scholarly journals Nonlocal-Interaction Equation on Graphs: Gradient Flow Structure and Continuum Limit

2021 ◽  
Author(s):  
Antonio Esposito ◽  
Francesco S. Patacchini ◽  
André Schlichting ◽  
Dejan Slepčev
Keyword(s):  
Gradient Flow ◽  
Wasserstein Distance ◽  
Convergence Result ◽  
Existence Theory ◽  
Nonlocal Interaction ◽  
Interaction Energies ◽  
Gradient Flows ◽  
Empirical Measures ◽  
Interaction Equation ◽  
The Continuum

AbstractWe consider dynamics driven by interaction energies on graphs. We introduce graph analogues of the continuum nonlocal-interaction equation and interpret them as gradient flows with respect to a graph Wasserstein distance. The particular Wasserstein distance we consider arises from the graph analogue of the Benamou–Brenier formulation where the graph continuity equation uses an upwind interpolation to define the density along the edges. While this approach has both theoretical and computational advantages, the resulting distance is only a quasi-metric. We investigate this quasi-metric both on graphs and on more general structures where the set of “vertices” is an arbitrary positive measure. We call the resulting gradient flow of the nonlocal-interaction energy the nonlocal nonlocal-interaction equation (NL$$^2$$ 2 IE). We develop the existence theory for the solutions of the NL$$^2$$ 2 IE as curves of maximal slope with respect to the upwind Wasserstein quasi-metric. Furthermore, we show that the solutions of the NL$$^2$$ 2 IE on graphs converge as the empirical measures of the set of vertices converge weakly, which establishes a valuable discrete-to-continuum convergence result.

scholarly journals PASSING TO THE LIMIT IN MAXIMAL SLOPE CURVES: FROM A REGULARIZED PERONA–MALIK EQUATION TO THE TOTAL VARIATION FLOW

2012 ◽  
Vol 22 (08) ◽  
pp. 1250017 ◽  
Author(s):  
MARIA COLOMBO ◽  
MASSIMO GOBBINO
Keyword(s):  
Total Variation ◽  
General Result ◽  
General Principle ◽  
Time Scale ◽  
Space Dimension ◽  
Gradient Flow ◽  
Convergence Result ◽  
Gradient Flows ◽  
Slow Time ◽  
Total Variation Flow

We prove that solutions of a mildly regularized Perona–Malik equation converge, in a slow time scale, to solutions of the total variation flow. The convergence result is global-in-time, and holds true in any space dimension. The proof is based on the general principle that "the limit of gradient-flows is the gradient-flow of the limit". To this end, we exploit a general result relating the Gamma-limit of a sequence of functionals to the limit of the corresponding maximal slope curves.

scholarly journals The nonlocal-interaction equation near attracting manifolds

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Francesco S. Patacchini ◽  
Dejan Slepčev
Keyword(s):  
Numerical Experiments ◽  
Compact Manifold ◽  
Attractive Potential ◽  
Nonlocal Interaction ◽  
Well Posedness ◽  
Gradient Flows ◽  
Compact Sets ◽  
Interaction Equation ◽  
Sets With Positive Reach ◽  
Regular Sets

<p style='text-indent:20px;'>We study the approximation of the nonlocal-interaction equation restricted to a compact manifold <inline-formula><tex-math id="M1">\begin{document}$ {\mathcal{M}} $\end{document}</tex-math></inline-formula> embedded in <inline-formula><tex-math id="M2">\begin{document}$ {\mathbb{R}}^d $\end{document}</tex-math></inline-formula>, and more generally compact sets with positive reach (i.e. prox-regular sets). We show that the equation on <inline-formula><tex-math id="M3">\begin{document}$ {\mathcal{M}} $\end{document}</tex-math></inline-formula> can be approximated by the classical nonlocal-interaction equation on <inline-formula><tex-math id="M4">\begin{document}$ {\mathbb{R}}^d $\end{document}</tex-math></inline-formula> by adding an external potential which strongly attracts to <inline-formula><tex-math id="M5">\begin{document}$ {\mathcal{M}} $\end{document}</tex-math></inline-formula>. The proof relies on the Sandier–Serfaty approach [<xref ref-type="bibr" rid="b23">23</xref>,<xref ref-type="bibr" rid="b24">24</xref>] to the <inline-formula><tex-math id="M6">\begin{document}$ \Gamma $\end{document}</tex-math></inline-formula>-convergence of gradient flows. As a by-product, we recover well-posedness for the nonlocal-interaction equation on <inline-formula><tex-math id="M7">\begin{document}$ {\mathcal{M}} $\end{document}</tex-math></inline-formula>, which was shown [<xref ref-type="bibr" rid="b10">10</xref>]. We also provide an another approximation to the interaction equation on <inline-formula><tex-math id="M8">\begin{document}$ {\mathcal{M}} $\end{document}</tex-math></inline-formula>, based on iterating approximately solving an interaction equation on <inline-formula><tex-math id="M9">\begin{document}$ {\mathbb{R}}^d $\end{document}</tex-math></inline-formula> and projecting to <inline-formula><tex-math id="M10">\begin{document}$ {\mathcal{M}} $\end{document}</tex-math></inline-formula>. We show convergence of this scheme, together with an estimate on the rate of convergence. Finally, we conduct numerical experiments, for both the attractive-potential-based and the projection-based approaches, that highlight the effects of the geometry on the dynamics.</p>

Equilibrium measure for a nonlocal dislocation energy with physical confinement

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Maria Giovanna Mora ◽  
Alessandro Scagliotti
Keyword(s):  
Equilibrium Measure ◽  
Symmetric Case ◽  
Nonlocal Interaction ◽  
Interaction Energies ◽  
Coulomb Interactions ◽  
Positive Edge ◽  
Dislocation Energy ◽  
Explicit Measure ◽  
Edge Dislocations

Abstract In this paper, we characterize the equilibrium measure for a family of nonlocal and anisotropic energies I α I_{\alpha} that describe the interaction of particles confined in an elliptic subset of the plane. The case α = 0 \alpha=0 corresponds to purely Coulomb interactions, while the case α = 1 \alpha=1 describes interactions of positive edge dislocations in the plane. The anisotropy into the energy is tuned by the parameter 𝛼 and favors the alignment of particles. We show that the equilibrium measure is completely unaffected by the anisotropy and always coincides with the optimal distribution in the case α = 0 \alpha=0 of purely Coulomb interactions, which is given by an explicit measure supported on the boundary of the elliptic confining domain. Our result does not seem to agree with the mechanical conjecture that positive edge dislocations at equilibrium tend to arrange themselves along “wall-like” structures. Moreover, this is one of the very few examples of explicit characterization of the equilibrium measure for nonlocal interaction energies outside the radially symmetric case.

scholarly journals A non-exponential extension of Sanov’s theorem via convex duality

2020 ◽  
Vol 52 (1) ◽  
pp. 61-101
Author(s):  
Daniel Lacker
Keyword(s):  
Optimal Transport ◽  
Large Deviation ◽  
Optimization Problems ◽  
Risk Measures ◽  
Approximate Solutions ◽  
Variational Problems ◽  
Dual Pair ◽  
Wasserstein Distance ◽  
Empirical Measures ◽  
Empirical Distributions

AbstractThis work is devoted to a vast extension of Sanov’s theorem, in Laplace principle form, based on alternatives to the classical convex dual pair of relative entropy and cumulant generating functional. The abstract results give rise to a number of probabilistic limit theorems and asymptotics. For instance, widely applicable non-exponential large deviation upper bounds are derived for empirical distributions and averages of independent and identically distributed samples under minimal integrability assumptions, notably accommodating heavy-tailed distributions. Other interesting manifestations of the abstract results include new results on the rate of convergence of empirical measures in Wasserstein distance, uniform large deviation bounds, and variational problems involving optimal transport costs, as well as an application to error estimates for approximate solutions of stochastic optimization problems. The proofs build on the Dupuis–Ellis weak convergence approach to large deviations as well as the duality theory for convex risk measures.

scholarly journals Finite thermoelastoplasticity and creep under small elastic strains

2018 ◽  
Vol 24 (4) ◽  
pp. 1161-1181 ◽  
Author(s):  
Tomáš Roubíček ◽  
Ulisse Stefanelli
Keyword(s):  
Convergence Result ◽  
Large Strains ◽  
Gradient Plasticity ◽  
Fourier Law ◽  
Large Plastic Strain ◽  
Rate Dependent ◽  
Galerkin Approximations ◽  
Elastic Strains ◽  
Rigorous Mathematical Analysis ◽  
The Continuum

A mathematical model for an elastoplastic continuum subject to large strains is presented. The inelastic response is modelled within the frame of rate-dependent gradient plasticity for non-simple materials. Heat diffuses through the continuum by the Fourier law in the actual deformed configuration. Inertia makes the nonlinear problem hyperbolic. The modelling assumption of small elastic Green–Lagrange strains is combined in a thermodynamically consistent way with the possibly large displacements and large plastic strain. The model is amenable to a rigorous mathematical analysis. The existence of suitably defined weak solutions and a convergence result for Galerkin approximations is proved.

scholarly journals A Review of Geometry, Construction and Modelling for Carbon Nanotori

2019 ◽  
Vol 9 (11) ◽  
pp. 2301 ◽  
Author(s):  
Pakhapoom Sarapat ◽  
James Hill ◽  
Duangkamon Baowan
Keyword(s):  
Carbon Nanotubes ◽  
Carbon Nanostructures ◽  
Single Walled Carbon Nanotubes ◽  
Continuum Approximation ◽  
Interaction Energies ◽  
Jones Potential ◽  
Lennard Jones ◽  
Walled Carbon Nanotubes ◽  
Analytical Expressions ◽  
The Continuum

After the discovery of circular formations of single walled carbon nanotubes called fullerene crop circles, their structure has become one of the most researched amongst carbon nanostructures due to their particular interesting physical properties. Several experiments and simulations have been conducted to understand these intriguing objects, including their formation and their hidden characteristics. It is scientifically conceivable that these crop circles, nowadays referred to as carbon nanotori, can be formed by experimentally bending carbon nanotubes into ring shaped structures or by connecting several sections of carbon nanotubes. Toroidal carbon nanotubes are likely to have many applications, especially in electricity and magnetism. In this review, geometry, construction, modelling and possible applications are discussed and the existing known analytical expressions, as obtained from the Lennard-Jones potential and the continuum approximation, for their interaction energies with other nanostructures are summarised.

scholarly journals On the total variation Wasserstein gradient flow and the TV-JKO scheme

2019 ◽  
Vol 25 ◽  
pp. 42
Author(s):  
Guillaume Carlier ◽  
Clarice Poon
Keyword(s):  
Total Variation ◽  
Nonlinear Evolution Equation ◽  
Nonlinear Evolution ◽  
Minimum Principle ◽  
Gradient Flow ◽  
Convergence Result ◽  
Symmetric Case ◽  
Qualitative Properties ◽  
Time Step ◽  
Dimension One

We study the JKO scheme for the total variation, characterize the optimizers, prove some of their qualitative properties (in particular a form of maximum principle and in some cases, a minimum principle as well). Finally, we establish a convergence result as the time step goes to zero to a solution of a fourth-order nonlinear evolution equation, under the additional assumption that the density remains bounded away from zero, this lower bound is shown in dimension one and in the radially symmetric case.

Γ-convergence and relaxations for gradient flows in metric spaces: a minimizing movement approach

2019 ◽  
Vol 25 ◽  
pp. 28
Author(s):  
Florentine Fleißner
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Gradient Flow ◽  
Gradient Flows ◽  
Flow Motion ◽  
Limit Functional ◽  
Γ Convergence

We present new abstract results on the interrelation between the minimizing movement scheme for gradient flows along a sequence of Γ-converging functionals and the gradient flow motion for the corresponding limit functional, in a general metric space. We are able to allow a relaxed form of minimization in each step of the scheme, and so we present new relaxation results too.

scholarly journals Large N scaling and factorization in SU(N) Yang-Mills theory

2018 ◽  
Vol 175 ◽  
pp. 11018 ◽  
Author(s):  
Miguel García Vera ◽  
Rainer Sommer
Keyword(s):  
Lattice Spacing ◽  
Continuum Limit ◽  
Finite Lattice ◽  
Gradient Flow ◽  
Scaling Up ◽  
Wilson Loops ◽  
Yang Mills ◽  
Large N ◽  
The Continuum

We present results for Wilson loops smoothed with the Yang-Mills gradient flow and matched through the scale t0. They provide renormalized and precise operators allowing to test the 1/N2 scaling both at finite lattice spacing and in the continuum limit. Our results show an excellent scaling up to 1/N = 1/3. Additionally, we obtain a very precise non-perturbative confirmation of factorization in the large N limit.

A convergence result for the Gradient Flow of ∫ |A|2 in Riemannian Manifolds

2015 ◽  
Vol 1 (1) ◽  
Author(s):  
Annibale Magni
Keyword(s):  
Riemannian Manifolds ◽  
Gradient Flow ◽  
Convergence Result

AbstractWe study the gradient flow of the L

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