A viscosity framework for computing Pogorelov solutions of the Monge-Ampere equation
Benamou, Jean-David; Froese, Brittany D. (2014), A viscosity framework for computing Pogorelov solutions of the Monge-Ampere equation. https://basepub.dauphine.fr/handle/123456789/18736
TypeDocument de travail / Working paper
Cahier de recherche CEREMADE, Université Paris-Dauphine
Series titleCahier de recherche CEREMADE, Université Paris-Dauphine
MetadataShow full item record
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Froese, Brittany D.
Departement of Mathematics [Austin]
Abstract (EN)We consider the Monge-Kantorovich optimal transportation problem between two measures, one of which is a weighted sum of Diracs. This problem is traditionally solved using expensive geometric methods. It can also be reformulated as an elliptic partial differential equation known as the Monge-Ampere equation. However, existing numerical methods for this non-linear PDE require the measures to have finite density. We introduce a new formulation that couples the viscosity and Aleksandrov solution definitions and show that it is equivalent to the original problem. Moreover, we describe a local reformulation of the subgradient measure at the Diracs, which makes use of one-sided directional derivatives. This leads to a consistent, monotone discretisation of the equation. Computational results demonstrate the correctness of this scheme when methods designed for conventional viscosity solutions fail.
Subjects / Keywordsoptimal transportation; Monge-Ampere equation; Aleksandrov solutions; viscosity solutions; finite difference methods
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