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

Annika Bach; Andrea Braides; Caterina Ida Zeppieri

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arXiv · 1807.05346

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

Annika Bach·Andrea Braides·Caterina Ida Zeppieri

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Abstract

Motivated by applications to image reconstruction, in this paper we analyse a \emph{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.

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Annika Bach, Andrea Braides, Caterina Ida Zeppieri. 2018-07-14. Quantitative analysis of finite-difference approximations of free-discontinuity problems. https://arxiv.org/abs/1807.05346

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