Cartesian convexity as the key notion in the variational existence theory for nonlocal supremal functionals

Kreisbeck, Carolin; Ritorto, Antonella; Zappale, Elvira

doi:10.1016/j.na.2022.113111

Cited by 3 publications

(1 citation statement)

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“…Motivated from both the practical and theoretical point of view, the study of nonlocal aspects in the calculus of variations has received widespread attention in the literature recently. From applications in peridynamics [35, 44], imaging processing [6, 8, 26] and machine learning [5, 30], to the abstract study of lower semicontinuity [12, 31, 32, 38] and localization [2, 10, 13] of various nonlocal functionals. Especially the introduction of the so-called Riesz fractional gradient by Shieh & Spector [42, 43], which for and is defined as has seen a dramatic rise in interest and has opened up the possibility to study new types of fractional problems.…”

Section: Introductionmentioning

confidence: 99%

Extending linear growth functionals to functions of bounded fractional variation

Schönberger

2023

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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In this paper we consider the minimization of a novel class of fractional linear growth functionals involving the Riesz fractional gradient. These functionals lack the coercivity properties in the fractional Sobolev spaces needed to apply the direct method. We therefore utilize the recently introduced spaces of bounded fractional variation and study the extension of the linear growth functional to these spaces through relaxation with respect to the weak* convergence. Our main result establishes an explicit representation for this relaxation, which includes an integral term accounting for the singular part of the fractional variation and features the quasiconvex envelope of the integrand. The role of quasiconvexity in this fractional framework is explained by a technique to switch between the fractional and classical settings. We complement the relaxation result with an existence theory for minimizers of the extended functional.

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Section: Introductionmentioning

confidence: 99%

Extending linear growth functionals to functions of bounded fractional variation

Schönberger

2023

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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Structural Changes in Nonlocal Denoising Models Arising Through Bi-Level Parameter Learning

et al. 2023

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We introduce a unified framework based on bi-level optimization schemes to deal with parameter learning in the context of image processing. The goal is to identify the optimal regularizer within a family depending on a parameter in a general topological space. Our focus lies on the situation with non-compact parameter domains, which is, for example, relevant when the commonly used box constraints are disposed of. To overcome this lack of compactness, we propose a natural extension of the upper-level functional to the closure of the parameter domain via Gamma-convergence, which captures possible structural changes in the reconstruction model at the edge of the domain. Under two main assumptions, namely, Mosco-convergence of the regularizers and uniqueness of minimizers of the lower-level problem, we prove that the extension coincides with the relaxation, thus admitting minimizers that relate to the parameter optimization problem of interest. We apply our abstract framework to investigate a quartet of practically relevant models in image denoising, all featuring nonlocality. The associated families of regularizers exhibit qualitatively different parameter dependence, describing a weight factor, an amount of nonlocality, an integrability exponent, and a fractional order, respectively. After the asymptotic analysis that determines the relaxation in each of the four settings, we finally establish theoretical conditions on the data that guarantee structural stability of the models and give examples of when stability is lost.

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A sufficient condition for the lower semicontinuity of nonlocal supremal functionals in the vectorial case

Gargiulo

Zappale

2023

European Journal of Mathematics

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We present a sufficient condition ensuring lower semicontinuity for nonlocal supremal functionals of the type "Equation missing"where $$\Omega $$ Ω is a bounded open subset of $${\mathbb {R}}^N$$ R N and $$W:\Omega \,{\times }\, \Omega \,{\times }\, {\mathbb {R}}^{d \times N}{\times }\, {\mathbb {R}}^{d \times N} \rightarrow {\mathbb {R}}.$$ W : Ω × Ω × R d × N × R d × N → R .

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Cartesian convexity as the key notion in the variational existence theory for nonlocal supremal functionals

Cited by 3 publications

References 44 publications

Extending linear growth functionals to functions of bounded fractional variation

Extending linear growth functionals to functions of bounded fractional variation

Structural Changes in Nonlocal Denoising Models Arising Through Bi-Level Parameter Learning

A sufficient condition for the lower semicontinuity of nonlocal supremal functionals in the vectorial case

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