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

We introduce a numerical method for extracting minimal geodesics along the group of volume preserving maps, equipped with the L metric, which as observed by Arnold [Arn66] solve the Euler equations of inviscid incompressible uids. The method relies on the generalized polar decomposition of Brenier [Bre91], 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 ow dimension de ned as the quantization dimension of Brenier's generalized ow is higher than the domain dimension, a striking and unavoidable consequence of this model [Shn94]. Our convergence results encompass this generalized model, and our numerical experiments illustrate it for the rst time in two space dimensions.