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Theses

Unbalanced Optimal Transport : Models, Numerical Methods, Applications

Abstract : This thesis generalizes optimal transport beyond the classical "balanced" setting of probability distributions. We define unbalanced optimal transport models between nonnegative measures, based either on the notion of interpolation or the notion of coupling of measures. We show relationships between these approaches. One of the outcomes of this framework is a generalization of the p-Wasserstein metrics. Secondly, we build numerical methods to solve interpolation and coupling-based models. We study, in particular, a new family of scaling algorithms that generalize Sinkhorn's algorithm. The third part deals with applications. It contains a theoretical and numerical study of a Hele-Shaw type gradient flow in the space of nonnegative measures. It also adresses the case of measures taking values in the cone of positive semi-definite matrices, for which we introduce a model that achieves a balance between geometrical accuracy and algorithmic efficiency.
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Lénaïc Chizat. Unbalanced Optimal Transport : Models, Numerical Methods, Applications. Numerical Analysis [math.NA]. Université Paris sciences et lettres, 2017. English. ⟨NNT : 2017PSLED063⟩. ⟨tel-01881166⟩

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