Nonlocal optimal design: A new perspective about the approximation of solutions in optimal design

This is a follow-up of a paper by , where the classical concept of H-convergence was extended to fractional p-Laplace type operators. In this short paper we provide an explicit characterization of this notion by demonstrating that the weak- * convergence of the coefficients is an equivalent condition for H-convergence of the sequence of nonlocal operators. This result takes advantage of nonlocality and is in stark contrast to the local p-Laplacian case.

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“…This result generalizes, giving a simpler proof, previous results for a related nonlocal situation in the linear case [3,Th. 6].…”

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confidence: 87%

A simple characterization of H-convergence for a class of nonlocal problems

Bellido

Evgrafov

2020

Rev Mat Complut

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“…All details for the derivation of these achievements can be found at papers. 30,31,38,42,43 The main result of this paper is the existence of nonlocal optimal design (Theorem 4). The proof in given in Section 3.…”

Section: Contributions and Organization Of The Papermentioning

confidence: 99%

“…[23][24][25][26][27][28][29] However, there is not much literature concerning nonlocal optimal design models. We refer, among others, Andrés and Muñoz, 30,31 Bonder and Spedaletti, 32 D'Elia and Gunzburger, 33,34 and Zhou and Du 35,36 mainly in the context of elliptic equations. Also, Bonder et al 37 is a valuable paper, where an H-convergence study has been carried out.…”

Section: Introductionmentioning

confidence: 99%

Existence and approximation of nonlocal optimal design problems driven by parabolic equations

Andrés

Múñoz

Rosado

2019

Math Methods in App Sciences

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This work is a follow-up to a series of articles by the authors where the same topic for the elliptic case is analyzed. In this article, a class of nonlocal optimal design problem driven by parabolic equations is examined. After a review of results concerning existence and uniqueness for the state equation, a detailed formulation of the nonlocal optimal design is given. The state equation is of nonlocal parabolic type, and the associated cost functional belongs to a broad class of nonlocal integrals. In the first part of the work, a general result on the existence of nonlocal optimal design is proved. The second part is devoted to analyzing the convergence of nonlocal optimal design problems toward the corresponding classical problem of optimal design. After a slight modification of the problem, either on the cost functional or by considering a new set of admissibility, the G-convergence for the state equation and, consequently, the convergence of the nonlocal optimal design problem are proved.

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“…where Ω ⊂ R n is an open subset, u : Ω → R d is in some Lebesgue space L p , and the integrand w : Ω × Ω × R d × R d → R ∪ {∞} has some measurability and continuity properties. This kind of functionals appears in many contexts in the mathematical modelling of some processes, whose common feature is their nonlocal nature; we mention here micromagnetics [38], phase transitions [4], peridynamics [39], pattern formation [25], image processing [27], population dispersal [20], diffusion [8] and optimal design [5]. It also has applications in the characterization of Sobolev spaces [16].…”

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confidence: 99%

“…Boundary conditions, coercivity and existence of minimizers. In this section we give conditions for the existence of minimizers of (5) I :…”

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confidence: 99%

Lower Semicontinuity and Relaxation Via Young Measures for Nonlocal Variational Problems and Applications to Peridynamics

Bellido¹,

Mora-Corral²

2018

SIAM J. Math. Anal.

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We study nonlocal variational problems in L p , like those that appear in peridynamics. The functional object of our study is given by a double integral. We establish characterizations of weak lower semicontinuity of the functional in terms of nonlocal versions of either a convexity notion of the integrand, or a Jensen inequality for Young measures. Existence results, obtained through the direct method of the Calculus of variations, are also established. We cover different boundary conditions, for which the coercivity is obtained from nonlocal Poincaré inequalities. Finally, we analyze the relaxation (that is, the computation of the lower semicontinuous envelope) for this problem when the lower semicontinuity fails. We state a general relaxation result in terms of Young measures and show, by means of two examples, the difficulty of having a relaxation in L p in an integral form. At the root of this difficulty lies the fact that, contrary to what happens for local functionals, non-positive integrands may give rise to positive nonlocal functionals.

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Nonlocal optimal design: A new perspective about the approximation of solutions in optimal design

Cited by 16 publications

References 34 publications

A simple characterization of H-convergence for a class of nonlocal problems

A simple characterization of H-convergence for a class of nonlocal problems

Existence and approximation of nonlocal optimal design problems driven by parabolic equations

Lower Semicontinuity and Relaxation Via Young Measures for Nonlocal Variational Problems and Applications to Peridynamics

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