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Minimal geodesics along volume preserving maps, through semi-discrete optimal transport

Authors
  • Mérigot, Quentin
  • Mirebeau, Jean-Marie
Type
Preprint
Publication Date
May 13, 2015
Submission Date
May 13, 2015
Identifiers
arXiv ID: 1505.03306
Source
arXiv
License
Yellow
External links

Abstract

We introduce a numerical method for extracting minimal geodesics along the group of volume preserving maps, equipped with the L2 metric, which as observed by Arnold solve Euler's equations of inviscid incompressible fluids. The method relies on the generalized polar decomposition of Brenier, numerically implemented through semi-discrete optimal transport. It is robust enough to extract non-classical, multi-valued solutions of Euler's equations, for which the flow dimension is higher than the domain dimension, a striking and unavoidable consequence of this model. Our convergence results encompass this generalized model, and our numerical experiments illustrate it for the first time in two space dimensions.

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