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In recent works - both experimental and theoretical - it has been shown how to use computational geometry to efficently construct approximations to the optimal transport map between two given probability measures on Euclidean space, by discretizing one of the measures.
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Jun Kitagawa, Quentin Mérigot, Boris Thibert: Convergence of a Newton algorithm for semi-discrete optimal transport. Journal of the European Math. Society. Published online first: 2019-04-24
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