Homogenisation of one-dimensional discrete optimal transport

Gladbach, Peter; Kopfer, Eva; Maas, Jan; Portinale, Lorenzo

doi:10.48550/arxiv.1905.05757

Cited by 2 publications

(3 citation statements)

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“…Here we do not pretend to prove any rigorous statement in this direction, and we shall be content with the following heuristics: Remark 11. Similar questions were investigated in [48,49], where it was shown that some homogeneity and uniformity of the space meshing is essential (as assumed here). Note that part of our statement in Claim 1 is that the limiting distance W does not "see" the potential U, while the discrete Wasserstein distance does depend on the whole kernel K N (hence a priori on U).…”

Section: The Large Population Limit N → ∞mentioning

confidence: 75%

“…The above expression should be understood to be +∞ whenever q ≪ π, and makes sense for general q. However, when q = w p w, p is obtained as the Q-process corresponding to some p with w, p = 0 (which propagates from t = 0 to later times as in the discrete case, see also (49) below), and recalling that π(x) = w(x)z(x), we abuse the notations and also express this same entropy in terms of the original p measure as…”

Section: Entropymentioning

confidence: 99%

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Gradient flow formulations of discrete and continuous evolutionary models: a unifying perspective

Chalub¹,

Monsaingeon²,

Ribeiro³

et al. 2019

Preprint

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We consider three classical models of biological evolution: (i) the Moran process, an example of a reducible Markov Chain; (ii) the Kimura Equation, a particular case of a degenerated Fokker-Planck Diffusion; (iii) the Replicator Equation, a paradigm in Evolutionary Game Theory. While these approaches are not completely equivalent, they are intimately connected, since (ii) is the diffusion approximation of (i), and (iii) is obtained from (ii) in an appropriate limit. It is well known that the Replicator Dynamics for two strategies is a gradient flow with respect to the celebrated Shahshahani distance. We reformulate the Moran process and the Kimura Equation as gradient flows and in the sequel we discuss conditions such that the associated gradient structures converge: (i) to (ii) and (ii) to (iii). This provides a geometric characterisation of these evolutionary processes and provides a reformulation of the above examples as time minimization of free energy functionals.

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Section: The Large Population Limit N → ∞mentioning

confidence: 75%

Section: Entropymentioning

confidence: 99%

Gradient flow formulations of discrete and continuous evolutionary models: a unifying perspective

Chalub¹,

Monsaingeon²,

Ribeiro³

et al. 2019

Preprint

View full text Add to dashboard Cite

show abstract

“…For quadratic dissipation, qualitative convergence results in 1-D using the underlying gradient structure are obtained in [DL15] looking at energy-dissipation mechanism, and in [GKMP19] proving convergence of the metric.…”

Section: 3mentioning

confidence: 94%

Consistency and convergence for a family of finite volume discretizations of the Fokker--Planck operator

Heida¹,

Kantner²,

Stephan³

2020

Preprint

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We introduce a family of various finite volume discretization schemes for the Fokker-Planck operator, which are characterized by different weight functions on the edges. This family particularly includes the well-established Scharfetter-Gummel discretization as well as the recently developed square-root approximation (SQRA) scheme. We motivate this family of discretizations both from the numerical and the modeling point of view and provide a uniform consistency and error analysis. Our main results state that the convergence order primarily depends on the quality of the mesh and in second place on the quality of the weights. We show by numerical experiments that for small gradients the choice of the optimal representative of the discretization family is highly non-trivial while for large gradients the Scharfetter-Gummel scheme stands out compared to the others.

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Homogenisation of one-dimensional discrete optimal transport

Cited by 2 publications

References 0 publications

Gradient flow formulations of discrete and continuous evolutionary models: a unifying perspective

Gradient flow formulations of discrete and continuous evolutionary models: a unifying perspective

Consistency and convergence for a family of finite volume discretizations of the Fokker--Planck operator

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