Scaling limits of discrete optimal transport

Gladbach, Peter; Kopfer, Eva; Maas, Jan

doi:10.48550/arxiv.1809.01092

Cited by 8 publications

(50 citation statements)

References 23 publications

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“…Semi-discretized problem [21] Γ-convergence [12] Gromov-Hausdorff convergence [13,14] Γ-convergence for finite difference under regularity assumption [9] This article Figure 1. Schematic representation of other convergence results present in the literature (dashed lines) and ours (solid line).…”

Section: Fully Discretized Problemmentioning

confidence: 99%

Unconditional convergence for discretizations of dynamical optimal transport

Lavenant¹

2019

Preprint

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The dynamical formulation of optimal transport, also known as Benamou-Brenier formulation or Computational Fluid Dynamic formulation, amounts to write the optimal transport problem as the optimization of a convex functional under a PDE constraint, and can handle a priori a vast class of cost functions and geometries. Several disretizations of this problem have been proposed, leading to computations on flat spaces as well as Riemannian manifolds, with extensions to mean field games and gradient flows in the Wasserstein space.In this article, we provide a framework which guarantees convergence under mesh refinement of the solutions of the space-time discretized problems to the one of the infinite-dimensional one for quadratic optimal transport. The convergence holds without condition on the ratio between spatial and temporal step sizes, and can handle arbitrary positive measures as input, while the underlying space can be a Riemannian manifold. Both the finite volume discretization proposed by Gladbach, Kopfer and Maas, as well as the discretization over triangulations of surfaces studied by the present author in collaboration with Claici, Chien and Solomon fit in this framework.

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Section: Fully Discretized Problemmentioning

confidence: 99%

Unconditional convergence for discretizations of dynamical optimal transport

Lavenant¹

2019

Preprint

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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%

“…We want to identify now the Kimura Equation ( 30) as a gradient flow, based on the formula (48). To this end we first need to retrieve the evolution equation for the rescaled Q-process q(t, x).…”

Section: Gradient Flow Formulationmentioning

confidence: 99%

“…Computing the first variation δH δq = κ 2 log q π − 1 of the entropy (40) and applying formula (48) for the Wasserstein gradient, we finally recognise the gradient flow structure…”

Section: Gradient Flow Formulationmentioning

confidence: 99%

“…Our previous formula (48) for Wasserstein gradients gives next, with the first variation δV δq = V , the gradient flow structure…”

Section: Gradient Flow Formulationmentioning

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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A Mixed Finite Element Discretization of Dynamical Optimal Transport

Natale

Todeschi

2022

J Sci Comput

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In this paper we introduce a new class of finite element discretizations of the quadratic optimal transport problem based on its dynamical formulation. These generalize to the finite element setting the finite difference scheme proposed by Papadakis et al.[SIAM J Imaging Sci, 7(1): 2014]. We solve the discrete problem using a proximal splitting approach and we show how to modify this in the presence of regularization terms which are relevant for physical data interpolation.

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Scaling limits of discrete optimal transport

Cited by 8 publications

References 23 publications

Unconditional convergence for discretizations of dynamical optimal transport

Unconditional convergence for discretizations of dynamical optimal transport

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

A Mixed Finite Element Discretization of Dynamical Optimal Transport

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