131 research outputs found

    Scaling Algorithms for Unbalanced Transport Problems

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    This article introduces a new class of fast algorithms to approximate variational problems involving unbalanced optimal transport. While classical optimal transport considers only normalized probability distributions, it is important for many applications to be able to compute some sort of relaxed transportation between arbitrary positive measures. A generic class of such "unbalanced" optimal transport problems has been recently proposed by several authors. In this paper, we show how to extend the, now classical, entropic regularization scheme to these unbalanced problems. This gives rise to fast, highly parallelizable algorithms that operate by performing only diagonal scaling (i.e. pointwise multiplications) of the transportation couplings. They are generalizations of the celebrated Sinkhorn algorithm. We show how these methods can be used to solve unbalanced transport, unbalanced gradient flows, and to compute unbalanced barycenters. We showcase applications to 2-D shape modification, color transfer, and growth models

    Fluctuations of Parabolic Equations with Large Random Potentials

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    In this paper, we present a fluctuation analysis of a type of parabolic equations with large, highly oscillatory, random potentials around the homogenization limit. With a Feynman-Kac representation, the Kipnis-Varadhan's method, and a quantitative martingale central limit theorem, we derive the asymptotic distribution of the rescaled error between heterogeneous and homogenized solutions under different assumptions in dimension d≥3d\geq 3. The results depend highly on whether a stationary corrector exits.Comment: 44 pages; reorganized the structure and extended the results; to appear in SPDE: Analysis and Computation

    Analysis and design of Multi-Agent Coverage and Transport algorithms

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    Els sistemes robòtics multi-agents són sistemes que presenten moltes aplicacions en ciència i enginyeria. En aquest treball estudiarem el control de la cobertura, que es centra en col·locar un grup de sensors per optimitzar la cobertura d’una densitat. Ens centrarem en el cas en què la densitat evoluciona en el temps i estudiarem l’ús de la teoría de perturbacions singulars per resoldre el problema. També considerarem grans eixams de robots, on podem fer servir models continus per analitzar el comportament dels agents. Recentment s'ha proposat models continus que incorporen idees de transport òptim en el problema de transport multi-agent. Presentarem aquests treballs i proveirem algunes modificacions.Los sistemas robóticos multi-agentes son sistemas que presentan muchas aplicaciones en ciencia y ingeniería. En este trabajo vamos a estudiar el control de la cobertura, que se centra en colocar un grupo de sensores para optimizar la cobertura de una densidad. Nos vamos a centrar en el casos en que la densidad evoluciona con el tiempo y estudiaremos el uso de la teoría de perturbaciones singulares para resolver el problema. También consideraremos grandes enjambres de robots, donde podemos utilizar modelos continuos para analizar el comportamiento del enjambre. Recientemente se ha propuesto el uso de modelos continuos que incorporan ideas de transporte òptimo para el problema de transporte multi-agente. Vamos a presentar dichos trabajos y proveeremos algunas modificaciones.Multi-agent robotic systems have shown to be useful and reliable solutions to many problems that arise in science and engineering. In this work we will study Coverage Control, that aims to achieve optimal coverage of a density. We will focus on the case when the density has a time dependence and we will study a Singular Perturbation Theory approach to solve the problem. We will also consider large swarms of agents, where we can develop continuous models to analyze the behaviour of the swarm. Recent work has focused on applying ideas from the theory of Optimal Transport to the Multi-Agent Transport problem. We will review the work and provide some modifications.Outgoin

    Transfer Operators from Optimal Transport Plans for Coherent Set Detection

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    The topic of this study lies in the intersection of two fields. One is related with analyzing transport phenomena in complicated flows.For this purpose, we use so-called coherent sets: non-dispersing, possibly moving regions in the flow's domain. The other is concerned with reconstructing a flow field from observing its action on a measure, which we address by optimal transport. We show that the framework of optimal transport is well suited for delivering the formal requirements on which a coherent-set analysis can be based on. The necessary noise-robustness requirement of coherence can be matched by the computationally efficient concept of unbalanced regularized optimal transport. Moreover, the applied regularization can be interpreted as an optimal way of retrieving the full dynamics given the extremely restricted information of an initial and a final distribution of particles moving according to Brownian motion
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