Calculus of Variations and Geometric Measure Theory

A. Bach - A. Braides - C. I. Zeppieri

Quantitative analysis of finite-difference approximations of free-discontinuity problems

created by zeppieri1 on 18 Jun 2018
modified by braidesa on 30 Dec 2020


Published Paper

Inserted: 18 jun 2018
Last Updated: 30 dec 2020

Journal: Interfaces Free Bound.
Volume: 22
Year: 2020
Doi: 10.4171/IFB/443


Motivated by applications to image reconstruction, in this paper we analyse a finite-difference discretisation of the Ambrosio-Tortorelli functional. Denoted by $\varepsilon$ the elliptic-approximation parameter and by $\delta$ the discretisation step-size, we fully describe the relative impact of $\varepsilon$ and $\delta$ in terms of $\Gamma$-limits for the corresponding discrete functionals, in the three possible scaling regimes. We show, in particular, that when $\varepsilon$ and $\delta$ are of the same order, the underlying lattice structure affects the $\Gamma$-limit which turns out to be an anisotropic free-discontinuity functional.

Keywords: Gamma-convergence, finite-difference discretisation, Ambrosio-Tortorelli functional, elliptic approximation, free-discontinuity functionals