Optimal Transportation

Theory and Applications

Author: Cedric Villani

Publisher: Cambridge University Press

ISBN: 110768949X

Category: Mathematics

Page: 316

View: 2818

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Lecture notes and research papers on optimal transportation, its applications, and interactions with other areas of mathematics.

Topics in Optimal Transportation

Author: Cédric Villani

Publisher: American Mathematical Soc.

ISBN: 082183312X

Category: Mathematics

Page: 370

View: 835

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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.

Optimal Transport

Theory and Applications

Author: Yann Ollivier,Hervé Pajot,Cedric Villani

Publisher: Cambridge University Press

ISBN: 1139993623

Category: Mathematics

Page: N.A

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The theory of optimal transportation has its origins in the eighteenth century when the problem of transporting resources at a minimal cost was first formalised. Through subsequent developments, particularly in recent decades, it has become a powerful modern theory. This book contains the proceedings of the summer school 'Optimal Transportation: Theory and Applications' held at the Fourier Institute in Grenoble. The event brought together mathematicians from pure and applied mathematics, astrophysics, economics and computer science. Part I of this book is devoted to introductory lecture notes accessible to graduate students, while Part II contains research papers. Together, they represent a valuable resource on both fundamental and advanced aspects of optimal transportation, its applications, and its interactions with analysis, geometry, PDE and probability, urban planning and economics. Topics covered include Ricci flow, the Euler equations, functional inequalities, curvature-dimension conditions, and traffic congestion.

Optimal Transport for Applied Mathematicians

Calculus of Variations, PDEs, and Modeling

Author: Filippo Santambrogio

Publisher: Birkhäuser

ISBN: 3319208284

Category: Mathematics

Page: 353

View: 6916

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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.

Geometric Science of Information

Third International Conference, GSI 2017, Paris, France, November 7-9, 2017, Proceedings

Author: Frank Nielsen,Frédéric Barbaresco

Publisher: Springer

ISBN: 3319684450

Category: Computers

Page: 877

View: 8344

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This book constitutes the refereed proceedings of the Third International Conference on Geometric Science of Information, GSI 2017, held in Paris, France, in November 2017. The 101 full papers presented were carefully reviewed and selected from 113 submissions and are organized into the following subjects: Statistics on non-linear data Shape Space Optimal Transport & Applications I (Data Science and Economics) Optimal Transport & Applications II (Signal and Image Processing) Topology and statistical learning Statistical Manifold & Hessian Information Geometry Monotone Embedding in Information Geometry Information Structure in Neuroscience Geometric Robotics & Tracking Geometric Mechanics & Robotics Stochastic Geometric Mechanics & Lie Group Thermodynamics Probability on Riemannian Manifolds Divergence Geometry Geometric Deep Learning First and second-order Optimization on Statistical Manifolds Non-parametric Information Geometry Geometry of quantum states Optimization on Manifold Computational Information Geometry Probability Density Estimation Geometry of Tensor-Valued Data Geometry and Inverse Problems Geometry in Vision, Learning and Dynamical Systems Lie Groups and Wavelets Geometry of metric measure spaces Geometry and Telecom Geodesic Methods with Constraints Applications of Distance Geometry

Optimal Transport

Old and New

Author: Cédric Villani

Publisher: Springer Science & Business Media

ISBN: 3540710507

Category: Mathematics

Page: 976

View: 5886

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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.

Random Matrices: High Dimensional Phenomena

Author: Gordon Blower

Publisher: Cambridge University Press

ISBN: 1139481959

Category: Mathematics

Page: 437

View: 8581

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This book focuses on the behaviour of large random matrices. Standard results are covered, and the presentation emphasizes elementary operator theory and differential equations, so as to be accessible to graduate students and other non-experts. The introductory chapters review material on Lie groups and probability measures in a style suitable for applications in random matrix theory. Later chapters use modern convexity theory to establish subtle results about the convergence of eigenvalue distributions as the size of the matrices increases. Random matrices are viewed as geometrical objects with large dimension. The book analyzes the concentration of measure phenomenon, which describes how measures behave on geometrical objects with large dimension. To prove such results for random matrices, the book develops the modern theory of optimal transportation and proves the associated functional inequalities involving entropy and information. These include the logarithmic Sobolev inequality, which measures how fast some physical systems converge to equilibrium.

Optimal Transportation and Applications

Lectures given at the C.I.M.E. Summer School held in Martina Franca, Italy, September 2–8, 2001

Author: Luigi Ambrosio,Yann Brenier,Giuseppe Buttazzo,Cédric Villani

Publisher: Springer

ISBN: 3540448578

Category: Mathematics

Page: 169

View: 6190

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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.

Noncommutative Geometry and Optimal Transport

Author: Pierre Martinetti,Jean-Christophe Wallet

Publisher: American Mathematical Soc.

ISBN: 1470422972

Category: Mathematical optimization

Page: 223

View: 9134

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The distance formula in noncommutative geometry was introduced by Connes at the end of the 1980s. It is a generalization of Riemannian geodesic distance that makes sense in a noncommutative setting, and provides an original tool to study the geometry of the space of states on an algebra. It also has an intriguing echo in physics, for it yields a metric interpretation for the Higgs field. In the 1990s, Rieffel noticed that this distance is a noncommutative version of the Wasserstein distance of order 1 in the theory of optimal transport. More exactly, this is a noncommutative generalization of Kantorovich dual formula of the Wasserstein distance. Connes distance thus offers an unexpected connection between an ancient mathematical problem and the most recent discovery in high energy physics. The meaning of this connection is far from clear. Yet, Rieffel's observation suggests that Connes distance may provide an interesting starting point for a theory of optimal transport in noncommutative geometry. This volume contains several review papers that will give the reader an extensive introduction to the metric aspect of noncommutative geometry and its possible interpretation as a Wasserstein distance on a quantum space, as well as several topic papers.

Optimal Transport Methods in Economics

Author: Alfred Galichon

Publisher: Princeton University Press

ISBN: 1400883598

Category: Business & Economics

Page: 184

View: 6154

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Optimal Transport Methods in Economics is the first textbook on the subject written especially for students and researchers in economics. Optimal transport theory is used widely to solve problems in mathematics and some areas of the sciences, but it can also be used to understand a range of problems in applied economics, such as the matching between job seekers and jobs, the determinants of real estate prices, and the formation of matrimonial unions. This is the first text to develop clear applications of optimal transport to economic modeling, statistics, and econometrics. It covers the basic results of the theory as well as their relations to linear programming, network flow problems, convex analysis, and computational geometry. Emphasizing computational methods, it also includes programming examples that provide details on implementation. Applications include discrete choice models, models of differential demand, and quantile-based statistical estimation methods, as well as asset pricing models. Authoritative and accessible, Optimal Transport Methods in Economics also features numerous exercises throughout that help you develop your mathematical agility, deepen your computational skills, and strengthen your economic intuition. The first introduction to the subject written especially for economists Includes programming examples Features numerous exercises throughout Ideal for students and researchers alike

Optimal Transport for Applied Mathematicians

Calculus of Variations, PDEs, and Modeling

Author: Filippo Santambrogio

Publisher: Birkhäuser

ISBN: 3319208284

Category: Mathematics

Page: 353

View: 5206

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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.

Sub-Riemannian Geometry and Optimal Transport

Author: Ludovic Rifford

Publisher: Springer Science & Business Media

ISBN: 331904804X

Category: Mathematics

Page: 140

View: 6989

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The book provides an introduction to sub-Riemannian geometry and optimal transport and presents some of the recent progress in these two fields. The text is completely self-contained: the linear discussion, containing all the proofs of the stated results, leads the reader step by step from the notion of distribution at the very beginning to the existence of optimal transport maps for Lipschitz sub-Riemannian structure. The combination of geometry presented from an analytic point of view and of optimal transport, makes the book interesting for a very large community. This set of notes grew from a series of lectures given by the author during a CIMPA school in Beirut, Lebanon.

Gradient Flows

In Metric Spaces and in the Space of Probability Measures

Author: Luigi Ambrosio,Nicola Gigli,Giuseppe Savare

Publisher: Springer Science & Business Media

ISBN: 9783764387228

Category: Mathematics

Page: 334

View: 5679

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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.

The Transport System and Transport Policy

An Introduction

Author: Bert van Wee,Jan Anne Annema,David Banister

Publisher: Edward Elgar Publishing

ISBN: 0857936905

Category: Business & Economics

Page: 399

View: 1229

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ÔThis very interesting book provides an excellent multi-disciplinary introduction into the functioning of transport systems and the interaction with their environments.Õ Ð Erik Verhoef, VU University Amsterdam, The Netherlands ÔThe editors of this important book have clearly identified that few writings on transport treat the transport system as a whole. Implicit in this is a need for a genuinely multidisciplinary approach. An impressive list of contributors ensures that the book draws on the latest research whilst providing new insights into some of the key challenges facing transport students and researchers, transport providers and policy makers.Õ Ð Roger Vickerman, University of Kent, UK ÔSince ancient times transportation has brought our world together. But the need for connectivity and accessibility in a spatially differentiated world has prompted the emergence of very complex transportation systems. This book offers a fresh and operational contribution to a better understanding of the complexity and manageability of a mobile world, by addressing in a balanced way both conceptual and applied or policy aspects of modern transportation systems.Õ Ð Peter Nijkamp, Free University of Amsterdam, The Netherlands Transport impacts on people and businesses in many different ways, and presents some of the key problems that decision-makers need to address. This comprehensive textbook introduces the transport system in a holistic and multidisciplinary way, bringing together the myriad components of transport. This textbook is written for an international readership of undergraduate and postgraduate students in transport and related subjects, as well as for professionals and policy decision-makers across both public and private sectors. Key features include: ¥ Discussion of the importance of transport accessibility and the impacts of transport on the environment and safety ¥ Policy issues relating to all of the discussed issues and prescribed future options. ¥ Transport evaluation methods and modelling approaches. ¥ Examples to highlight the linkages between components of the transport system Ð for example infrastructures, land-use, vehicle technologies Ð and the relevance of these linkages for decision making.

Inequalities for Graph Eigenvalues

Author: Zoran Stanić

Publisher: Cambridge University Press

ISBN: 1316395758

Category: Mathematics

Page: N.A

View: 5800

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Written for mathematicians working with the theory of graph spectra, this book explores more than 400 inequalities for eigenvalues of the six matrices associated with finite simple graphs: the adjacency matrix, Laplacian matrix, signless Laplacian matrix, normalized Laplacian matrix, Seidel matrix, and distance matrix. The book begins with a brief survey of the main results and selected applications to related topics, including chemistry, physics, biology, computer science, and control theory. The author then proceeds to detail proofs, discussions, comparisons, examples, and exercises. Each chapter ends with a brief survey of further results. The author also points to open problems and gives ideas for further reading.

Geometric Mechanics and Symmetry

The Peyresq Lectures

Author: James Montaldi,Tudor Ratiu

Publisher: Cambridge University Press

ISBN: 9780521539579

Category: Mathematics

Page: 402

View: 7371

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Geometric mechanics lies on the border of pure and applied mathematics and incorporates such disciplines as differential geometry, Hamiltonian mechanics and integrable systems. The editors organised a summer school on Geometric Mechanics and Symmetry from which the main courses have been written up and published here. The book was written with a significant input from the participants at the conference. This means that the lecture notes are thoroughly geared towards the needs of a graduate student and take great care to explain concepts at the correct level.

Computer Vision

Algorithms and Applications

Author: Richard Szeliski

Publisher: Springer

ISBN: 9781848829466

Category: Computers

Page: 812

View: 4242

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Humans perceive the three-dimensional structure of the world with apparent ease. However, despite all of the recent advances in computer vision research, the dream of having a computer interpret an image at the same level as a two-year old remains elusive. Why is computer vision such a challenging problem and what is the current state of the art? Computer Vision: Algorithms and Applications explores the variety of techniques commonly used to analyze and interpret images. It also describes challenging real-world applications where vision is being successfully used, both for specialized applications such as medical imaging, and for fun, consumer-level tasks such as image editing and stitching, which students can apply to their own personal photos and videos. More than just a source of “recipes,” this exceptionally authoritative and comprehensive textbook/reference also takes a scientific approach to basic vision problems, formulating physical models of the imaging process before inverting them to produce descriptions of a scene. These problems are also analyzed using statistical models and solved using rigorous engineering techniques Topics and features: structured to support active curricula and project-oriented courses, with tips in the Introduction for using the book in a variety of customized courses; presents exercises at the end of each chapter with a heavy emphasis on testing algorithms and containing numerous suggestions for small mid-term projects; provides additional material and more detailed mathematical topics in the Appendices, which cover linear algebra, numerical techniques, and Bayesian estimation theory; suggests additional reading at the end of each chapter, including the latest research in each sub-field, in addition to a full Bibliography at the end of the book; supplies supplementary course material for students at the associated website, http://szeliski.org/Book/. Suitable for an upper-level undergraduate or graduate-level course in computer science or engineering, this textbook focuses on basic techniques that work under real-world conditions and encourages students to push their creative boundaries. Its design and exposition also make it eminently suitable as a unique reference to the fundamental techniques and current research literature in computer vision.

The H-Function

Theory and Applications

Author: A.M. Mathai,Ram Kishore Saxena,Hans J. Haubold

Publisher: Springer Science & Business Media

ISBN: 9781441909169

Category: Science

Page: 268

View: 6162

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TheH-function or popularly known in the literature as Fox’sH-function has recently found applications in a large variety of problems connected with reaction, diffusion, reaction–diffusion, engineering and communication, fractional differ- tial and integral equations, many areas of theoretical physics, statistical distribution theory, etc. One of the standard books and most cited book on the topic is the 1978 book of Mathai and Saxena. Since then, the subject has grown a lot, mainly in the elds of applications. Due to popular demand, the authors were requested to - grade and bring out a revised edition of the 1978 book. It was decided to bring out a new book, mostly dealing with recent applications in statistical distributions, pa- way models, nonextensive statistical mechanics, astrophysics problems, fractional calculus, etc. and to make use of the expertise of Hans J. Haubold in astrophysics area also. It was decided to con ne the discussion toH-function of one scalar variable only. Matrix variable cases and many variable cases are not discussed in detail, but an insight into these areas is given. When going from one variable to many variables, there is nothing called a unique bivariate or multivariate analogue of a givenfunction. Whatever be the criteria used, there may be manydifferentfunctions quali ed to be bivariate or multivariate analogues of a given univariate function. Some of the bivariate and multivariateH-functions, currently in the literature, are also questioned by many authors.

Recent Advances in Algebraic Geometry

Author: Christopher D. Hacon,Mircea Mustaţă,Mihnea Popa

Publisher: Cambridge University Press

ISBN: 110764755X

Category: Mathematics

Page: 447

View: 9539

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A comprehensive collection of expository articles on cutting-edge topics at the forefront of research in algebraic geometry.

Tensors

Geometry and Applications

Author: J. M. Landsberg

Publisher: American Mathematical Soc.

ISBN: 0821869078

Category: Mathematics

Page: 439

View: 2712

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Tensors are ubiquitous in the sciences. The geometry of tensors is both a powerful tool for extracting information from data sets, and a beautiful subject in its own right. This book has three intended uses: a classroom textbook, a reference work for researchers in the sciences, and an account of classical and modern results in (aspects of) the theory that will be of interest to researchers in geometry. For classroom use, there is a modern introduction to multilinear algebra and to the geometry and representation theory needed to study tensors, including a large number of exercises. For researchers in the sciences, there is information on tensors in table format for easy reference and a summary of the state of the art in elementary language. This is the first book containing many classical results regarding tensors. Particular applications treated in the book include the complexity of matrix multiplication, P versus NP, signal processing, phylogenetics, and algebraic statistics. For geometers, there is material on secant varieties, G-varieties, spaces with finitely many orbits and how these objects arise in applications, discussions of numerous open questions in geometry arising in applications, and expositions of advanced topics such as the proof of the Alexander-Hirschowitz theorem and of the Weyman-Kempf method for computing syzygies.