Nonlocal cross-interaction systems on graphs: Nonquadratic Finslerian structure and nonlinear mobilities

Heinze, Georg; Pietschmann, Jan‐Frederik; Schmidtchen, Markus

doi:10.48550/arxiv.2107.11289

Cited by 4 publications

(21 citation statements)

References 29 publications

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“…This free energy was proposed in the above literature to derive the long-time dynamics of the system in dimension d = 2. We also refer to [6,14,16,34] for other recent multi-species models in the literature where our methods could be applied.…”

Section: Introductionmentioning

confidence: 99%

Heterogeneous gradient flows in the topology of fibered optimal transport

Peszek¹,

Poyato²

2022

Preprint

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We introduce an optimal transport topology on the space of probability measures over a fiber bundle, which penalizes the transport cost from one fiber to another. For simplicity, we illustrate our construction in the Euclidean case R d × R d , where we penalize the quadratic cost in the second component. Optimal transport becomes then constrained to happen along fixed fibers. Despite the degeneracy of the infinitely-valued and discontinuous cost, we prove that the space of probability measures (P2,ν(R 2d ), W2,ν) with fixed marginal ν ∈ P(R d ) in the second component becomes a Polish space under the fibered transport distance, which enjoys a weak Riemannian structure reminiscent of the one proposed by F. Otto for the classical quadratic Wasserstein space. Three fundamental issues are addressed: 1) We develop an abstract theory of gradient flows with respect to the new topology; 2) We show applications that identify a novel fibered gradient flow structure on a large class of evolution PDEs with heterogeneities; 3) We exploit our method to derive long-time behavior and global-in-time mean-field limits in a multidimensional Cucker-Smale-type alignment model with weakly singular coupling.

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Section: Introductionmentioning

confidence: 99%

Heterogeneous gradient flows in the topology of fibered optimal transport

Peszek¹,

Poyato²

2022

Preprint

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“…In order to cope with this structural property, one needs to consider suitable interpolating functions so that to be able to describe the flux, hence the dynamics. We refer to [29,28,40] for more details, as well as related works [54,56,16]. Another important aspect is to deal with a large number of entities, for instance individuals or data; hence it is crucial to consider discrete and continuum models.…”

mentioning

confidence: 99%

“…The relation constituting the flux depends on a -finite absolutely continuous measure ∈ M + (R ), as in [29,40,28], wherein acts as an abstract notion of vertices of a graph. More precisely, we associate to a nonlocal time-dependent velocity field : → R the induced flux by using an upwind interpolation as follows d ( , ) = ( , ) + d( ⊗ ) ( , ) − ( , ) − d( ⊗ ) ( , ).…”

mentioning

confidence: 99%

“…Due to the loss of symmetry the underlying structure of P (R ) does not have the formal Riemannian structure, but Finslerian instead. We refer the reader to [29] and [40,41] for further details.…”

mentioning

confidence: 99%

“…Depending on the interpolation chosen, one can notice structural differences with the more standard Euclidean space. In [40], the analysis of [29] is extended to two species with cross-interactions as well as nonlinear mobilities and -homogeneous flux-velocity relations for > 0, by generalizing the underlying Finslerian structure and notion of gradient. In [41] the generalized system is explored both in analytical case studies and numerical simulations on finite graphs of varying shapes and connectivity, leading to the observation of various patterns, including mixing of the two species, partial engulfment, or phase separation.…”

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confidence: 99%

See 2 more Smart Citations

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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Graph‐to‐local limit for a multi‐species nonlocal cross‐interaction system

Esposito,

Heinze,

Pietschmann

et al. 2023

Proc Appl Math and Mech

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In this note we continue the study of nonlocal interaction dynamics on a sequence of infinite graphs, extending the results of Benamou and Brenier to an arbitrary number of species. Our analysis relies on the observation that the graph dynamics form a gradient flow with respect to a non‐symmetric Finslerian gradient structure. Keeping the nonlocal interaction energy fixed, while localizing the graph structure, we are able to prove evolutionary Γ‐convergence to an Otto‐Wassertein‐type gradient flow with a tensor‐weighted, yet symmetric, inner product. As a byproduct this implies the existence of solutions to the multi‐species non‐local (cross‐)interaction system on the tensor‐weighted Euclidean space.

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Nonlocal cross-interaction systems on graphs: Nonquadratic Finslerian structure and nonlinear mobilities

Cited by 4 publications

References 29 publications

Heterogeneous gradient flows in the topology of fibered optimal transport

Heterogeneous gradient flows in the topology of fibered optimal transport

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

Graph‐to‐local limit for a multi‐species nonlocal cross‐interaction system

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