Topological Optimization And Optimal Transport

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Topological Optimization and Optimal Transport

Maïtine Bergounioux,Édouard Oudet,Martin Rumpf,Guillaume Carlier,Thierry Champion,Filippo Santambrogio
Topological Optimization and Optimal Transport Book PDF

Author : Maïtine Bergounioux,Édouard Oudet,Martin Rumpf,Guillaume Carlier,Thierry Champion,Filippo Santambrogio
Publisher : Walter de Gruyter GmbH & Co KG
Page : 432 pages
File Size : 43,9 Mb
Release : 2017-08-07
Category : Mathematics
ISBN : 9783110430417

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Topological Optimization and Optimal Transport by Maïtine Bergounioux,Édouard Oudet,Martin Rumpf,Guillaume Carlier,Thierry Champion,Filippo Santambrogio Pdf

By discussing topics such as shape representations, relaxation theory and optimal transport, trends and synergies of mathematical tools required for optimization of geometry and topology of shapes are explored. Furthermore, applications in science and engineering, including economics, social sciences, biology, physics and image processing are covered. Contents Part I Geometric issues in PDE problems related to the infinity Laplace operator Solution of free boundary problems in the presence of geometric uncertainties Distributed and boundary control problems for the semidiscrete Cahn–Hilliard/Navier–Stokes system with nonsmooth Ginzburg–Landau energies High-order topological expansions for Helmholtz problems in 2D On a new phase field model for the approximation of interfacial energies of multiphase systems Optimization of eigenvalues and eigenmodes by using the adjoint method Discrete varifolds and surface approximation Part II Weak Monge–Ampere solutions of the semi-discrete optimal transportation problem Optimal transportation theory with repulsive costs Wardrop equilibria: long-term variant, degenerate anisotropic PDEs and numerical approximations On the Lagrangian branched transport model and the equivalence with its Eulerian formulation On some nonlinear evolution systems which are perturbations of Wasserstein gradient flows Pressureless Euler equations with maximal density constraint: a time-splitting scheme Convergence of a fully discrete variational scheme for a thin-film equatio Interpretation of finite volume discretization schemes for the Fokker–Planck equation as gradient flows for the discrete Wasserstein distance

Topological Optimization and Optimal Transport

Maïtine Bergounioux,Édouard Oudet,Martin Rumpf,Guillaume Carlier,Thierry Champion,Filippo Santambrogio
Topological Optimization and Optimal Transport Book PDF

Author : Maïtine Bergounioux,Édouard Oudet,Martin Rumpf,Guillaume Carlier,Thierry Champion,Filippo Santambrogio
Publisher : Walter de Gruyter GmbH & Co KG
Page : 432 pages
File Size : 53,8 Mb
Release : 2017-08-07
Category : Mathematics
ISBN : 9783110430509

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Topological Optimization and Optimal Transport by Maïtine Bergounioux,Édouard Oudet,Martin Rumpf,Guillaume Carlier,Thierry Champion,Filippo Santambrogio Pdf

By discussing topics such as shape representations, relaxation theory and optimal transport, trends and synergies of mathematical tools required for optimization of geometry and topology of shapes are explored. Furthermore, applications in science and engineering, including economics, social sciences, biology, physics and image processing are covered. Contents Part I Geometric issues in PDE problems related to the infinity Laplace operator Solution of free boundary problems in the presence of geometric uncertainties Distributed and boundary control problems for the semidiscrete Cahn–Hilliard/Navier–Stokes system with nonsmooth Ginzburg–Landau energies High-order topological expansions for Helmholtz problems in 2D On a new phase field model for the approximation of interfacial energies of multiphase systems Optimization of eigenvalues and eigenmodes by using the adjoint method Discrete varifolds and surface approximation Part II Weak Monge–Ampere solutions of the semi-discrete optimal transportation problem Optimal transportation theory with repulsive costs Wardrop equilibria: long-term variant, degenerate anisotropic PDEs and numerical approximations On the Lagrangian branched transport model and the equivalence with its Eulerian formulation On some nonlinear evolution systems which are perturbations of Wasserstein gradient flows Pressureless Euler equations with maximal density constraint: a time-splitting scheme Convergence of a fully discrete variational scheme for a thin-film equatio Interpretation of finite volume discretization schemes for the Fokker–Planck equation as gradient flows for the discrete Wasserstein distance

Optimal Transportation

Yann Ollivier,Hervé Pajot,Cedric Villani
Optimal Transportation Book PDF

Author : Yann Ollivier,Hervé Pajot,Cedric Villani
Publisher : Cambridge University Press
Page : 317 pages
File Size : 42,5 Mb
Release : 2014-08-07
Category : Mathematics
ISBN : 9781107689497

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Optimal Transportation by Yann Ollivier,Hervé Pajot,Cedric Villani Pdf

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

Optimal Urban Networks via Mass Transportation

Giuseppe Buttazzo,Aldo Pratelli,Sergio Solimini,Eugene Stepanov
Optimal Urban Networks via Mass Transportation Book PDF

Author : Giuseppe Buttazzo,Aldo Pratelli,Sergio Solimini,Eugene Stepanov
Publisher : Springer Science & Business Media
Page : 161 pages
File Size : 41,8 Mb
Release : 2008-12-03
Category : Mathematics
ISBN : 9783540857983

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Optimal Urban Networks via Mass Transportation by Giuseppe Buttazzo,Aldo Pratelli,Sergio Solimini,Eugene Stepanov Pdf

Recently much attention has been devoted to the optimization of transportation networks in a given geographic area. One assumes the distributions of population and of services/workplaces (i.e. the network's sources and sinks) are known, as well as the costs of movement with/without the network, and the cost of constructing/maintaining it. Both the long-term optimization and the short-term, "who goes where," optimization are considered. These models can also be adapted for the optimization of other types of networks, such as telecommunications, pipeline or drainage networks. In the monograph we study the most general problem settings, namely, when neither the shape nor even the topology of the network to be constructed is known a priori.

Optimal Transportation and Applications

Luigi Ambrosio,Luis A. Caffarelli,Yann Brenier,Giuseppe Buttazzo,Cédric Villani
Optimal Transportation and Applications Book PDF

Author : Luigi Ambrosio,Luis A. Caffarelli,Yann Brenier,Giuseppe Buttazzo,Cédric Villani
Publisher : Springer
Page : 169 pages
File Size : 44,5 Mb
Release : 2003-07-03
Category : Mathematics
ISBN : 9783540448570

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Optimal Transportation and Applications by Luigi Ambrosio,Luis A. Caffarelli,Yann Brenier,Giuseppe Buttazzo,Cédric Villani Pdf

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.

Optimal Transport

Cédric Villani
Optimal Transport Book PDF

Author : Cédric Villani
Publisher : Unknown
Page : 318 pages
File Size : 40,8 Mb
Release : 2014-09-02
Category : Combinatorial analysis
ISBN : 1316005143

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Optimal Transport by Cédric Villani Pdf

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

Optimal Transport

Cédric Villani
Optimal Transport Book PDF

Author : Cédric Villani
Publisher : Springer Science & Business Media
Page : 970 pages
File Size : 43,7 Mb
Release : 2008-10-26
Category : Mathematics
ISBN : 9783540710509

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Optimal Transport by Cédric Villani Pdf

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

Filippo Santambrogio
Optimal Transport for Applied Mathematicians Book PDF

Author : Filippo Santambrogio
Publisher : Birkhäuser
Page : 353 pages
File Size : 49,7 Mb
Release : 2015-10-17
Category : Mathematics
ISBN : 9783319208282

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Optimal Transport for Applied Mathematicians by Filippo Santambrogio Pdf

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

Gabriel Peyre,Marco Cuturi
Computational Optimal Transport Book PDF

Author : Gabriel Peyre,Marco Cuturi
Publisher : Foundations and Trends(r) in M
Page : 272 pages
File Size : 53,5 Mb
Release : 2019-02-12
Category : Computers
ISBN : 1680835505

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Computational Optimal Transport by Gabriel Peyre,Marco Cuturi Pdf

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.

Regularity of Optimal Transport Maps and Applications

Guido Philippis
Regularity of Optimal Transport Maps and Applications Book PDF

Author : Guido Philippis
Publisher : Springer Science & Business Media
Page : 190 pages
File Size : 53,9 Mb
Release : 2013-06-28
Category : Mathematics
ISBN : 9788876424588

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Regularity of Optimal Transport Maps and Applications by Guido Philippis Pdf

In this thesis, we study the regularity of optimal transport maps and its applications to the semi-geostrophic system. The first two chapters survey the known theory, in particular there is a self-contained proof of Brenier’ theorem on existence of optimal transport maps and of Caffarelli’s Theorem on Holder continuity of optimal maps. In the third and fourth chapter we start investigating Sobolev regularity of optimal transport maps, while in Chapter 5 we show how the above mentioned results allows to prove the existence of Eulerian solution to the semi-geostrophic equation. In Chapter 6 we prove partial regularity of optimal maps with respect to a generic cost functions (it is well known that in this case global regularity can not be expected). More precisely we show that if the target and source measure have smooth densities the optimal map is always smooth outside a closed set of measure zero.

Optimal Transport

Gershon Wolansky
Optimal Transport Book PDF

Author : Gershon Wolansky
Publisher : Walter de Gruyter GmbH & Co KG
Page : 235 pages
File Size : 50,9 Mb
Release : 2021-01-18
Category : Mathematics
ISBN : 9783110633177

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Optimal Transport by Gershon Wolansky Pdf

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 Transport Methods in Economics

Alfred Galichon
Optimal Transport Methods in Economics Book PDF

Author : Alfred Galichon
Publisher : Princeton University Press
Page : 184 pages
File Size : 40,5 Mb
Release : 2018-08-14
Category : Business & Economics
ISBN : 9780691183466

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Optimal Transport Methods in Economics by Alfred Galichon Pdf

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

Topics in Optimal Transportation

Cédric Villani
Topics in Optimal Transportation Book PDF

Author : Cédric Villani
Publisher : American Mathematical Soc.
Page : 370 pages
File Size : 53,7 Mb
Release : 2021-08-25
Category : Education
ISBN : 9781470467265

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Topics in Optimal Transportation by Cédric Villani Pdf

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.

Sub-Riemannian Geometry and Optimal Transport

Ludovic Rifford
Sub Riemannian Geometry and Optimal Transport Book PDF

Author : Ludovic Rifford
Publisher : Springer Science & Business Media
Page : 140 pages
File Size : 55,5 Mb
Release : 2014-04-03
Category : Mathematics
ISBN : 9783319048048

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Sub-Riemannian Geometry and Optimal Transport by Ludovic Rifford Pdf

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.

Optimal Transport Networks in Nature

Natalya Kizilova
Optimal Transport Networks in Nature Book PDF

Author : Natalya Kizilova
Publisher : World Scientific Publishing Company
Page : 200 pages
File Size : 45,8 Mb
Release : 2010
Category : Medical
ISBN : 9812838732

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Optimal Transport Networks in Nature by Natalya Kizilova Pdf

This unique book presents a broad range of data on geometry and topology of long-distance liquid transport networks in nature including circulatory and respiratory systems of mammals, trophic fluid transport systems of animals, and conducting systems of higher plants. It is the very first book where evidence of the common design principles and optimal properties of the transportation networks of vascular plants and animals is provided. The book also provides a comprehensive comparative study of the recent measurement results and data analysis, including unique data obtained by the author to conduct systems of plant leaves of different shapes, sizes, venation types and evolutionary ages. It was shown that the mathematical solutions of the optimization problem for the animal and plant conducting systems lead to the same design principles, despite different physical conditions of the fluid transport.