Calculus of Variations and Geometric Measure Theory

A. Bach - M. Cicalese - M. Ruf

Random finite-difference discretizations of the Ambrosio-Tortorelli functional with optimal mesh size

created by ruf on 25 Feb 2019
modified by cicalese on 25 Jun 2022


Published Paper

Inserted: 25 feb 2019
Last Updated: 25 jun 2022

Journal: SIAM J. Math. Anal.
Volume: 53
Number: 2
Pages: 2275–2318
Year: 2021

ArXiv: 1902.08437 PDF


We propose and analyze a finite-difference discretization of the Ambrosio-Tortorelli functional. It is known that if the discretization is made with respect to an underlying periodic lattice of spacing $\delta$, the discretized functionals $\Gamma$-converge to the Mumford-Shah functional only if $\delta\ll\varepsilon$, $\varepsilon$ being the elliptic approximation parameter of the Ambrosio-Tortorelli functional. Discretizing with respect to stationary, ergodic and isotropic random lattices we prove this $\Gamma$-convergence result also for $\delta\sim\varepsilon$, a regime at which the discretization with respect to a periodic lattice converges instead to an anisotropic version of the Mumford-Shah functional.