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Optimal Transportation and Action-Minimizing Measures

Author : Alessio Figalli

Publisher : Edizioni della Normale

Page : 0 pages

File Size : 31,69 MB

Release : 2008-07-17

Category : Mathematics

ISBN : 9788876423307

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Book Description

In this book we describe recent developments in the theory of optimal transportation, and some of its applications to fluid dynamics. Moreover we explore new variants of the original problem, and we try to figure out some common (and sometimes unexpected) features in this emerging variety of problems . In Chapter 1 we study the optimal transportation problem on manifolds with geometric costs coming from Tonelli Lagrangians, while in Chapter 2 we consider a generalization of the classical transportation problem called the optimal irrigation problem. Then, Chapter 3 is about the Brenier variational theory of incompressible flows, which concerns a weak formulation of the Euler equations viewed as a geodesic equation in the space of measure-preserving diffeomorphism. Chapter 4 is devoted to the study of regularity and uniqueness of solutions of Hamilton-Jacobi equations applying the Aubry-Mather theory. Finally, the last chapter deals with a DiPerna-Lions theory for martingale solutions of stochastic differential equations.



Optimal Transport

Author : Cédric Villani

Publisher : Springer Science & Business Media

Page : 970 pages

File Size : 11,21 MB

Release : 2008-10-26

Category : Mathematics

ISBN : 3540710507

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Book Description

At the close of the 1980s, the independent contributions of Yann Brenier, Mike Cullen and John Mather launched a revolution in the venerable field of optimal transport founded by G. Monge in the 18th century, which has made breathtaking forays into various other domains of mathematics ever since. The author presents a broad overview of this area, supplying complete and self-contained proofs of all the fundamental results of the theory of optimal transport at the appropriate level of generality. Thus, the book encompasses the broad spectrum ranging from basic theory to the most recent research results. PhD students or researchers can read the entire book without any prior knowledge of the field. A comprehensive bibliography with notes that extensively discuss the existing literature underlines the book’s value as a most welcome reference text on this subject.



Optimal Transport for Applied Mathematicians

Author : Filippo Santambrogio

Publisher : Birkhäuser

Page : 376 pages

File Size : 12,58 MB

Release : 2015-10-17

Category : Mathematics

ISBN : 3319208284

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Book Description

This monograph presents a rigorous mathematical introduction to optimal transport as a variational problem, its use in modeling various phenomena, and its connections with partial differential equations. Its main goal is to provide the reader with the techniques necessary to understand the current research in optimal transport and the tools which are most useful for its applications. Full proofs are used to illustrate mathematical concepts and each chapter includes a section that discusses applications of optimal transport to various areas, such as economics, finance, potential games, image processing and fluid dynamics. Several topics are covered that have never been previously in books on this subject, such as the Knothe transport, the properties of functionals on measures, the Dacorogna-Moser flow, the formulation through minimal flows with prescribed divergence formulation, the case of the supremal cost, and the most classical numerical methods. Graduate students and researchers in both pure and applied mathematics interested in the problems and applications of optimal transport will find this to be an invaluable resource.



Topics in Optimal Transportation

Author : Cédric Villani

Publisher : American Mathematical Soc.

Page : 370 pages

File Size : 37,75 MB

Release : 2021-08-25

Category : Education

ISBN : 1470467267

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Book Description

This is the first comprehensive introduction to the theory of mass transportation with its many—and sometimes unexpected—applications. In a novel approach to the subject, the book both surveys the topic and includes a chapter of problems, making it a particularly useful graduate textbook. In 1781, Gaspard Monge defined the problem of “optimal transportation” (or the transferring of mass with the least possible amount of work), with applications to engineering in mind. In 1942, Leonid Kantorovich applied the newborn machinery of linear programming to Monge's problem, with applications to economics in mind. In 1987, Yann Brenier used optimal transportation to prove a new projection theorem on the set of measure preserving maps, with applications to fluid mechanics in mind. Each of these contributions marked the beginning of a whole mathematical theory, with many unexpected ramifications. Nowadays, the Monge-Kantorovich problem is used and studied by researchers from extremely diverse horizons, including probability theory, functional analysis, isoperimetry, partial differential equations, and even meteorology. Originating from a graduate course, the present volume is intended for graduate students and researchers, covering both theory and applications. Readers are only assumed to be familiar with the basics of measure theory and functional analysis.



Computational Optimal Transport

Author : Gabriel Peyre

Publisher : Foundations and Trends(r) in M

Page : 272 pages

File Size : 15,13 MB

Release : 2019-02-12

Category : Computers

ISBN : 9781680835502

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Book Description

The goal of Optimal Transport (OT) is to define geometric tools that are useful to compare probability distributions. Their use dates back to 1781. Recent years have witnessed a new revolution in the spread of OT, thanks to the emergence of approximate solvers that can scale to sizes and dimensions that are relevant to data sciences. Thanks to this newfound scalability, OT is being increasingly used to unlock various problems in imaging sciences (such as color or texture processing), computer vision and graphics (for shape manipulation) or machine learning (for regression, classification and density fitting). This monograph reviews OT with a bias toward numerical methods and their applications in data sciences, and sheds lights on the theoretical properties of OT that make it particularly useful for some of these applications. Computational Optimal Transport presents an overview of the main theoretical insights that support the practical effectiveness of OT before explaining how to turn these insights into fast computational schemes. Written for readers at all levels, the authors provide descriptions of foundational theory at two-levels. Generally accessible to all readers, more advanced readers can read the specially identified more general mathematical expositions of optimal transport tailored for discrete measures. Furthermore, several chapters deal with the interplay between continuous and discrete measures, and are thus targeting a more mathematically-inclined audience. This monograph will be a valuable reference for researchers and students wishing to get a thorough understanding of Computational Optimal Transport, a mathematical gem at the interface of probability, analysis and optimization.



Gradient Flows

Author : Luigi Ambrosio

Publisher : Springer Science & Business Media

Page : 333 pages

File Size : 17,85 MB

Release : 2008-10-29

Category : Mathematics

ISBN : 376438722X

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Book Description

The book is devoted to the theory of gradient flows in the general framework of metric spaces, and in the more specific setting of the space of probability measures, which provide a surprising link between optimal transportation theory and many evolutionary PDE's related to (non)linear diffusion. Particular emphasis is given to the convergence of the implicit time discretization method and to the error estimates for this discretization, extending the well established theory in Hilbert spaces. The book is split in two main parts that can be read independently of each other.



Lectures on Optimal Transport

Author : Luigi Ambrosio

Publisher : Springer Nature

Page : 250 pages

File Size : 40,31 MB

Release : 2021-07-22

Category : Mathematics

ISBN : 3030721620

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Book Description

This textbook is addressed to PhD or senior undergraduate students in mathematics, with interests in analysis, calculus of variations, probability and optimal transport. It originated from the teaching experience of the first author in the Scuola Normale Superiore, where a course on optimal transport and its applications has been given many times during the last 20 years. The topics and the tools were chosen at a sufficiently general and advanced level so that the student or scholar interested in a more specific theme would gain from the book the necessary background to explore it. After a large and detailed introduction to classical theory, more specific attention is devoted to applications to geometric and functional inequalities and to partial differential equations.



Optimal Transport

Author : Gershon Wolansky

Publisher : Walter de Gruyter GmbH & Co KG

Page : 235 pages

File Size : 20,53 MB

Release : 2021-01-18

Category : Mathematics

ISBN : 3110633175

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Book Description

The series is devoted to the publication of high-level monographs which cover the whole spectrum of current nonlinear analysis and applications in various fields, such as optimization, control theory, systems theory, mechanics, engineering, and other sciences. One of its main objectives is to make available to the professional community expositions of results and foundations of methods that play an important role in both the theory and applications of nonlinear analysis. Contributions which are on the borderline of nonlinear analysis and related fields and which stimulate further research at the crossroads of these areas are particularly welcome. Editor-in-Chief J rgen Appell, W rzburg, Germany Honorary and Advisory Editors Catherine Bandle, Basel, Switzerland Alain Bensoussan, Richardson, Texas, USA Avner Friedman, Columbus, Ohio, USA Umberto Mosco, Worcester, Massachusetts, USA Louis Nirenberg, New York, USA Alfonso Vignoli, Rome, Italy Editorial Board Manuel del Pino, Bath, UK, and Santiago, Chile Mikio Kato, Nagano, Japan Wojciech Kryszewski, Toruń, Poland Vicenţiu D. Rădulescu, Krak w, Poland Simeon Reich, Haifa, Israel Please submit book proposals to J rgen Appell. Titles in planning include Lucio Damascelli and Filomena Pacella, Morse Index of Solutions of Nonlinear Elliptic Equations (2019) Tomasz W. Dlotko and Yejuan Wang, Critical Parabolic-Type Problems (2019) Rafael Ortega, Periodic Differential Equations in the Plane: A Topological Perspective (2019) Ireneo Peral Alonso and Fernando Soria, Elliptic and Parabolic Equations Involving the Hardy-Leray Potential (2020) Cyril Tintarev, Profile Decompositions and Cocompactness: Functional-Analytic Theory of Concentration Compactness (2020) Takashi Suzuki, Semilinear Elliptic Equations: Classical and Modern Theories (2021)



Optimal Transportation Networks

Author : Marc Bernot

Publisher : Springer Science & Business Media

Page : 204 pages

File Size : 14,19 MB

Release : 2009

Category : Business & Economics

ISBN : 3540693149

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Book Description

The transportation problem can be formalized as the problem of finding the optimal way to transport a given measure into another with the same mass. In contrast to the Monge-Kantorovitch problem, recent approaches model the branched structure of such supply networks as minima of an energy functional whose essential feature is to favour wide roads. Such a branched structure is observable in ground transportation networks, in draining and irrigation systems, in electrical power supply systems and in natural counterparts such as blood vessels or the branches of trees. These lectures provide mathematical proof of several existence, structure and regularity properties empirically observed in transportation networks. The link with previous discrete physical models of irrigation and erosion models in geomorphology and with discrete telecommunication and transportation models is discussed. It will be mathematically proven that the majority fit in the simple model sketched in this volume.



Transportation Decision Making

Author : Kumares C. Sinha

Publisher : John Wiley & Sons

Page : 576 pages

File Size : 21,26 MB

Release : 2011-09-09

Category : Technology & Engineering

ISBN : 1118169662

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Book Description

This pioneering text provides a holistic approach to decisionmaking in transportation project development and programming, whichcan help transportation professionals to optimize their investmentchoices. The authors present a proven set of methodologies forevaluating transportation projects that ensures that all costs andimpacts are taken into consideration. The text's logical organization gets readers started with asolid foundation in basic principles and then progressively buildson that foundation. Topics covered include: Developing performance measures for evaluation, estimatingtravel demand, and costing transportation projects Performing an economic efficiency evaluation that accounts forsuch factors as travel time, safety, and vehicle operatingcosts Evaluating a project's impact on economic development and landuse as well as its impact on society and culture Assessing a project's environmental impact, including airquality, noise, ecology, water resources, and aesthetics Evaluating alternative projects on the basis of multipleperformance criteria Programming transportation investments so that resources can beoptimally allocated to meet facility-specific and system-widegoals Each chapter begins with basic definitions and concepts followedby a methodology for impact assessment. Relevant legislation isdiscussed and available software for performing evaluations ispresented. At the end of each chapter, readers are providedresources for detailed investigation of particular topics. Theseinclude Internet sites and publications of international anddomestic agencies and research institutions. The authors alsoprovide a companion Web site that offers updates, data foranalysis, and case histories of project evaluation and decisionmaking. Given that billions of dollars are spent each year ontransportation systems in the United States alone, and that thereis a need for thorough and rational evaluation and decision makingfor cost-effective system preservation and improvement, this textshould be on the desks of all transportation planners, engineers,and educators. With exercises in every chapter, this text is anideal coursebook for the subject of transportation systems analysisand evaluation.