Scaling Algorithms for Unbalanced Transport Problems

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

Large-Scale Multi-Agent Transport: Theory, Algorithms and Analysis

The problem of transport of multi-agent systems has received much attention in a wide range of engineering and biological contexts, such as spatial coverage optimization, collective migration, estimation and mapping of unknown environments. In particular, the emphasis has been on the search for scalable decentralized algorithms that are applicable to large-scale multi-agent systems.For large multi-agent collectives, it is appropriate to describe the configuration of the collective and its evolution using macroscopic quantities, while actuation rests at the microscopic scale at the level of individual agents. Moreover, the control problem faces a multitude of information constraints imposed by the multi-agent setting, such as limitations in sensing, communication and localization. Viewed in this way, the problem naturally extends across scales and this motivates a search for algorithms that respect information constraints at the microscopic level while guaranteeing performance at the macroscopic level.We address the above concerns in this dissertation on three fronts: theory, algorithms and analysis. We begin with the development of a multiscale theory of gradient descent-based multi-agent transport that bridges the microscopic and macroscopic perspectives and sets out a general framework for the design and analysis of decentralized algorithms for transport. We then consider the problem of optimal transport of multi-agent systems, wherein the objective is the minimization of the net cost of transport under constraints of distributed computation. This is followed by a treatment of multi-agent transport under constraints on sensing and communication, in the absence of location information, where we study the problem of self-organization in swarms of agents. Motivated by the problem of multi-agent navigation and tracking of moving targets, we then present a study of moving-horizon estimation of nonlinear systems viewed as a transport of probability measures. Finally, we investigate the robustness of multi-agent networks to agent failure, via the problem of identifying critical nodes in large-scale networks

Fluctuations of Parabolic Equations with Large Random Potentials

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 $d\geq 3$. 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

Applications of Optimal Transportation in the Natural Sciences

The aim of this workshop was to gather a mixed group of experts and young researchers from different areas of applied mathematics in which optimal transport plays a central role. Applications in one of the classical areas of natural sciences, like physics, chemistry and (mathematical) biology were the main focus of the workshop

Variational Methods for Evolution

Many evolutionary systems, as for example gradient flows or Hamiltonian systems, can be formulated in terms of variational principles or can be approximated using time-incremental minimization. Hence they can be studied using the mathematical techniques of the field of calculus of variations. This viewpoint has led to many discoveries and rapid expansion of the field over the last two decades. Relevant applications arise in mechanics of fluids and solids, in reaction-diffusion systems, in biology, in many-particle models, as well as in emerging uses in data science. This workshop brought together a broad spectrum of researchers from calculus of variations, partial differential equations, metric geometry, and stochastics, as well as applied and computational scientists to discuss and exchange ideas. It focused on variational tools such as minimizing movement schemes, Gamma convergence, optimal transport, gradient flows, and large-deviation principles for time-continuous Markov processes

Classical and Quantum Mechanical Models of Many-Particle Systems

This workshop was dedicated to the presentation of recent results in the field of the mathematical study of kinetic theory and its naturalextensions (statistical physics and fluid mechanics). The main models are the Vlasov(-Poisson) equation and the Boltzmann equation, which are obtainedas limits of many-body equations (Newton’s equations in the classical case and Schrödinger’s equation in the quantum case) thanks to the mean-field and Boltzmann-Grad scalings. Numerical aspects and applications to mechanics, physics, engineering and biology were also discussed

Analysis and design of Multi-Agent Coverage and Transport algorithms

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

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