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Preprints, Working Papers, ... Year : 2014

A viscosity framework for computing Pogorelov solutions of the Monge-Ampere equation

Abstract

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.

Dates and versions

hal-01053454 , version 1 (31-07-2014)

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Cite

Jean-David Benamou, Brittany D. Froese. A viscosity framework for computing Pogorelov solutions of the Monge-Ampere equation. 2014. ⟨hal-01053454⟩
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