Localization of nonlocal gradients in various topologies

Mengesha, Tadele; Spector, Daniel

doi:10.1007/s00526-014-0711-3

Cited by 42 publications

(86 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Here we prove several results analogous to the first order case, as in [MS14], for the generalizations involving generic radial weights that satisfy (1.2)-(1.4). Of primary importance is to first establish that the distributional non-local Hessian defined by equation (1.10) is, in fact, a distribution.…”

Section: Analysis Of the Non-local Hessianmentioning

confidence: 62%

“…Analogously with the first order case in [MS14], we can define a second order non-local divergence that corresponds to H n , and we can also derive a second order non-local integration by parts formula which is an essential tool for the proofs of this section. The second order non-local divergence is defined for a function φ = (φ ij )…”

Section: Consider Now a Sequence (Vmentioning

confidence: 99%

“…Then, one defines the non-local total variation as the L 1 norm of (1.5). The localization analysis of the non-local gradient (1.5) has been performed by Mengesha and Spector in [MS14], where a more general (and technical) distributional definition is utilized. Their first observation is that the definition of the non-local gradient via the Lebesgue integral (1.5) extends to spaces of weakly differentiable functions.…”

Section: Introductionmentioning

confidence: 99%

“…From this perspective, the development of a non-local higherorder method promises to be interesting. One approach to a non-local higher-order method would be to approximate the gradients using non-local gradients as developed in [MS14]. For example, one can define a form of non-local Hessian as…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Analysis and Application of a Nonlocal Hessian

Lellmann

Papafitsoros

Schönlieb

et al. 2015

SIAM J. Imaging Sci.

Self Cite

View full text Add to dashboard Cite

Abstract. In this work we introduce a formulation for a non-local Hessian that combines the ideas of higher-order and non-local regularization for image restoration, extending the idea of non-local gradients to higher-order derivatives. By carefully choosing the weights, the model allows to improve on the current state of the art higher-order method, Total Generalized Variation, with respect to overall quality and particularly close to jumps in the data. In the spirit of recent work by Brezis et al., our formulation also has analytic implications: for a suitable choice of weights, it can be shown to converge to classical second-order regularizers, and in fact allows a novel characterization of higher-order Sobolev and BV spaces.

show abstract

Section: Analysis Of the Non-local Hessianmentioning

confidence: 62%

Section: Consider Now a Sequence (Vmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Analysis and Application of a Nonlocal Hessian

Lellmann

Papafitsoros

Schönlieb

et al. 2015

SIAM J. Imaging Sci.

Self Cite

View full text Add to dashboard Cite

show abstract

“…The Theorem 1.2 and the Theorem 1.3 are an alternative characterization of the spaces of Orlicz Sobolev using non local relations, such relations are at the base of numerous results for not local functionals with standard growths and for functionals defined on fractional Sobolev space, see (Bourgain et al, 2001;Di Castro et al, 2016;Di Castro et al, 2014;Maz'ya & Shaposhnikova, 2002;Mengesha & Spector;Milman, 2005;Ponce, 2004;Schikarra et al;. The generalization of these theorems seems to point out the possibility to also extend such results in more general cases.…”

Section: Introductionmentioning

confidence: 92%

Characterization of Orlicz Sobolev Spaces

Granucci¹

2018

JMR

View full text Add to dashboard Cite

show abstract

Quasistatic elastoplasticity via Peridynamics: existence and localization

Kružík

Mora-Corral

Stefanelli

2018

Continuum Mech. Thermodyn.

View full text Add to dashboard Cite

Peridynamics is a nonlocal continuum-mechanical theory based on minimal regularity on the deformations. Its key trait is that of replacing local constitutive relations featuring spacial differential operators with integrals over differences of displacement fields over a suitable positive interaction range. The advantage of such perspective is that of directly including nonregular situations, in which discontinuities in the displacement field may occur. In the linearized elastic setting, the mechanical foundation of the theory and its mathematical amenability have been thoroughly analyzed in the last years.We present here the extension of Peridynamics to linearized elastoplasticity. This calls for considering the time evolution of elastic and plastic variables, as the effect of a combination of elastic energy storage and plastic energy dissipation mechanisms. The quasistatic evolution problem is variationally reformulated and solved by time discretization. In addition, by a rigorous evolutive Γ-convergence argument we prove that the nonlocal peridynamic model converges to classic local elastoplasticity as the interaction range goes to zero. 2

show abstract

Localization of nonlocal gradients in various topologies

Cited by 42 publications

References 19 publications

Analysis and Application of a Nonlocal Hessian

Analysis and Application of a Nonlocal Hessian

Characterization of Orlicz Sobolev Spaces

Quasistatic elastoplasticity via Peridynamics: existence and localization

Contact Info

Product

Resources

About