Published Paper

Inserted: 18 jun 2018
Last Updated: 30 dec 2020

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

Abstract:

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

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