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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

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1407.1300.pdf (2.047Mb)
Type
Document de travail / Working paper
Date
2014
Publisher
Cahier de recherche CEREMADE, Université Paris-Dauphine
Series title
Cahier de recherche CEREMADE, Université Paris-Dauphine
Published in
Paris
Pages
24
Metadata
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Author(s)
Benamou, Jean-David
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 / Keywords
optimal transportation; Monge-Ampere equation; Aleksandrov solutions; viscosity solutions; finite difference methods

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