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Interfaces and Free Boundaries

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Volume 22, Issue 3, 2020, pp. 317–381
DOI: 10.4171/IFB/443

Published online: 2020-09-01

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

Annika Bach[1], Andrea Braides[2] and Caterina Ida Zeppieri[3]

(1) Technische Universität München, Garching, Germany
(2) Università di Roma Tor Vergata, Italy
(3) Universität Münster, Germany

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: Finite-difference discretisation, Ambrosio–Tortorelli functional, $\Gamma$-convergence, elliptic approximation, free-discontinuity

Bach Annika, Braides Andrea, Zeppieri Caterina Ida: Quantitative analysis of finite-difference approximations of free-discontinuity problems. Interfaces Free Bound. 22 (2020), 317-381. doi: 10.4171/IFB/443