Lectures On Optimal Transport
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Lectures On Optimal Transport
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Author : Luigi Ambrosio
language : en
Publisher: Springer Nature
Release Date : 2021-07-22
Lectures On Optimal Transport written by Luigi Ambrosio and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-07-22 with Mathematics categories.
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 Transportation Networks
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Author : Marc Bernot
language : en
Publisher: Springer Science & Business Media
Release Date : 2009
Optimal Transportation Networks written by Marc Bernot and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009 with Business & Economics categories.
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.
Lectures On Optimal Transport
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Author : Luigi Ambrosio
language : en
Publisher: Springer Nature
Release Date : 2024-12-28
Lectures On Optimal Transport written by Luigi Ambrosio and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024-12-28 with Mathematics categories.
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. This is the second edition of the book, first published in 2018. It includes refinement of proofs, an updated bibliography and a more detailed discussion of minmax principles, with the aim of giving two fully self-contained proofs of Kantorovich duality.
Lectures On Optimal Transport
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Author : Luigi Ambrosio
language : en
Publisher:
Release Date : 2024
Lectures On Optimal Transport written by Luigi Ambrosio and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024 with Calculus of variations categories.
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. This is the second edition of the book, first published in 2018. It includes refinement of proofs, an updated bibliography and a more detailed discussion of minmax principles, with the aim of giving two fully self-contained proofs of Kantorovich duality.
Optimal Transportation And Applications
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Author : Luigi Ambrosio
language : en
Publisher: Springer
Release Date : 2003-01-01
Optimal Transportation And Applications written by Luigi Ambrosio and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2003-01-01 with Mathematics categories.
Leading researchers in the field of Optimal Transportation, with different views and perspectives, contribute to this Summer School volume: Monge-Ampère and Monge-Kantorovich theory, shape optimization and mass transportation are linked, among others, to applications in fluid mechanics granular material physics and statistical mechanics, emphasizing the attractiveness of the subject from both a theoretical and applied point of view. The volume is designed to become a guide to researchers willing to enter into this challenging and useful theory.
Topics In Optimal Transportation
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Author : Cédric Villani
language : en
Publisher: American Mathematical Soc.
Release Date : 2003
Topics In Optimal Transportation written by Cédric Villani and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2003 with Monge-Ampère equations categories.
Cedric Villani's book is a lucid and very readable documentation of the tremendous recent analytic progress in ``optimal mass transportation'' theory and of its diverse and unexpected applications in optimization, nonlinear PDE, geometry, and mathematical physics. --Lawrence C. Evans, University of California at Berkeley 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 at once an introduction to the field of optimal transportation and a survey of the research on the topic over the last 15 years. The book is intended for graduate students and researchers, and it covers both theory and applications. Readers are only assumed to be familiar with the basics of measure theory and functional analysis.
Gradient Flows
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Author : Luigi Ambrosio
language : en
Publisher: Springer Science & Business Media
Release Date : 2008-10-29
Gradient Flows written by Luigi Ambrosio and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008-10-29 with Mathematics categories.
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.
Optimal Transport For Applied Mathematicians
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Author : Filippo Santambrogio
language : en
Publisher: Birkhäuser
Release Date : 2015-10-17
Optimal Transport For Applied Mathematicians written by Filippo Santambrogio and has been published by Birkhäuser this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-10-17 with Mathematics categories.
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.
Computational Optimal Transport
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Author : Gabriel Peyre
language : en
Publisher: Foundations and Trends(r) in M
Release Date : 2019-02-12
Computational Optimal Transport written by Gabriel Peyre and has been published by Foundations and Trends(r) in M this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-02-12 with Computers categories.
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
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Author : Luigi Ambrosio
language : en
Publisher: Springer Science & Business Media
Release Date : 2005-01-28
Gradient Flows written by Luigi Ambrosio and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2005-01-28 with Mathematics categories.
This book is devoted to a theory of gradient ?ows in spaces which are not nec- sarily endowed with a natural linear or di?erentiable structure. It is made of two parts, the ?rst one concerning gradient ?ows in metric spaces and the second one 2 1 devoted to gradient ?ows in the L -Wasserstein space of probability measures on p a separable Hilbert space X (we consider the L -Wasserstein distance, p? (1,?), as well). The two parts have some connections, due to the fact that the Wasserstein space of probability measures provides an important model to which the “metric” theory applies, but the book is conceived in such a way that the two parts can be read independently, the ?rst one by the reader more interested to Non-Smooth Analysis and Analysis in Metric Spaces, and the second one by the reader more oriented to theapplications in Partial Di?erential Equations, Measure Theory and Probability.