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

## TL;DR

This paper investigates finite-difference discretizations of the Ambrosio-Tortorelli functional, demonstrating that using random lattices allows for $ ext{Gamma}$-convergence to the Mumford-Shah functional even when the mesh size is comparable to the approximation parameter.

## Contribution

It extends the understanding of discretization schemes by proving $ ext{Gamma}$-convergence for random lattices at optimal mesh sizes, unlike traditional periodic lattice discretizations.

## Key findings

- Random lattices enable $ ext{Gamma}$-convergence at $ ext{mesh size} \, ext{comparable to} \, ext{parameter}$
- Periodic lattice discretizations require $ ext{mesh size} \, ext{much smaller than}$ \, $ ext{parameter}$ for convergence
- Discretization on random lattices converges to an isotropic Mumford-Shah functional

## Abstract

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.