IPAM GBM Tutorials March 27 – April 6 , 2001 Level Set Methods : An Overview and Some Recent Results

The level set method was devised by Osher and Sethian in [64] as a simple and versatile method for computing and analyzing the motion of an interface Γ in two or three dimensions. Γ bounds a (possibly multiply connected) region Ω. The goal is to compute and analyze the subsequent motion of Γ under a velocity field v. This velocity can depend on position, time, the geometry of the interface and the external physics. The interface is captured for later time as the zero level set of a smooth (at least Lipschitz continuous) function φ( x, t), i.e., Γ(t) = { x|φ( x, t) = 0}. φ is positive inside Ω, negative outside Ω and is zero on Γ(t). Topological merging and breaking are well defined and easily performed. In this review article we discuss recent variants and extensions, including the motion of curves in three dimensions, the Dynamic Surface Extension method, fast methods for steady state problems, diffusion generated motion and the variational level set approach. We also give a user’s guide to the level set dictionary and technology, couple the method to a wide variety of problems involving external physics, such as compressible and incompressible (possibly reacting) flow, Stefan problems, kinetic crystal growth, epitaxial growth of thin films, ∗Research supported in part by ONR N00014-97-1-0027, DARPA/NSF VIP grant NSF DMS9615854, AFOSR FQ8671-9801346, NSF DMS 9706827 and ARO DAAG 55-98-10323. †Department of Mathematics, University of California Los Angeles, Los Angeles, California 90095 ‡Computer Science Department, Stanford University, Stanford, California 94305.