Second order derivative of a functional associated to an optimal transport map

In this article we investigate the second order differentiability of a functional associated to a Monge’s optimal transportation problem, namely the case of the quadratic cost, in its dual formulation. The application problem that motivates the present research is an algorithm for warping of images...

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Bibliographic Details
Published in:Monatshefte für Mathematik 2023-07, Vol.201 (3), p.943-959
Main Authors: Wences, Giovanni, Delgado, Joaquín
Format: Article
Language:English
Subjects:
Algorithms
Mathematics
Mathematics and Statistics
Transportation problem
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Summary:In this article we investigate the second order differentiability of a functional associated to a Monge’s optimal transportation problem, namely the case of the quadratic cost, in its dual formulation. The application problem that motivates the present research is an algorithm for warping of images that uses the first derivative of this functional for a gradient-descent method which is proposed in Chartrand (Appl Math Sci 3:1071–1080, 2009). We prove the second order Gâteaux differentiability of the functional at the Monge’s potential. With this result we also prove that this functional is first order differentiable in the strong sense, that is, Fréchet differentiable. Our results can be used for a further improvement of a Newton-like algorithm in this and other problems.
ISSN:0026-9255
1436-5081
DOI:10.1007/s00605-023-01856-9