Geodesic
A Hopf–Lax formula for the time evolution of the level-set equation and a new approach to shape sensitivity analysis
Daniel Kraft1
1Universität Graz, Austria
Interfaces and free boundaries, Tome 18 (2016) no. 3, pp. 317-353

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DOI

The level-set method is used in many different applications to describe the propagation of shapes and domains. When scalar speed fields are used to encode the desired shape evolution, this leads to the classical level-set equation. We present a concise Hopf–Lax representation formula that can be used to characterise the evolved domains at arbitrary times. This result is also applicable for the case of speed fields without a fixed sign, even though the level-set equation has a non-convex Hamiltonian in these situations. The representation formula is based on the same idea that underpins the Fast-Marching Method, and it provides a strong theoretical justification for a generalised Composite Fast-Marching method.
DOI : 10.4171/ifb/366
Classification : 49-XX
Mots-clés : Level-set method, shape optimisation, Hopf–Lax formula, viscosity solutions, nonfattening, shape-sensitivity analysis

Daniel Kraft  1

1 Universität Graz, Austria 
Daniel Kraft. A Hopf–Lax formula for the time evolution of the level-set equation and a new approach to shape sensitivity analysis. Interfaces and free boundaries, Tome 18 (2016) no. 3, pp. 317-353. doi: 10.4171/ifb/366
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     pages = {317--353},
     year = {2016},
     volume = {18},
     number = {3},
     doi = {10.4171/ifb/366},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/ifb/366/}
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