On the Time-Continuous Mass Transport Problem and Its Approximation by Augmented Lagrangian Techniques

In [J. D. Benamou and Y. Brenier}, Numer. Math., 84 (2000), pp. 375--393], a computational fluid dynamic approach was introduced for computing the optimal map occurring in the Monge--Kantorovich problem. Though the described augmented Lagrangian method involves a Hilbertian framework, the discussion was purely formal. Taking advantage of the recent progress in optimal transport theory [L. A. Caffarelli, Comm. Pure Appl. Math., 45 (1992), pp. 1141--1151], [L. A. Caffarelli, Ann. of Math. (2), 144 (1996), pp. 453--496], [D. Cordero-Erausquin, C. R. Acad. Sci. Paris Ser. I Math., 329 (1999), pp. 199--202], [R. J. McCann, Geom. Funct. Anal., 11 (2001), pp. 589--608] and despite the lack of coercivity of the Hilbertian problem, we establish an existence result. Then, under a reasonable assumption of positivity for the density, we prove the existence of saddle-points for both Lagrangians defined in Benamou and Brenier, and finally prove the convergence of the numerical method.