In applied mathematics, the fast sweeping method is a

numerical method In numerical analysis, a numerical method is a mathematical tool designed to solve numerical problems. The implementation of a numerical method with an appropriate convergence check in a programming language is called a numerical algorithm. Mathem ...

for solving

boundary value problem In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to t ...

s of the Eikonal equation. :

, \nabla u(\mathbf),  = \frac 1  \text \mathbf \in \Omega

u(\mathbf) = 0 \text \mathbf \in \partial \Omega

where

\Omega

is an open set in

\mathbb^n

f(\mathbf)

is a function with positive values,

\partial \Omega

is a well-behaved boundary of the open set and

, \cdot,

is the

Euclidean norm Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean s ...

. The fast sweeping method is an iterative method which uses upwind difference for discretization and uses Gauss–Seidel iterations with alternating sweeping ordering to solve the discretized Eikonal equation on a rectangular grid. The origins of this approach lie in control theory. Although fast sweeping methods have existed in control theory, it was first proposed for Eikonal equations by Hongkai Zhao, an applied mathematician at the University of California, Irvine. Sweeping algorithms are highly efficient for solving Eikonal equations when the corresponding characteristic curves do not change direction very often.A. Chacon and A. Vladimirsky. Fast two-scale methods for Eikonal equations. SIAM J. on Scientific Computing 34/2: A547-A578, 2012

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References

See also