1,165 citations
1,016 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2025 | 59 |
| 2024 | 81 |
| 2023 | 145 |
| 2022 | 407 |
| 2021 | 125 |
| 2020 | 108 |