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Generalized incompressible flows, multi-marginal transport and Sinkhorn algorithm

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Abstract

Starting from Brenier’s relaxed formulation of the incompressible Euler equation in terms of geodesics in the group of measure-preserving diffeomorphisms, we propose a numerical method based on Sinkhorn’s algorithm for the entropic regularization of optimal transport. We also make a detailed comparison of this entropic regularization with the so-called Bredinger entropic interpolation problem (see Arnaudon et al. in An entropic interpolation problem for incompressible viscid fluids, 2017, arXiv preprint arXiv:1704.02126). Numerical results in dimension one and two illustrate the feasibility of the method.

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Notes

  1. We adopt here the terminology of [1] where the name Bredinger is introduced as a contraction of Brenier and Schrödinger.

  2. However, as we shall see, one way to overcome this problem is by penalizing in the cost the condition \(x_N=X_T(x_0)\).

  3. In connection with Navier–Stokes.

  4. This is formal since existence of Lagrange multipliers for the continuous problem cannot be taken for granted in infinite dimensions unless a demanding qualification-like assumption is met requiring that \(\pi _{0,T}\) somehow lies in the interior of the domain of the relative entropy. Nevertheless, once discretized in space, the problem becomes a finite-dimensional smooth convex minimization with linear constraints so that the existence of such multipliers is guaranteed. To reduce the amount of notation here, we use the same notations for the continuous problem (5.1) as for the discretized one where integrals are replaced by finite sums.

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Acknowledgements

It is our pleasure to thank Christian Léonard and Yann Brenier for many fruitful discussions, we are also grateful to Christian Léonard for sharing a preliminary version of [1] with us. The authors are grateful to the Agence Nationale de La Recherche through the projects ISOTACE and MAGA.

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Authors and Affiliations

  1. INRIA-Paris, MOKAPLAN, rue Simone Iff, 75012, Paris, France

    Jean-David Benamou

  2. CEREMADE (UMR CNRS 7534), Université Paris-Dauphine, PSL Research University, 75016, Paris, France

    Guillaume Carlier

  3. CEREMADE (UMR CNRS 7534), CNRS and Université Paris-Dauphine, PSL Research University, 75016, Paris, France

    Luca Nenna

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  1. Jean-David Benamou
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  2. Guillaume Carlier
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  3. Luca Nenna
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Correspondence to Jean-David Benamou.

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Benamou, JD., Carlier, G. & Nenna, L. Generalized incompressible flows, multi-marginal transport and Sinkhorn algorithm. Numer. Math. 142, 33–54 (2019). https://doi.org/10.1007/s00211-018-0995-x

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  • DOI: https://doi.org/10.1007/s00211-018-0995-x

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