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>

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 Computations of Eigenvalues and Resonances on Perturbed Hyperbolic Surfaces with Cusps

2019 ◽  
Author(s):  
Michael Levitin ◽  
Alexander Strohmaier
Keyword(s):  
Finite Element Method ◽  
Numerical Experiments ◽  
Compact Manifold ◽  
Good Accuracy ◽  
Scattering Matrix ◽  
Direct Computation ◽  
Simple Method ◽  
Manifold With Boundary ◽  
Arithmetic Surfaces ◽  
Genus One

Abstract In this paper we describe a simple method that allows for a fast direct computation of the scattering matrix for a surface with hyperbolic cusps from the Neumann-to-Dirichlet map on the compact manifold with boundary obtained by removing the cusps. We illustrate that even if the Neumann-to-Dirichlet map is obtained by a finite element method (FEM) one can achieve good accuracy for the scattering matrix. We give various interesting examples of how this can be used to investigate the behaviour of resonances under conformal perturbations or when moving in Teichmüller space. For example, based on numerical experiments we rediscover the four arithmetic surfaces of genus one with one cusp. This demonstrates that it is possible to identify arithmetic objects using FEM. All the videos accompanying this paper are available with its online version, or externally either at michaellevitin.net/hyperbolic.html or as a dedicated YouTubeplaylist.

scholarly journals MULTIVARIABLE CONTROL OF SOLAR BATTERY POWER: ELECTROTECHNICAL COMPLEX AS OBJECT WITH HESSIAN-DRIVEN GRADIENT FLOWS

2021 ◽  
Vol 21 (3) ◽  
pp. 57-65
Author(s):  
A. Sologubov ◽  
◽  
I. Kirpichnikova
Keyword(s):  
Initial Conditions ◽  
Small Neighborhood ◽  
Extremum Seeking ◽  
Transient Processes ◽  
Gradient Flows ◽  
Compact Sets ◽  
Tuning Parameters ◽  
Solar Tracking ◽  
Urgent Task ◽  
The Cost

The paper presents certain development results for the novel extremum seeking controller based on Nesterov’s gradient flows for solar tracking systems. It achieves convergence to an arbitrarily small neighborhood of the set of the cost function optimizers. Our results evident ate that for arbitrarily large compact sets of initial conditions, and arbitrarily small neighborhoods of the optimizer, the controller can be tuned to guarantee convergence taking into account the influence of the Hessian, as well as with tuning parameters that have a fairly clear physical meaning. The influence of the Hessian as a vector field, which is a reflection of the distortion of transient processes in the system, and taking it into account is an urgent task, since it allows for a more flexible impact on the speed of transient processes, and by endowing the system with some damping and smoothing, also for its improved quality.

scholarly journals The High Order Conservative Method for the Parameters Estimation in a PM2.5 Transport Adjoint Model

2017 ◽  
Vol 2017 ◽  
pp. 1-13 ◽  
Author(s):  
Ning Li ◽  
Yongzhi Liu ◽  
Xianqing Lv ◽  
Jicai Zhang ◽  
Kai Fu
Keyword(s):  
Numerical Experiments ◽  
High Order ◽  
Parameters Estimation ◽  
Well Posedness ◽  
Initial Condition ◽  
Adjoint Model ◽  
Simulation Results ◽  
Piecewise Parabolic ◽  
Source And Sink ◽  
Good Agreement

We propose to apply Piecewise Parabolic Method (PPM), a high order and conservative interpolation, for the parameters estimation in a PM2.5 transport adjoint model. Numerical experiments are taken to show the accuracy of PPM in space and its ability to increase the well-posedness of the inverse problem. Based on the obtained results, the PPM provides better interpolation quality by employing much fewer independent points. Meanwhile, this method is still well-behaved in the case of insufficient observations. In twin experiments, two prescribed parameters, including the initial condition (IC) and the source and sink (SS), are successfully estimated by the PPM with lower interpolation errors than the Cressman interpolation. In practical experiments, simulation results show good agreement with the observations of the period when the 21th APEC summit took place.

scholarly journals Well-posedness and numerical treatment of the Blackstock equation in nonlinear acoustics

2018 ◽  
Vol 28 (13) ◽  
pp. 2557-2597 ◽  
Author(s):  
Marvin Fritz ◽  
Vanja Nikolić ◽  
Barbara Wohlmuth
Keyword(s):  
Numerical Experiments ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Energy Estimates ◽  
Well Posedness ◽  
Sound Waves ◽  
Small Initial Data ◽  
Fixed Point Technique ◽  
Linearized Model ◽  
Homogeneous Dirichlet Boundary Conditions

We study the Blackstock equation which models the propagation of nonlinear sound waves through dissipative fluids. Global well-posedness of the model with homogeneous Dirichlet boundary conditions is shown for small initial data. To this end, we employ a fixed-point technique coupled with well-posedness results for a linearized model and appropriate energy estimates. Furthermore, we obtain exponential decay for the energy of the solution. We present additionally a finite element-based method for solving the Blackstock equation and illustrate the behavior of solutions through several numerical experiments.

scholarly journals Nonlocal-interaction equations on uniformly prox-regular sets

2015 ◽  
Vol 36 (3) ◽  
pp. 1209-1247
Author(s):  
Lijiang Wu ◽  
Dejan Slepčev ◽  
José A. Carrillo
Keyword(s):  
Nonlocal Interaction ◽  
Regular Sets

A proof of the well posedness of discretized wave equation with an absorbing boundary condition

2014 ◽  
Vol 22 (4) ◽  
Author(s):  
I. Alonso-Mallo ◽  
A.M. Portillo
Keyword(s):  
Wave Equation ◽  
Numerical Experiments ◽  
Necessary Conditions ◽  
Absorbing Boundary Conditions ◽  
Well Posedness ◽  
Initial Value ◽  
Absorbing Boundary ◽  
One Dimensional ◽  
The One ◽  
Fifth Order

Abstract- Local absorbing boundary conditions with fifth order of absorption to approximate the solution of an initial value problem, for the spatially discretized wave equation, are considered. For the one dimensional case, it is proved necessary conditions for well posedness. Numerical experiments confirming well posedness and showing good results of absorption are included.

scholarly journals Inferring interaction rules from observations of evolutive systems I: The variational approach

2017 ◽  
Vol 27 (05) ◽  
pp. 909-951 ◽  
Author(s):  
Mattia Bongini ◽  
Massimo Fornasier ◽  
Markus Hansen ◽  
Mauro Maggioni
Keyword(s):  
Probability Distribution ◽  
Variational Approach ◽  
Initial Conditions ◽  
Mean Field ◽  
Convergence Result ◽  
Least Square ◽  
Nonlocal Interaction ◽  
Arbitrary Sequence ◽  
Compact Sets ◽  
Initial Probability Distribution

In this paper, we are concerned with the learnability of nonlocal interaction kernels for first-order systems modeling certain social interactions, from observations of realizations of their dynamics. This paper is the first of a series on learnability of nonlocal interaction kernels and presents a variational approach to the problem. In particular, we assume here that the kernel to be learned is bounded and locally Lipschitz continuous and that the initial conditions of the systems are drawn identically and independently at random according to a given initial probability distribution. Then the minimization over a rather arbitrary sequence of (finite-dimensional) subspaces of a least square functional measuring the discrepancy from observed trajectories produces uniform approximations to the kernel on compact sets. The convergence result is obtained by combining mean-field limits, transport methods, and a [Formula: see text]-convergence argument. A crucial condition for the learnability is a certain coercivity property of the least square functional, defined by the majorization of an [Formula: see text]-norm discrepancy to the kernel with respect to a probability measure, depending on the given initial probability distribution by suitable push forwards and transport maps. We illustrate the convergence result by means of several numerical experiments.

scholarly journals A variational formulation of the BDF2 method for metric gradient flows

2019 ◽  
Vol 53 (1) ◽  
pp. 145-172 ◽  
Author(s):  
Daniel Matthes ◽  
Simon Plazotta
Keyword(s):  
Variational Formulation ◽  
Metric Spaces ◽  
Compact Riemannian Manifold ◽  
Energy Functional ◽  
Convergence Order ◽  
Well Posedness ◽  
Gradient Flows ◽  
Wasserstein Metric ◽  
Discrete Approximations ◽  
Variational Form

We propose a variational form of the BDF2 method as an alternative to the commonly used minimizing movement scheme for the time-discrete approximation of gradient flows in abstract metric spaces. Assuming uniform semi-convexity – but no smoothness – of the augmented energy functional, we prove well-posedness of the method and convergence of the discrete approximations to a curve of steepest descent. In a smooth Hilbertian setting, classical theory would predict a convergence order of two in time, we prove convergence order of one-half in the general metric setting and under our weak hypotheses. Further, we illustrate these results with numerical experiments for gradient flows on a compact Riemannian manifold, in a Hilbert space, and in the L2-Wasserstein metric.

scholarly journals A QUADRATIC C0 INTERIOR PENALTY METHOD FOR THE QUAD-CURL PROBLEM

2020 ◽  
Vol 25 (2) ◽  
pp. 208-225
Author(s):  
Zhengjia Sun ◽  
Fuzheng Gao ◽  
Chao Wang ◽  
Yi Zhang
Keyword(s):  
Numerical Experiments ◽  
Penalty Method ◽  
Numerical Scheme ◽  
Optimal Error Estimates ◽  
Well Posedness ◽  
Optimal Error ◽  
Discrete Energy ◽  
Interior Penalty ◽  
Processing Procedure ◽  
Interior Penalty Method

In this paper we study the C0 interior penalty method for a quad-curl problem arising from magnetohydrodynamics model on bounded polygons or polyhedrons. We prove the well-posedness of the numerical scheme and then derive the optimal error estimates in a discrete energy norm. A post-processing procedure that can produce C1 approximations is also presented. The performance of the method is illustrated by numerical experiments.

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