Augmented Lagrangian Methods for Transport Optimization, Mean Field Games and Degenerate Elliptic Equations

Article

Abstract

Many problems from mass transport can be reformulated as variational problems under a prescribed divergence constraint (static problems) or subject to a time-dependent continuity equation, which again can be formulated as a divergence constraint but in time and space. The variational class of mean field games, introduced by Lasry and Lions, may also be interpreted as a generalization of the time-dependent optimal transport problem. Following Benamou and Brenier, we show that augmented Lagrangian methods are well suited to treat such convex but non-smooth problems. They include in particular Monge historic optimal transport problem. A finite-element discretization and implementation of the method are used to provide numerical simulations and a convergence study.

Keywords

Augmented Lagrangian Optimal transport Monge problem Mean field games Degenerate elliptic PDEs 

Mathematics Subject Classification

49A50 49Q20 65M60 60K30 

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.MOKAPLAN, INRIALe Chesnay CedexFrance
  2. 2.CeremadeU. Paris Dauphine Université Paris DauphineParis Cedex 16France

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