The **nonlocal** **interactions** are set constant all along calculation leading to unacceptable damage profile at the end. A damage equal to 1 corresponds actu- ally to a crack and should be concentrated on a line, whereas a band is obtained. Moreover, in the vicinity of a boundary, the weight function is chopped off and normalised. This results to a damage field preferably developed near the boundaries. It has already been pointed out that the regularisation technique should be altered close to boundaries (Krayani et al. 2009). Besides and in a more general sense, the apparition of a crack can be seen as the creation of a new bound- ary and thus, the **interactions** between points on both sides of this crack should vanish. A crack separates two areas that can no more interact with each other.

En savoir plus
An efficient and accurate numerical method via the NUFFT was proposed for the fast evaluation of dif- ferent **nonlocal** **interactions** including the Coulomb **interactions** in 3D/2D and the interaction kernel taken as either the Green’s function of the Laplace operator in 3D/2D/1D or **nonlocal** interaction kernels in 2D/1D obtained from the 3D Schr¨odinger-Poisson system under strongly external confining potentials via dimension reduction. The method was compared extensively with those existing numerical methods and was demon- strated that it can achieve much more accurate numerical results, especially on a smaller computational domain and/or with anisotropic interaction density. Efficient and accurate numerical methods were then presented for computing the ground state and dynamics of the nonlinear Schr¨odinger equation with nonlo- cal **interactions** by combining the normalized gradient flow with the backward Euler Fourier pseudospectral discretization and time-splitting Fourier pseudospectral method, respectively, together with the fast and accurate NUFFT method for evaluating the **nonlocal** **interactions**. Extensive numerical comparisons were carried out between the proposed numerical methods and other existing methods for studying ground state and dynamics of the NLSE with different **nonlocal** **interactions**. Numerical results showed that the meth- ods via the NUFFT perform much better than those existing methods in terms of accuracy and efficiency, especially when the computational domain is chosen smaller and/or the solution is anisotropic.

En savoir plus
* nikolaj.bittner@unifr.ch † philipp.werner@unifr.ch
features have been studied using exact diagonalization of small lattice systems [ 15 , 16 ] and very recently also with the time-dependent density matrix renormalization group [ 17 ]. These excitonic features can originate either from the **nonlocal** **interactions** [ 16 – 19 ] or the modification of the spin back- ground [ 10 ]. Methods for infinite lattices based on single-site dynamical mean field theory (DMFT) [ 20 ], such as extended DMFT [ 21 – 24 ] or the combination of GW and extended DMFT [ 25 – 27 ], can capture the dynamical screening of the local Coulomb interaction resulting from **nonlocal** interac- tions, but they cannot describe exciton formation. Here, we combine the two approaches by implementing a cluster exten- sion of nonequilibrium DMFT [ 8 , 28 ], with local and **nonlocal** **interactions** on the cluster. Specifically, we consider the U -V Hubbard model on the square lattice and the so-called dynam- ical cluster approximation (DCA) [ 29 , 30 ] with a periodized cluster of four sites. For this model, a recent equilibrium study has demonstrated a fast convergence of the results with cluster size [ 31 ]. The nonequilibrium DCA approach allows us to measure electron-hole correlations on the periodized four-site cluster in photodoped states, and to connect these results with other observables such as the photoemission spectrum.

En savoir plus
We use Newtonian potentials W 1 = W 2 = 0.1N , K = N for inter and intra-specific
**interactions**, with a mobility β = 0.3. At the beginning of the simulation, we observe that the predator is getting closer to the preys. When the group of preys is close, the preys create a circular pattern around the the predators in order to run away from him.

, (2)
where M is the mobility of species assumed to be order parameter independent [ 9 ] and the fluctuations of the order parameter are neglected [ 1 , 10 ]. Elastic or coulombic **nonlocal** **interactions** deriving from an effective pairwise interaction potential, W (x − x ), can be additively introduced in F (η(x,t)) [ 8 , 11 ], as shown by Leibler [ 12 ], and Ohta and Kawasaki [ 8 ] for the case of block copolymers at equilibrium. For an evolution governed by the CH equation, the nature, the number, and the energy barriers associated with the equilibrium phases depend only on the extrema of F (η(x,t)). Besides, previous works clearly show that the existence of ordered and disordered phases at thermodynamic equilibrium can be determined solely from the study of the quadratic term of this free-energy functional [ 1 , 2 , 8 , 9 , 13 ].

En savoir plus
1 INTRODUCTION
During the development of nonlinearities, the soften- ing behavior of some materials (e.g. ductile failure in metals, quasi-brittle failure in concrete, etc.) leads to the appearance of a localization process zone finite in size. Several theories were proposed to provide a description of these phenomena (G. & Bazant 1987, Pijaudier-Cabot & Benallal 1993, Baˇzant & Jir´asek 2002, Fr´emond & Nedjar 1996, Peerlings, Geers, de Borst, & Brekelmans 2001, Miehe, Welschinger, & Hofacker 2010, Mo¨es, Stolz, Bernard, & Chevau- geon 2011). Their common feature consists in the in- troduction of an internal length expressing **nonlocal** **interactions** in the localization process zone. Further- more, these methods allow to avoid problems of non- objective results (mesh dependency) that can appear when using a finite element method for the solution of the quasi-static boundary value problem.

En savoir plus
I have so far only investigate the one dimensional case. Maximum and com- parison principles in multi-dimension for various type of **nonlocal** operators are currently under investigation and appears to be largely be an open question.
As a first consequence of this investigation on maximum principles, I obtain a generalized version of Theorem 1.4. More precisely, I prove the following result. Theorem 1.11. Let Ω = (r, R) for some real r < 0 < R, g an increasing function and J be such that [−b, −a] ∪ [a, b] ⊂ supp(J ) ∩ Ω for some constant 0 ≤ a < b. Then any smooth solution u of

En savoir plus
In this thesis we present a few contributions to geometric variational problems involving **nonlocal** energies. Those contributions may be sorted into two main different topics: the first one is the geometry and singularities of fractional harmonic maps, and the second one is an isoperimetric problem with a **nonlocal** repulsive potential, in particular the study of its large mass minimizers. Each chapter is self-contained: Chapter 1 is a submitted paper written in collaboration with V. Millot, Chapter 2 is a paper to be submitted, in collaboration with V. Millot and A. Schikorra, and Chapter 3 is a work conducted by myself in parallel with the other two. Chapters 1 and 2 are devoted to the topic of fractional harmonic maps, while Chapter 3 is devoted to the aforementioned isoperimetric problem. In Appendix A , we collect some well-known facts about the fractional laplacian, which we define in a general, distributional setting which encompasses the functional setting adopted in Chapters 1 and 2 , and give rigorous proofs of elliptic regularity for the distributional fractional laplacian.

En savoir plus
218 En savoir plus

Solvability subject to a **nonlocal** Robin BC has been studied in [ 11–13 ]. Numerical study based on monotonicity methods has been performed in [ 14 ].
The paper is organized as follows. First we perform the time discretization based on Rothe’s method. The nonlinear problem on each time point is solved using a relaxation technique. In this way a solution to the nonlinear problem is approached by a sequence of linear BVPs with a **nonlocal** BC. The well-posedness of these linear problems is based on the superposition principle - cf. [ 13 ]. We show the convergence of the relaxation iterations. Further we perform the stability analysis and finally we prove the existence of a solution to the original nonlinear parabolic problem. The last section is devoted to some numerical experiments.

En savoir plus
[17] W. H. Fleming and P. E. Souganidis. PDE-viscosity solution approach to some problems of large deviations. Ann. Scuola Norm. Sup. Pisa Cl. Sci., 4:171–192, 1986.
[18] S. Genieys, V. Volpert, and P. Auger. Pattern and waves for a model in population dynamics with **nonlocal** consumption of resources. Math. Model. Nat. Phenom., 1(1):63–80, 2006. [19] P.-E. Jabin and G. Raoul. Selection dynamics with competition. J. Math. Biol., To appear. [20] P. L. Lions. Generalized solutions of Hamilton-Jacobi equations, volume 69 of Research

Briefly, the idea in [7] is that, on a particle model with discrete space variable, the entropy structure is obtained by imposing that a pair of particles is jumping together with a suitable rate. Trying to use this idea for a **nonlocal** approximation, we intuitively want to make pairs of particle with a given distance jump together. In order to identify the pairs, we therefore take the convolution reflected between the two species.

robust and reliable automatic extraction of anatomical structures. Recently, template-warping methods incorporating a label fusion strategy have demonstrated high accuracy in segmenting cerebral structures. In this study, we propose a novel patch-based method using expert segmentation priors to achieve this task. Inspired by recent work in image denoising, the proposed **nonlocal** patch-based label fusion produces accurate and robust segmentation. During our experiments, the hippocampi of 80 healthy subjects were segmented. The influence on segmentation accuracy of different parameters such as patch size or number of training subjects was also studied. Moreover, a comparison with an appearance-based method and a template-based method was carried out. The highest median kappa value obtained with the proposed method was 0.884, which is competitive compared with recently published methods.

En savoir plus
Presented by ...
Abstract
We study a generalization of the porous medium equation involving **nonlocal** terms. More precisely, explicit self- similar solutions with compact support generalizing the Barenblatt solutions are constructed. We also present a formal argument to get the L p decay of weak solutions of the corresponding Cauchy problem.

any substitution that acts as the identity on S, the same property holds also true of ∆(φi(xT )), from which the Lemma follows.
3 Some remarkable Hopf algebras
Our favorite examples in view of applications to quantum chemistry, QFT and solid state physics are simple ones. However the generality chosen (allowing for example n2 6= 0) is natural to handle **nonlocal** interaction terms in the La- grangians (think for example of the quantum chemistry approach with Coulomb interaction). Our general approach also paves the way to a unification of QFT techniques (Feynman-type diagrammatics), umbral calculus (duality and lin- ear forms on polynomial algebras) and combinatorics of (possibly ordered) set partitions.

En savoir plus
with sign-changing coefficients. Numerical results indicate that the proposed **nonlocal** model has some key advantages over the local one.
I. I NTRODUCTION
In electromagnetism, one can model materials that exhibit (almost) real-valued and strictly negative electric permittivity and/or magnetic permeability, within given frequency ranges. These so-called metamaterials, or left- handed materials, raise unusual questions. Among others, proving the existence of electromagnetic fields, and computing them, is a challenging issue for a problem set in a domain Ω ⊂ R n (n = 2, 3) divided into a classical dielectric material and a metamaterial, when the frequency is in the above mentioned frequency range (see for in- stance [11, 12]). The main issue is that the problem is ill-posed on some situations (see § II.), and as a consequence its numerical solution is unstable. To adress this difficulty, we propose to reformulate/transform the problem by using a **nonlocal** framework (§ III.), and then to study the numerical approximation of the **nonlocal** problem (§ IV.). Another approach would be to add a small, fixed imaginary part to the sign-changing coefficient that appears in the principal part of the PDE, in regions where it takes negative values [8].

En savoir plus
I. INTRODUCTION A. **Nonlocal** rheology
Gravity-driven particulate flows are not only a central issue in many geophysical processes but also of great concern in a variety of technological domains involving chemical, phar- maceutical, food, metallurgical, or construction industries. The crucial difficulty to overcome to bring a coherent and useful conceptual vision of these flows still lies in a poor understanding of the passage between flow and blockade. In earth processes, these questions pertain as a missing piece necessary to provide a full understanding in many situations such as dune migration [ 1 ], landslides [ 2 – 4 ], snow avalanches [ 5 ], underwater gravity currents [ 6 ], and coastal geomorphology [ 7 , 8 ], which also are encumbered with many unresolved questions around the mechanics of the dense solid fraction motion and its possible mobilization by a particulate flow.

En savoir plus
Introduction
The effect and the formulation of boundary conditions – such as free edges, notches and initial cracks – remain an open question for **nonlocal** models. The main drawback of the classical **nonlocal** integral theory [1] consists in the nonphysical interaction, through the **nonlocal** averaging process, of points across a crack or a hole. The definition of natural boundary conditions of vanishing strain normal derivative at a free edge is still under discussion for gradient formulations [2, 3]. The continuous nucleation of a crack of zero thickness is not so simple as the thickness of a localization band is more or less proportional to the internal length introduced. Local behavior along free edges – i.e. with a vanishing internal length – has been obtained by some authors [4]. The consideration of an internal length evolving with damage [5, 6] seems a way to properly bridge Damage Mechanics and Fracture Mechanics as the internal length may then vanish for large values of damage.

En savoir plus
An originality of our work lies in the non-local spatial dispersion operator L which allows to consider non-connected domains. While the role of the **nonlocal** dispersion and the fragmentation of the environ- ment is significant in many situations, as in the adaptation of forest trees to the climate change because of the effect of the wind on the seeds or the pollens, very few theoretical works take it into account [21]. The non-local dispersion may have antagonistic effects on the population dynamics. On the one hand, it may allow the population to reach new favorable geographic regions which are not accessible by a local diffusion. On the other hand, it may also impede local adaptation by bringing individuals with locally maladapted traits from other regions.

En savoir plus