The Weighted Ambrosio--Tortorelli Approximation Scheme

Fonseca, Irene; Liu, Pan

doi:10.1137/16m1088661

Cited by 9 publications

(9 citation statements)

References 29 publications

(40 reference statements)

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“…Even for the single-well potential if v is close to zero around some interface then it is expected that E ε still approximates the surface area of the interface. This observation enables us to prove that for σ > 0, the Gamma limit of E ε (u, v) in the convergence in measure is a Mumford-Shah functional; see [2,3,12]. If E ε (v ε ) is bounded for small ε > 0, then it is rather clear that v ε → 1 in L 1 as ε → 0, so that v ε → 1 almost everywhere by taking a suitable subsequence.…”

Section: Introductionmentioning

confidence: 95%

“…For the Ambrosio-Tortorelli functional, it is enough to consider L 1 ×L 1 converges since v ε (x) → 1 except finitely many points where lim inf [2,3,12].) Here lim inf * denotes the relaxed liminf and we shall give its definition in Section 2.…”

Section: Introductionmentioning

confidence: 99%

“…A single-well Modica-Mortola functional is first used in [2] to approximate the Mumford-Shah functional. The Gamma limit of the Ambrosio-Tortorelli functional is by now well studied ( [2,3,12]). However, convergence of critical points is studied only in one dimension ( [10]).…”

Section: Introductionmentioning

confidence: 99%

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A finer singular limit of a single-well Modica--Mortola functional and its applications to the Kobayashi--Warren--Carter energy

Giga,

Okamoto,

Uesaka

2020

Preprint

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An explicit representation of the Gamma limit of a single-well Modica-Mortola functional is given for one-dimensional space under the graph convergence which is finer than conventional L 1 -convergence or convergence in measure. As an application, an explicit representation of a singular limit of the Kobayashi-Warren-Carter energy, which is popular in materials science, is given. Some compactness under the graph convergence is also established. Such formulas as well as compactness is useful to characterize the limit of minimizers the Kobayashi-Warren-Carter energy. To characterize the Gamma limit under the graph convergence, a new idea which is especially useful for one-dimensional problem is introduced. It is a change of parameter of the variable by arc-length parameter of its graph, which is called unfolding by the arc-length parameter in this paper.

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Section: Introductionmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

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A finer singular limit of a single-well Modica--Mortola functional and its applications to the Kobayashi--Warren--Carter energy

Giga,

Okamoto,

Uesaka

2020

Preprint

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“…as in [AT,AT2,FL], where h is a given L 2 function and F (v) = (v − 1) 2 . This problem can be handled in L 1 topology, and its limit is known to be the Mumford-Shah functional…”

Section: Introductionmentioning

confidence: 99%

On a singular limit of the Kobayashi--Warren--Carter energy

Giga¹,

Okamoto²,

Sakakibara³

et al. 2022

Preprint

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By introducing a new topology, a representation formula of the Gamma limit of the Kobayashi-Warren-Carter energy is given in a multidimensional domain. A key step is to study the Gamma limit of a single-well Modica-Mortola functional. The convergence introduced here is called the sliced graph convergence, which is finer than conventional L 1 convergence, and the problem is reduced to a one-dimensional setting by a slicing argument.

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“…• spatially dependent differential operators and multi-layer training schemes. This will allow to specialize the regularization according to the position in the image, providing a more accurate analysis of complex textures and of images alternating areas with finer details with parts having sharpest contours (see also [18]).…”

Section: Introductionmentioning

confidence: 99%

Adaptive image processing: first order PDE constraint regularizers and a bilevel training scheme

Davoli¹,

Fonseca²,

Liu³

2019

Preprint

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The Weighted Ambrosio--Tortorelli Approximation Scheme

Abstract: The Ambrosio-Tortorelli approximation scheme with weighted underlying metric is investigated. It is shown that it Γ-converges to a Mumford-Shah image segmentation functional depending on the weight ω dx, where ω ∈ SBV (Ω), and on its value ω − .

Cited by 9 publications

References 29 publications

A finer singular limit of a single-well Modica--Mortola functional and its applications to the Kobayashi--Warren--Carter energy

A finer singular limit of a single-well Modica--Mortola functional and its applications to the Kobayashi--Warren--Carter energy

On a singular limit of the Kobayashi--Warren--Carter energy

Adaptive image processing: first order PDE constraint regularizers and a bilevel training scheme

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