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Minimal convex extensions and finite difference discretisation of the quadratic Monge–Kantorovich problem

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MABV2final.pdf (3.260Mb)
Date
2018
Dewey
Analyse
Sujet
Optimal transport; Monge-Ampère equation; finite-difference scheme
Journal issue
European Journal of Applied Mathematics
Publication date
09-2018
Article pages
1-38
Publisher
Cambridge University Press
DOI
http://dx.doi.org/10.1017/S0956792518000451
URI
https://basepub.dauphine.fr/handle/123456789/19907
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  • CEREMADE : Publications
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Author
Duval, Vincent
60 CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Benamou, Jean-David
60 CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Type
Article accepté pour publication ou publié
Abstract (EN)
We present an adaptation of the Monge–Ampère (MA) lattice basis reduction scheme to the MA equation with second boundary value condition, provided the target is a convex set. This yields a fast adaptive method to numerically solve the optimal transport (OT) problem between two absolutely continuous measures, the second of which has convex support. The proposed numerical method actually captures a specific Brenier solution which is minimal in some sense. We prove the convergence of the method as the grid step size vanishes and show with numerical experiments that it is able to reproduce subtle properties of the OT problem.

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