Résumé : In this talk we present a numerical method to solve Brenier’s variational models for incompressible Euler equations. These models give rise to a relaxation in the space of measure-preserving plans of Arnold’s interpretation in terms of geodesics.
The relaxation of Euler equations proposed by Brenier can be understood as requiring the resolution of a multi-marginal
transportation with an infinite number of marginals.
When discretizing Brenier’s problem with $K$ steps in time, one thus faces the resolution of a $K$ marginals OT problem (1) :
$ \inf \int \sum_i=1^K-1\frac1K\lvert x_i+1-x_i\rvert^2d\gamma(x_1,\cdots,x_K)$
$ s.t.\quad \gamma\geq0,\quad \int\gamma=1,\quad (e_i)_\sharp \gamma=Leb,i=1,\cdots,K\quad (e_1,e_K)_\sharp \gamma=(s_\star,s^\star)_\sharp Leb, $
where the constraints $(e_i)_\sharp \gamma=Leb$ stand for the incompressibility of the fluid ($Leb$ is the Lebesgue measure on $[0,1]^d$) and $(e_1,e_K)_\sharp \gamma=(s_\star,s^\star)_\sharp Leb$
expresses that moving fluid particles from the initial configuration $s_\star$ (one usually has $s_\star=Id$) to the final one $s^\star$ yields equivalent transport plans.
We regularise problem (1) by adding an entropy term $\mathcalE(\gamma)=\int\gamma(\log(\gamma)-1)$ and then, once discretised also in space, we can re-write (1)
as the minimization of the Kullback-Leibler distance.
The new problem can be solved by using an extension of the Sinkhorn algorithm (we will also focus on some convergence results).
Finally, we present some numerical results for different final configurations $s^\star$ in dimension $d\geq 1$. This is a joint work with JD Benamou and G. Carlier.