On the convergence of augmented Lagrangian method for optimal transport between nonnegative densities

The dynamical formulation of the optimal transport problem, introduced by J. D. Benamou and Y. Brenier, corresponds to the time-space search of a density and a momentum minimizing a transport energy between two densities. In order to solve this problem, an algorithm has been proposed to estimate a saddle point of a Lagrangian. We will study the convergence of this algorithm to a saddle point of the Lagrangian, in the most general conditions, particularly in cases where initial and final densities vanish on some areas of the transportation domain. The principal difficulty of our study will consist in the proof, under these conditions, of the existence of a saddle point, and especially in the uniqueness of the density-momentum component. Indeed, these conditions imply to have to deal with non-regular optimal transportation maps. For these reasons, a detailed study of the properties of the velocity field associated to an optimal transportation map in quadratic space is required.

[1]  C. Villani.  Topics in Optimal Transportation , 2003 .

[2]  P. Lions,et al.  Ordinary differential equations, transport theory and Sobolev spaces , 1989 .

[3]  Heinz H. Bauschke,et al.  Convex Analysis and Monotone Operator Theory in Hilbert Spaces , 2011, CMS Books in Mathematics.

[4]  Radu Ioan Bot,et al.  Inducing strong convergence into the asymptotic behaviour of proximal splitting algorithms in Hilbert spaces , 2016, Optim. Methods Softw..

[5]  L. Kantorovitch,et al.  On the Translocation of Masses , 1958 .

[6]  Éléments de distributions et d'équations aux dérivées partielles : cours et problèmes rèsolus , 2005 .

[7]  G. Peyré,et al.  Unbalanced Optimal Transport: Geometry and Kantorovich Formulation , 2015 .

[8]  Heinz H. Bauschke.  The Approximation of Fixed Points of Compositions of Nonexpansive Mappings in Hilbert Space , 1996 .

[9]  Adam M. Oberman,et al.  Numerical solution of the second boundary value problem for the Elliptic Monge-Amp ere equation , 2012 .

[10]  Yann Brenier,et al.  A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem , 2000, Numerische Mathematik.

[11]  L. Evans.  Measure theory and fine properties of functions , 1992 .

[12]  Z. Opial.  Weak convergence of the sequence of successive approximations for nonexpansive mappings , 1967 .

[13]  Vivette Girault,et al.  Finite Element Methods for Navier-Stokes Equations - Theory and Algorithms , 1986, Springer Series in Computational Mathematics.

[14]  L. Caffarelli.  Boundary regularity of maps with convex potentials – II , 1996 .

[15]  F. Santambrogio.  Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling , 2015 .

[16]  Adam M. Oberman,et al.  Numerical solution of the Optimal Transportation problem using the Monge-Ampère equation , 2012, J. Comput. Phys..

[17]  A variational proof of partial regularity for optimal transportation maps , 2017, 1704.05339.

[18]  J.-C. Pesquet,et al.  A Douglas–Rachford Splitting Approach to Nonsmooth Convex Variational Signal Recovery , 2007, IEEE Journal of Selected Topics in Signal Processing.

[19]  Gabriel Peyré,et al.  Computational Optimal Transport , 2018, Found. Trends Mach. Learn..

[20]  G. Peyré,et al.  Unbalanced optimal transport: Dynamic and Kantorovich formulations , 2015, Journal of Functional Analysis.

[21]  K. Guittet.  On the Time-Continuous Mass Transport Problem and Its Approximation by Augmented Lagrangian Techniques , 2003, SIAM J. Numer. Anal..

[22]  M. Fortin,et al.  Augmented Lagrangian methods : applications to the numerical solution of boundary-value problems , 1983 .

[23]  Dario Cordero-Erausquin.  Sur le transport de mesures périodiques , 1999 .

[24]  L. Ambrosio,et al.  Gradient Flows: In Metric Spaces and in the Space of Probability Measures , 2005 .

[25]  L. Caffarelli.  Boundary regularity of maps with convex potentials , 1992 .

[26]  P. Lions,et al.  Splitting Algorithms for the Sum of Two Nonlinear Operators , 1979 .

[27]  D. Lombardi,et al.  EULERIAN MODELS AND ALGORITHMS FOR UNBALANCED OPTIMAL TRANSPORT , 2015 .

[28]  Gabriel Peyré,et al.  Optimal Transport with Proximal Splitting , 2013, SIAM J. Imaging Sci..

[29]  R. McCann.  A Convexity Principle for Interacting Gases , 1997 .

[30]  N. Papadakis,et al.  Multi-physics optimal transportation and image interpolation , 2015 .