Entropy regularized optimal transport and its multi-marginal generalization have attracted increasing attention in various applications, in particular due to efficient Sinkhorn-like algorithms for computing optimal transport plans. However, it is often desirable that the marginals of the optimal transport plan do not match the given measures exactly, which led to the introduction of the so-called unbalanced optimal transport. Since unbalanced methods were not examined for the multi-marginal setting so far, we address this topic in the present paper. More precisely, we introduce the unbalanced multi-marginal optimal transport problem and its dual, and show that a unique optimal transport plan exists under mild assumptions. Further, we generalize the Sinkhorn algorithm for regularized unbalanced optimal transport to the multi-marginal setting and prove its convergence. If the cost function decouples according to a tree, the iterates can be computed efficiently. At the end, we discuss three applications of our framework, namely two barycenter problems and a transfer operator approach, where we establish a relation between the barycenter problem and the multi-marginal optimal transport with an appropriate tree-structured cost function.

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Entropy-regularized optimal transport and its multi-marginal generalization have attracted increasing attention in various applications, in particular due to efficient Sinkhorn-like algorithms for computing optimal transport plans. However, it is often desirable that the marginals of the optimal transport plan do not match the given measures exactly, which led to the introduction of the so-called unbalanced optimal transport. Since unbalanced methods were not examined for the multi-marginal setting so far, we address this topic in the present paper. More precisely, we introduce the unbalanced multi-marginal optimal transport problem and its dual and show that a unique optimal transport plan exists under mild assumptions. Furthermore, we generalize the Sinkhorn algorithm for regularized unbalanced optimal transport to the multi-marginal setting and prove its convergence. For cost functions decoupling according to a tree, the iterates can be computed efficiently. At the end, we discuss three applications of our framework, namely two barycenter problems and a transfer operator approach, where we establish a relation between the barycenter problem and the multi-marginal optimal transport with an appropriate tree-structured cost function.

Long-lived flow patterns in the atmosphere such as weather fronts, mid-latitude blockings or tropical cyclones often induce extreme weather conditions. As a consequence, their description, detection, and tracking has received increasing attention in recent years. Similar objectives also arise in diverse fields such as turbulence and combustion research, image analysis, and medical diagnostics under the headlines of "feature tracking", "coherent structure detection" or "image registration" -to name just a few. A host of different approaches to addressing the underlying, often very similar, tasks have been developed and successfully used. Here, several typical examples of such approaches are summarized, further developed and applied to meteorological data sets. Common abstract operational steps form the basis for a unifying framework for the specification of "persistent structures" involving the definition of the physical state of a system, the features of interest, and means of measuring their persistence.

Computing Wasserstein barycenters of discrete measures has recently attracted considerable attention due to its wide variety of applications in data science. In general, this problem is NP-hard, calling for practical approximative algorithms. In this paper, we analyze two straightforward algorithms for approximating barycenters, which produce sparse support solutions and show promising numerical results. These algorithms require N − 1 and N (N − 1)/2 standard two-marginal OT computations between the N input measures, respectively, so that they are fast, in particular the first algorithm, as well as memory-efficient and easy to implement. Further, they can be used with any OT solver as a black box. Based on relations of the barycenter problem to the multi-marginal optimal transport problem, which are interesting on their own, we prove sharp upper bounds for the relative approximation error. In the second algorithm, this upper bound can be evaluated specifically for the given problem, which always guaranteed an error of at most a few percent in our numerical experiments.