Fast sweeping method

In applied mathematics, the fast sweeping method is a numerical method for solving boundary value problems of the Eikonal equation.

|\nabla u(\mathbf {x} )|={\frac {1}{f(\mathbf {x} )}}{\text{ for }}\mathbf {x} \in \Omega

u(\mathbf {x} )=0{\text{ for }}\mathbf {x} \in \partial \Omega

where $\Omega$ is an open set in $\mathbb {R} ^{n}$ and $f(\mathbf {x} )$ is a function with positive values and $\partial \Omega$ is a well-behaved boundary of the open set and $|\cdot |$ is the $L^{2}$ norm.

Introduction

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^[1] by Hongkai Zhao, an applied mathematician at the University of California, Irvine.

References

↑ Zhao, Hongkai (2005-01-01). "A fast sweeping method for Eikonal equations". Mathematics of Computation. 74 (250): 603–627. doi:10.1090/S0025-5718-04-01678-3. ISSN 0025-5718.

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