applied mathematics Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathemati ...

, the fast sweeping method is a numerical method for solving boundary value problems of the

Eikonal equation An eikonal equation (from Greek εἰκών, image) is a non-linear first-order partial differential equation that is encountered in problems of wave propagation. The classical eikonal equation in geometric optics is a differential equation of ...

. :

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

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

where

\Omega

is an

open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are su ...

\mathbb^n

f(\mathbf)

is a

function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...

with positive values,

\partial \Omega

is a well-behaved

boundary Boundary or Boundaries may refer to: * Border, in political geography Entertainment * ''Boundaries'' (2016 film), a 2016 Canadian film * ''Boundaries'' (2018 film), a 2018 American-Canadian road trip film *Boundary (cricket), the edge of the pla ...

of the open set and

, \cdot,

is the Euclidean norm. 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 Control theory is a field of mathematics that deals with the control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to a ...

. 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 The University of California, Irvine (UCI or UC Irvine) is a public land-grant research university in Irvine, California. One of the ten campuses of the University of California system, UCI offers 87 undergraduate degrees and 129 graduate and p ...

. 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