Introduction: Big data and partial differential equations

Gennip, Yves van; Schönlieb, Carola-Bibiane

doi:10.1017/s0956792517000304

Cited by 2 publications

(1 citation statement)

References 38 publications

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“…In the second case, the family of approximating functionals we consider is the one introduced by Slepĉev and García-Trillos in the context of point clouds (see [32]), and that are of wide interest for the community ( [3,7,9,10,19,20,28,29,30,31,33]). The main technical result in studying this case is a wriggling result for the weighted total variation functional (see Proposition 5.8), that allows us to use Theorem 3.10 to identify the Γ-limit in Proposition 5.14.…”

Section: Introductionmentioning

confidence: 99%

On the Gamma Convergence of Functionals Defined Over Pairs of Measures and Energy-Measures

Caroccia

Cristoferi

2020

J Nonlinear Sci

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A novel general framework for the study of Γ-convergence of functionals defined over pairs of measures and energy-measures is introduced. This theory allows us to identify the Γ-limit of these kind of functionals by knowing the Γ-limit of the underlying energies. In particular, the interaction between the functionals and the underlying energies results, in the case these latter converge to a non continuous energy, in an additional effect in the relaxation process. This study was motivated by a question in the context of epitaxial growth evolution with adatoms. Interesting cases of application of the general theory are also presented.

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Section: Introductionmentioning

confidence: 99%

On the Gamma Convergence of Functionals Defined Over Pairs of Measures and Energy-Measures

Caroccia

Cristoferi

2020

J Nonlinear Sci

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Large data limit for a phase transition model with the p-Laplacian on point clouds

Cristoferi

Thorpe

2018

Eur. J. Appl. Math

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The consistency of a nonlocal anisotropic Ginzburg-Landau type functional for data classification and clustering is studied. The Ginzburg-Landau objective functional combines a double well potential, that favours indicator valued functions, and the p-Laplacian, that enforces regularity. Under appropriate scaling between the two terms minimisers exhibit a phase transition on the order of ε = ε n where n is the number of data points. We study the large data asymptotics, i.e. as n → ∞, in the regime where ε n → 0. The mathematical tool used to address this question is Γ-convergence. It is proved that the discrete model converges to a weighted anisotropic perimeter. procedure in order to obtain some desired features of the classification. Furthermore, understanding the large data limits can open up new algorithms. This paper is part of an ongoing project aimed at justifying analytically the consistency of several models for soft labelling used by practitioners. Here we consider a generalization of the approach introduced by Bertozzi and Flenner in [7] (see also [17], for an introduction on this topic see [56]), where a Ginzburg-Landau (or Modica-Mortola, see [40,41]) type functional is used as the underlining energy to minimize in the context of the soft classification problem. The functional we consider is a discretisation of the non-local Ginzburg-Landau functional studied by Alberti and Bellettini [1,2] with the generalisation that we consider non-uniform densities and p (rather than 2 ) cost on finite differences. Our goal is to prove the consistency of the model.There are multiple extensions to the approach we consider here; for instance our Ginzburg-Landau functional is based on the p-Laplacian, one can also consider the normalised p-Laplacian or the random walk Laplacian (see [42,47]). Further open problems concern the extention to multi-phase labelling (see [30]) and convergence of the associated gradient flows.The paper is organized as follows: in the following subsection we define the discrete model, and in Subsection 1.2 we define the continuum limiting problem. The main results are given in Section 1.3 with the proofs presented in Sections 4 and 5. In Section 1.4 we give an overview on the related literature. In Sections 1.5 and 1.6 we include two examples with the purpose of demonstrating key properties of our functional; in particular how the choice of p effects minimizers of our Ginzburg-Landau functional and an example to image segmentation. Section 2 contains some prelimimary material we include for the convenience of the reader. Finally, Section 3 is devoted to the proofs of some technical results that are of interest in their own right, and are later used in the proofs in Section 4.The consistency of the model is studied using Γ-convergence (see Section 2.4), a very important tool introduced by De Giorgi in the 70's to understand the limiting behavior of a sequence of functionals (see [21]). This kind of variational convergence gives, almost immediately, convergence of minimizers.

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Introduction: Big data and partial differential equations

Cited by 2 publications

References 38 publications

On the Gamma Convergence of Functionals Defined Over Pairs of Measures and Energy-Measures

On the Gamma Convergence of Functionals Defined Over Pairs of Measures and Energy-Measures

Large data limit for a phase transition model with the p-Laplacian on point clouds

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