*Abstract*
We review recent work on level set methods for following the
evolution of complex interfaces. These techniques are based on solving an
initial value partial differential equations for a level set functions,
using 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.
The methodology results in robust, accurate, and efficient numerical
algorithms for propagating interfaces in highly complex settings.
We review the basic theory and approximations, describe a hierarchy of
fast methods, including an extremely fast marching level set scheme for
monotonically advancing fronts, based on a stationary formulation of
the problem, and discuss extensions to multiple
interfaces and triple points. Finally, we demonstrate the
technique applied to a series of examples from geometry, material science and
computer vision, including mean curvature flow, minimal surfaces,
grid generation, fluid mechanics, and combustion.

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