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Nonlocal-Interaction Equation on Graphs: Gradient Flow Structure and Continuum Limit
Archive for Rational Mechanics and Analysis ( IF 2.5 ) Pub Date : 2021-03-15 , DOI: 10.1007/s00205-021-01631-w
Antonio Esposito , Francesco S. Patacchini , André Schlichting , Dejan Slepčev

We 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\(^2\)IE). We develop the existence theory for the solutions of the NL\(^2\)IE as curves of maximal slope with respect to the upwind Wasserstein quasi-metric. Furthermore, we show that the solutions of the NL\(^2\)IE on graphs converge as the empirical measures of the set of vertices converge weakly, which establishes a valuable discrete-to-continuum convergence result.



中文翻译:

图上的非局部相互作用方程:梯度流结构和连续极限

我们考虑由图上的相互作用能驱动的动力学。我们介绍了连续非局部相互作用方程的图类似物,并将它们解释为相对于图Wasserstein距离的梯度流。我们考虑的特定Wasserstein距离来自Benamou–Brenier公式的图形类似物,其中图形连续性方程使用逆风插值来定义沿边的密度。尽管此方法在理论和计算上均具有优势,但所得距离仅是准度量。我们在图和更普遍的结构(其中“顶点”集是任意正度量)上研究这种准度量。我们将所产生的非局部相互作用能量的梯度流称为非局部非局部相互作用方程(NL \(^ 2 \)IE)。我们发展了NL \(^ 2 \) IE解的存在性理论,该解是关于上风Wasserstein准度量的最大斜率曲线。此外,我们证明了图上的NL \(^ 2 \) IE的解收敛于顶点集的经验测度弱收敛,从而建立了有价值的离散到连续的收敛结果。

更新日期:2021-03-15
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