A Hopf-Lax formula for the time evolution of the level-set equation and a new approach to shape sensitivity analysis
type: Item
instance of: scholarly article
title: A Hopf-Lax formula for the time evolution of the level-set equation and a new approach to shape sensitivity analysis
zbMATH Open document ID: 1353.49054
DOI: 10.4171/IFB/366
author: Daniel Kraft
published in: Interfaces and Free Boundaries
publication date: 2016-12-02T00:00:00Z
review text: Summary: 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 characterize 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 generalized composite fast-marching method. Based on our Hopf-Lax formula, we are also able to present new theoretical results. In particular, we show non-fattening of the zero level set in a measure-theoretic sense, derive a very general shape sensitivity calculus that does not require the usual regularity assumptions on the domains, prove optimal Lipschitz constants for the evolved level-set function and discuss the effect of perturbations in both the speed field and the initial geometry.
Mathematics Subject Classification ID: 49Q10
Mathematics Subject Classification ID: 49Q12
Mathematics Subject Classification ID: 49L25
zbMATH DE Number: 6659247
zbMATH Keywords: level-set method
zbMATH Keywords: shape optimisation
zbMATH Keywords: Hopf-Lax formula
zbMATH Keywords: viscosity solutions
zbMATH Keywords: shape sensitivity analysis
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.4171/ifb/366
OpenAlex ID: W2530970253
description: scientific article; zbMATH DE number 6659247