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Minimal Geodesics Along Volume-Preserving Maps, Through Semidiscrete Optimal Transport

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Date
2016
Link to item file
https://arxiv.org/abs/1505.03306v1
Dewey
Probabilités et mathématiques appliquées
Sujet
Euler equation; optimal transport; calculus of variation; quantization
Journal issue
SIAM Journal on Numerical Analysis
Volume
54
Number
6
Publication date
2016
Article pages
3465-3492
Publisher
SIAM
DOI
http://dx.doi.org/10.1137/15M1017235
URI
https://basepub.dauphine.fr/handle/123456789/16357
Collections
  • CEREMADE : Publications
Metadata
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Author
Mérigot, Quentin
Mirebeau, Jean-Marie
Type
Article accepté pour publication ou publié
Abstract (EN)
We introduce a numerical method for extracting minimal geodesics along the group of volume-preserving maps, equipped with the $L^2$ metric, which as observed by Arnold [Ann. Inst. Fourier (Grenoble), 16 (1966), pp. 319--361] solve the Euler equations of inviscid incompressible fluids. The method relies on the generalized polar decomposition of Brenier [Comm. Pure Appl. Math., 44 (1991), pp. 375--417], numerically implemented through semidiscrete optimal transport. It is robust enough to extract nonclassical, multivalued solutions of Euler's equations, for which the flow dimension---defined as the quantization dimension of Brenier's generalized flow---is higher than the domain dimension, a striking and unavoidable consequence of thismodel [A. I. Shnirelman, Geom. Funct. Anal., 4 (1994), pp. 586--620]. Our convergence results encompass this generalizedmodel, and our numerical experiments illustrate it for the first time in two space dimensions.

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