Evolutionary $\Gamma$-Convergence of Entropic Gradient Flow Structures for Fokker--Planck Equations in Multiple Dimensions

Forkert, Dominik; Maas, Jan; Portinale, Lorenzo

doi:10.1137/21m1410968

Cited by 9 publications

(5 citation statements)

References 34 publications

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“…Another difference is that the method used relies on a generalized gradient structure of the forward Kolmogorov equation with respect to the relative entropy -the so called cosh gradient structure to be precise. This result on tessellations is related to [21,32], where a similar problem is considered, though the gradient structure is the so called quadratic gradient structure. The latter manuscripts concern the convergence of discrete optimal transport distances to their continuous counterparts, see also [38,39].…”

mentioning

confidence: 86%

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

Esposito

Patacchini

Schlichting

et al. 2021

Arch Rational Mech Anal

122

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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$$^2$$ 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.

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mentioning

confidence: 86%

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

Esposito

Patacchini

Schlichting

et al. 2021

Arch Rational Mech Anal

122

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“…Recently, [19,18] study graphs as nonlocal approximation of nonlocal interaction equations on graphs. This is linked to discrete-to-continuum evolution problems using tessellations, [16,24,30,29]. Structures resembling graphs have been also noticed in collision dynamics in kinetic theory, see [17].…”

Section: Introductionmentioning

confidence: 97%

“…and(24), uniformly in n, and ω :[0, T ] × M M T V (R d ) × R 2d → R d satisfy (ω1)-(ω3). Assume ((x, y) → V [•](•, x, y)) ∈ C 0 (R 2d ) and ((x, y) → ω t [•](•, x, y)) ∈ C 0 (R 2d ).…”

mentioning

confidence: 99%

On evolution PDEs on co-evolving graphs

Esposito,

Mikolás

2024

DCDS

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We provide a well-posedness theory for a class of nonlocal continuity equations on co-evolving graphs. We describe the connection among vertices through an edge weight function and we let it evolve in time, coupling its dynamics with the dynamics on the graph. This is relevant in applications to opinion dynamics and transportation networks. Existence and uniqueness of suitably defined solutions is obtained by exploiting the Banach fixed-point theorem. We consider different time scales for the evolution of the weight function: faster and slower than the flow defined on the graph. The former leads to graphs whose weight functions depend nonlocally on the density configuration at the vertices, while the latter induces static graphs. Furthermore, we prove a discrete-to-continuum limit for the PDEs under study as the number of vertices converges to infinity.

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“…Another related question concerns the convergence of discrete optimal transport distances to its continuous counterpart, cf. [14,17,18]. Similarly, the variational convergence of discretisation for evolution problems is investigated in [21].…”

Section: Introductionmentioning

confidence: 99%

On a class of nonlocal continuity equations on graphs

2023

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Motivated by applications in data science, we study partial differential equations on graphs. By a classical fixed-point argument, we show existence and uniqueness of solutions to a class of nonlocal continuity equations on graphs. We consider general interpolation functions, which give rise to a variety of different dynamics, for example, the nonlocal interaction dynamics coming from a solution-dependent velocity field. Our analysis reveals structural differences with the more standard Euclidean space, as some analogous properties rely on the interpolation chosen.

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Evolutionary $\Gamma$-Convergence of Entropic Gradient Flow Structures for Fokker--Planck Equations in Multiple Dimensions

Cited by 9 publications

References 34 publications

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

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

On evolution PDEs on co-evolving graphs

On a class of nonlocal continuity equations on graphs

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