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178
Small volume fraction limit of the diblock copolymer problem: II. Diffuseinterface functional
, 2010
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Recent analytical developments in micromagnetics.
 The Science of Hysteresis II: Physical Modeling, Micromagnetics, and Magnetization Dynamics. Elsevier,
, 2006
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Continuous Multiclass Labeling Approaches and Algorithms
 SIAM J. Imag. Sci
, 2011
"... We study convex relaxations of the image labeling problem on a continuous domain with regularizers based on metric interaction potentials. The generic framework ensures existence of minimizers and covers a wide range of relaxations of the originally combinatorial problem. We focus on two specific r ..."
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Cited by 28 (5 self)
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We study convex relaxations of the image labeling problem on a continuous domain with regularizers based on metric interaction potentials. The generic framework ensures existence of minimizers and covers a wide range of relaxations of the originally combinatorial problem. We focus on two specific relaxations that differ in flexibility and simplicity – one can be used to tightly relax any metric interaction potential, while the other one only covers Euclidean metrics but requires less computational effort. For solving the nonsmooth discretized problem, we propose a globally convergent DouglasRachford scheme, and show that a sequence of dual iterates can be recovered in order to provide a posteriori optimality bounds. In a quantitative comparison to two other firstorder methods, the approach shows competitive performance on synthetical and realworld images. By combining the method with an improved binarization technique for nonstandard potentials, we were able to routinely recover discrete solutions within 1%–5 % of the global optimum for the combinatorial image labeling problem. 1 Problem Formulation The multiclass image labeling problem consists in finding, for each pixel x in the image domain Ω ⊆ Rd, a label `(x) ∈ {1,..., l} which assigns one of l class labels to x so that the labeling function ` adheres to some local data fidelity as well as nonlocal spatial coherency constraints. This problem class occurs in many applications, such as segmentation, multiview reconstruction, stitching, and inpainting [PCF06]. We consider the variational formulation inf `:Ω→{1,...,l} f(`), f(`):= Ω s(x, `(x))dx ︸ ︷ ︷ ︸ data term + J(`). ︸ ︷ ︷ ︸ regularizer
FROM A LARGEDEVIATIONS PRINCIPLE TO THE WASSERSTEIN GRADIENT FLOW: A NEW MICROMACRO PASSAGE
, 2010
"... We study the connection between a system of many independent Brownian particles on one hand and the deterministic diffusion equation on the other. For a fixed time step h> 0, a largedeviations rate functional Jh characterizes the behaviour of the particle system at t = h in terms of the initial ..."
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Cited by 28 (18 self)
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We study the connection between a system of many independent Brownian particles on one hand and the deterministic diffusion equation on the other. For a fixed time step h> 0, a largedeviations rate functional Jh characterizes the behaviour of the particle system at t = h in terms of the initial distribution at t = 0. For the diffusion equation, a single step in the timediscretized entropyWasserstein gradient flow is characterized by the minimization of a functional Kh. We establish a new connection between these systems by proving that Jh and Kh are equal up to second order in h as h → 0. This result gives a microscopic explanation of the origin of the entropyWasserstein gradient flow formulation of the diffusion equation. Simultaneously, the limit passage presented here gives a physically natural description of the underlying particle system by describing it as an entropic gradient flow.
Action minimization and sharpinterface limits for the stochastic AllenCahn equation
 Commun. Pure Appl. Math
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Surface energies in nonconvex discrete systems
"... Nonconvex interactions in lattice systems lead to a number of interesting phenomena that can be translated into a variety of energies within their limit continuum description as the lattice size tends to zero. These effects may be due to different superposed causes. When only nearestneighbour inter ..."
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Cited by 25 (5 self)
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Nonconvex interactions in lattice systems lead to a number of interesting phenomena that can be translated into a variety of energies within their limit continuum description as the lattice size tends to zero. These effects may be due to different superposed causes. When only nearestneighbour interactions are taken into account,
A handbook of Γconvergence
 in “Handbook of Differential Equations – Stationary Partial Differential Equations
"... The notion of Γconvergence has become, over the more than thirty years after its introduction by Ennio De Giorgi, the commonlyrecognized notion of convergence for variational problems, and it would be difficult nowadays to think of any other ‘limit ’ than a Γlimit when talking about asymptotic an ..."
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Cited by 25 (13 self)
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The notion of Γconvergence has become, over the more than thirty years after its introduction by Ennio De Giorgi, the commonlyrecognized notion of convergence for variational problems, and it would be difficult nowadays to think of any other ‘limit ’ than a Γlimit when talking about asymptotic analysis in a general variational setting (even though special convergences may fit better specific problems, as Moscoconvergence, twoscale convergence, G and Hconvergence, etc.). This short presentation is meant as an introduction to the many applications of this theory to problems in Partial Differential Equations, both as an effective method for solving asymptotic and approximation issues and as a means of expressing results that are derived by other techniques. A complete introduction to the general theory of Γconvergence is the bynowclassical book by Gianni Dal Maso [85], while a userfriendly introduction can be found in my book ‘for beginners ’ [46], where also simplified onedimensional versions of many of the problems in this article are treated. These notes are addressed to an audience of experienced mathematicians, with some background and interest in Partial Differential Equations, and are meant to direct the reader to what I regard as the most interesting features of this theory. The style of the exposition is how I would present the subject to a colleague in a neighbouring field or to an interested PhD student: the issues that I think
Nonperiodic finiteelement formulation of orbitalfree density functional theory
 Journal of the Mechanics and Physics of Solids 55 (4), 669
"... We propose an approach to perform orbitalfree density functional theory calculations in a nonperiodic setting using the finiteelement method. We consider this a step towards constructing a seamless multiscale approach for studying defects like vacancies, dislocations and cracks that require qua ..."
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Cited by 23 (6 self)
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We propose an approach to perform orbitalfree density functional theory calculations in a nonperiodic setting using the finiteelement method. We consider this a step towards constructing a seamless multiscale approach for studying defects like vacancies, dislocations and cracks that require quantum mechanical resolution at the core and are sensitive to long range continuum stresses. In this paper, we describe a local real space variational formulation for orbitalfree density functional theory, including the electrostatic terms and prove existence results. We prove the convergence of the finiteelement approximation including numerical quadratures for our variational formulation. Finally, we demonstrate our method using examples.
Phase and antiphase boundaries in binary discrete systems: a variational viewpoint
, 2005
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Equilibrium configurations of epitaxially strained crystalline films: existence and regularity results
 Arch. Rational Mech. Anal
"... Strained epitaxial films grown on a relatively thick substrate are considered in the context of plane linear elasticity. The total free energy of the system is assumed to be the sum of the energy of the free surface of the film and the strain energy. Because of the lattice mismatch between film and ..."
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Cited by 23 (5 self)
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Strained epitaxial films grown on a relatively thick substrate are considered in the context of plane linear elasticity. The total free energy of the system is assumed to be the sum of the energy of the free surface of the film and the strain energy. Because of the lattice mismatch between film and substrate, flat configurations are in general energetically unfavourable and a corrugated or islanded morphology is the preferred growth mode of the strained film. After specifying the functional setup where the existence problem can be properly framed, a study of the qualitative properties of the solutions is undertaken. New regularity results for volume constrained local minimizers of the total free energy are established, leading, as a byproduct, to a rigorous proof of the zero contactangle condition between islands and wetting layers. 1