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Ordered line integral methods for solving the eikonal equation. (English) Zbl 1434.65307

Summary: We present a family of fast and accurate Dijkstra-like solvers for the eikonal equation and factored eikonal equation which compute solutions on a regular grid by solving local variational minimization problems. Our methods converge linearly but compute significantly more accurate solutions than competing first order methods. In 3D, we present two different families of algorithms which significantly reduce the number of FLOPs needed to obtain an accurate solution to the eikonal equation. One method employs a fast search using local characteristic directions to prune unnecessary updates, and the other uses the theory of constrained optimization to achieve the same end. The proposed solvers are more efficient than the standard fast marching method in terms of the relationship between error and CPU time. We also modify our method for use with the additively factored eikonal equation, which can be solved locally around point sources to maintain linear convergence. We conduct extensive numerical simulations and provide theoretical justification for our approach. A library that implements the proposed solvers is available on GitHub.

MSC:

65N99 Numerical methods for partial differential equations, boundary value problems
65Y20 Complexity and performance of numerical algorithms
65K10 Numerical optimization and variational techniques
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65D32 Numerical quadrature and cubature formulas
49L20 Dynamic programming in optimal control and differential games
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
35L70 Second-order nonlinear hyperbolic equations
76Q05 Hydro- and aero-acoustics
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