Browsing by Author "Van Goethem, Nicolas"

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    Currents and dislocations at the continuum scale
    (SISSA, 2013-07-03) Scala, Riccardo; Van Goethem, Nicolas; Mathematics
    A striking geometric property of elastic bodies with dislocations is their non-Riemannian nature in the sense that the deformation cannot be written as the gradient of a one-to-one immersion, and hence that no displacement field can properly be defined as model variable. In fact, the deformation curl equals to the density of dislocations which is a concentrated Radon measure in the dislocation lines. In this paper we consider a countable family of dislocations, discuss the mathematical properties of such constraint deformations and study a variational problem in finite-strain elasticity. It turns out that both the deformation and the dislocation lines may be modelled by means of the mathematical theory of currents. In particular, Cartesian maps allow one to consider deformations in L^p with 1\leq p<2, which are appropriate for dislocation-induced strain singularities. Moreover, integer-multiplicity currents are perfectly suited to describe either the static or the dynamics of families of dislocations which mutually interact, and possibly form complex structures such as clusters. Though the evolution of dislocations is not considered in this paper, it is the main motivation of our approach. As a matter of fact, this work describes a conservative ground state where dislocations are assumed to obey energy minimization principles, and over which any relevant effect resorting to thermodynamics outside equilibrium might be added in a subsequent step within the proposed mathematical formalism.
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    Minimal partitions and image classification using a gradient-free perimeter approximation
    (SISSA, 2013-07-03) Amstutz, Samuel; Novotny, Antonio André; Van Goethem, Nicolas; Mathematics
    In this paper a new mathematically-founded method for the optimal partitioning of domains, with applications to the classification of greyscale and color images, is proposed. Since optimal partition problems are in general ill-posed, some regularization strategy is required. Here we regularize by a non-standard approximation of the total interface length, which does not involve the gradient of approximate characteristic functions, in contrast to the classical Modica-Mortola approximation. Instead, it involves a system of uncoupled linear partial differential equations and nevertheless shows $\Gamma$-convergence properties in appropriate function spaces. This approach leads to an alternating algorithm that ensures a decrease of the objective function at each iteration, and which always provides a partition, even during the iterations. The efficiency of this algorithm is illustrated by various numerical examples. Among them we consider binary and multilabel minimal partition problems including supervised or automatic image classification, inpainting, texture pattern identification and deblurring.