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From Knothe’s transport to Brenier’s map and a continuation method for optimal transport

Abstract : A simple procedure to map two probability measures in ℝd is the so-called \emph{Knothe-Rosenblatt rearrangement}, which consists in rearranging monotonically the marginal distributions of the last coordinate, and then the conditional distributions, iteratively. We show that this mapping is the limit of solutions to a class of Monge-Kantorovich mass transportation problems with quadratic costs, with the weights of the coordinates asymptotically dominating one another. This enables us to design a continuation method for numerically solving the optimal transport problem.
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Guillaume Carlier, Alfred Galichon, Filippo Santambrogio. From Knothe’s transport to Brenier’s map and a continuation method for optimal transport. SIAM Journal on Mathematical Analysis, Society for Industrial and Applied Mathematics, 2008, 41 (6), pp.2554 - 2576. ⟨hal-03473711⟩

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