Sub Riemannian Geometry And Optimal Transport

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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 : 146 pages
File Size : 48,7 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.

A Comprehensive Introduction to Sub-Riemannian Geometry

Andrei Agrachev,Davide Barilari,Ugo Boscain
A Comprehensive Introduction to Sub Riemannian Geometry Book PDF

Author : Andrei Agrachev,Davide Barilari,Ugo Boscain
Publisher : Cambridge University Press
Page : 765 pages
File Size : 48,9 Mb
Release : 2019-10-31
Category : Mathematics
ISBN : 9781108476355

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A Comprehensive Introduction to Sub-Riemannian Geometry by Andrei Agrachev,Davide Barilari,Ugo Boscain Pdf

Provides a comprehensive and self-contained introduction to sub-Riemannian geometry and its applications. For graduate students and researchers.

Optimal Transportation

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

Author : Yann Ollivier,Hervé Pajot,Cédric Villani
Publisher : Cambridge University Press
Page : 317 pages
File Size : 51,6 Mb
Release : 2014-08-07
Category : Mathematics
ISBN : 9781107689497

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

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

Noncommutative Geometry and Optimal Transport

Pierre Martinetti,Jean-Christophe Wallet
Noncommutative Geometry and Optimal Transport Book PDF

Author : Pierre Martinetti,Jean-Christophe Wallet
Publisher : American Mathematical Soc.
Page : 223 pages
File Size : 40,8 Mb
Release : 2016-10-26
Category : Mathematical optimization
ISBN : 9781470422974

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Noncommutative Geometry and Optimal Transport by Pierre Martinetti,Jean-Christophe Wallet Pdf

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.

Control of Nonholonomic Systems: from Sub-Riemannian Geometry to Motion Planning

Frédéric Jean
Control of Nonholonomic Systems from Sub Riemannian Geometry to Motion Planning Book PDF

Author : Frédéric Jean
Publisher : Springer
Page : 112 pages
File Size : 55,6 Mb
Release : 2014-07-17
Category : Science
ISBN : 9783319086903

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Control of Nonholonomic Systems: from Sub-Riemannian Geometry to Motion Planning by Frédéric Jean Pdf

Nonholonomic systems are control systems which depend linearly on the control. Their underlying geometry is the sub-Riemannian geometry, which plays for these systems the same role as Euclidean geometry does for linear systems. In particular the usual notions of approximations at the first order, that are essential for control purposes, have to be defined in terms of this geometry. The aim of these notes is to present these notions of approximation and their application to the motion planning problem for nonholonomic systems.

Curvature: A Variational Approach

A. Agrachev,D. Barilari,L. Rizzi
Curvature A Variational Approach Book PDF

Author : A. Agrachev,D. Barilari,L. Rizzi
Publisher : American Mathematical Soc.
Page : 142 pages
File Size : 40,5 Mb
Release : 2019-01-08
Category : Curvature
ISBN : 9781470426460

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Curvature: A Variational Approach by A. Agrachev,D. Barilari,L. Rizzi Pdf

The curvature discussed in this paper is a far reaching generalization of the Riemannian sectional curvature. The authors give a unified definition of curvature which applies to a wide class of geometric structures whose geodesics arise from optimal control problems, including Riemannian, sub-Riemannian, Finsler and sub-Finsler spaces. Special attention is paid to the sub-Riemannian (or Carnot–Carathéodory) metric spaces. The authors' construction of curvature is direct and naive, and similar to the original approach of Riemann. In particular, they extract geometric invariants from the asymptotics of the cost of optimal control problems. Surprisingly, it works in a very general setting and, in particular, for all sub-Riemannian spaces.

Introduction to Geometric Control

Yuri Sachkov
Introduction to Geometric Control Book PDF

Author : Yuri Sachkov
Publisher : Springer Nature
Page : 176 pages
File Size : 52,5 Mb
Release : 2022-07-02
Category : Technology & Engineering
ISBN : 9783031020704

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Introduction to Geometric Control by Yuri Sachkov Pdf

This text is an enhanced, English version of the Russian edition, published in early 2021 and is appropriate for an introductory course in geometric control theory. The concise presentation provides an accessible treatment of the subject for advanced undergraduate and graduate students in theoretical and applied mathematics, as well as to experts in classic control theory for whom geometric methods may be introduced. Theory is accompanied by characteristic examples such as stopping a train, motion of mobile robot, Euler elasticae, Dido's problem, and rolling of the sphere on the plane. Quick foundations to some recent topics of interest like control on Lie groups and sub-Riemannian geometry are included. Prerequisites include only a basic knowledge of calculus, linear algebra, and ODEs; preliminary knowledge of control theory is not assumed. The applications problems-oriented approach discusses core subjects and encourages the reader to solve related challenges independently. Highly-motivated readers can acquire working knowledge of geometric control techniques and progress to studying control problems and more comprehensive books on their own. Selected sections provide exercises to assist in deeper understanding of the material. Controllability and optimal control problems are considered for nonlinear nonholonomic systems on smooth manifolds, in particular, on Lie groups. For the controllability problem, the following questions are considered: controllability of linear systems, local controllability of nonlinear systems, Nagano–Sussmann Orbit theorem, Rashevskii–Chow theorem, Krener's theorem. For the optimal control problem, Filippov's theorem is stated, invariant formulation of Pontryagin maximum principle on manifolds is given, second-order optimality conditions are discussed, and the sub-Riemannian problem is studied in detail. Pontryagin maximum principle is proved for sub-Riemannian problems, solution to the sub-Riemannian problems on the Heisenberg group, the group of motions of the plane, and the Engel group is described.

Geometric Control Theory and Sub-Riemannian Geometry

Gianna Stefani,Ugo Boscain,Jean-Paul Gauthier,Andrey Sarychev,Mario Sigalotti
Geometric Control Theory and Sub Riemannian Geometry Book PDF

Author : Gianna Stefani,Ugo Boscain,Jean-Paul Gauthier,Andrey Sarychev,Mario Sigalotti
Publisher : Springer
Page : 385 pages
File Size : 45,9 Mb
Release : 2014-06-05
Category : Mathematics
ISBN : 9783319021324

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Geometric Control Theory and Sub-Riemannian Geometry by Gianna Stefani,Ugo Boscain,Jean-Paul Gauthier,Andrey Sarychev,Mario Sigalotti Pdf

Honoring Andrei Agrachev's 60th birthday, this volume presents recent advances in the interaction between Geometric Control Theory and sub-Riemannian geometry. On the one hand, Geometric Control Theory used the differential geometric and Lie algebraic language for studying controllability, motion planning, stabilizability and optimality for control systems. The geometric approach turned out to be fruitful in applications to robotics, vision modeling, mathematical physics etc. On the other hand, Riemannian geometry and its generalizations, such as sub-Riemannian, Finslerian geometry etc., have been actively adopting methods developed in the scope of geometric control. Application of these methods has led to important results regarding geometry of sub-Riemannian spaces, regularity of sub-Riemannian distances, properties of the group of diffeomorphisms of sub-Riemannian manifolds, local geometry and equivalence of distributions and sub-Riemannian structures, regularity of the Hausdorff volume, etc.

Optimal Transport

Cédric Villani
Optimal Transport Book PDF

Author : Cédric Villani
Publisher : Springer Science & Business Media
Page : 970 pages
File Size : 51,7 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.

An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem

Luca Capogna,Donatella Danielli,Scott D. Pauls,Jeremy Tyson
An Introduction to the Heisenberg Group and the Sub Riemannian Isoperimetric Problem Book PDF

Author : Luca Capogna,Donatella Danielli,Scott D. Pauls,Jeremy Tyson
Publisher : Springer Science & Business Media
Page : 224 pages
File Size : 50,6 Mb
Release : 2007-08-08
Category : Mathematics
ISBN : 9783764381332

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An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem by Luca Capogna,Donatella Danielli,Scott D. Pauls,Jeremy Tyson Pdf

This book gives an up-to-date account of progress on Pansu's celebrated problem on the sub-Riemannian isoperimetric profile of the Heisenberg group. It also serves as an introduction to the general field of sub-Riemannian geometric analysis. It develops the methods and tools of sub-Riemannian differential geometry, nonsmooth analysis, and geometric measure theory suitable for attacks on Pansu's problem.

Control Theory from the Geometric Viewpoint

Andrei A. Agrachev,Yuri Sachkov
Control Theory from the Geometric Viewpoint Book PDF

Author : Andrei A. Agrachev,Yuri Sachkov
Publisher : Springer Science & Business Media
Page : 415 pages
File Size : 41,9 Mb
Release : 2013-03-14
Category : Science
ISBN : 9783662064047

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Control Theory from the Geometric Viewpoint by Andrei A. Agrachev,Yuri Sachkov Pdf

This book presents some facts and methods of the Mathematical Control Theory treated from the geometric point of view. The book is mainly based on graduate courses given by the first coauthor in the years 2000-2001 at the International School for Advanced Studies, Trieste, Italy. Mathematical prerequisites are reduced to standard courses of Analysis and Linear Algebra plus some basic Real and Functional Analysis. No preliminary knowledge of Control Theory or Differential Geometry is required. What this book is about? The classical deterministic physical world is described by smooth dynamical systems: the future in such a system is com pletely determined by the initial conditions. Moreover, the near future changes smoothly with the initial data. If we leave room for "free will" in this fatalistic world, then we come to control systems. We do so by allowing certain param eters of the dynamical system to change freely at every instant of time. That is what we routinely do in real life with our body, car, cooker, as well as with aircraft, technological processes etc. We try to control all these dynamical systems! Smooth dynamical systems are governed by differential equations. In this book we deal only with finite dimensional systems: they are governed by ordi nary differential equations on finite dimensional smooth manifolds. A control system for us is thus a family of ordinary differential equations. The family is parametrized by control parameters.

Second Order Analysis on $(\mathscr {P}_2(M),W_2)$

Nicola Gigli
Second Order Analysis on mathscr P 2 M W 2 Book PDF

Author : Nicola Gigli
Publisher : American Mathematical Soc.
Page : 173 pages
File Size : 47,7 Mb
Release : 2012-02-22
Category : Mathematics
ISBN : 9780821853092

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Second Order Analysis on $(\mathscr {P}_2(M),W_2)$ by Nicola Gigli Pdf

The author develops a rigorous second order analysis on the space of probability measures on a Riemannian manifold endowed with the quadratic optimal transport distance $W_2$. The discussion includes: definition of covariant derivative, discussion of the problem of existence of parallel transport, calculus of the Riemannian curvature tensor, differentiability of the exponential map and existence of Jacobi fields. This approach does not require any smoothness assumption on the measures considered.

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 : 51,7 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

Issues in Algebra, Geometry, and Topology: 2011 Edition

Anonim
Issues in Algebra Geometry and Topology 2011 Edition Book PDF

Author : Anonim
Publisher : ScholarlyEditions
Page : 395 pages
File Size : 52,5 Mb
Release : 2012-01-09
Category : Mathematics
ISBN : 9781464966354

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Issues in Algebra, Geometry, and Topology: 2011 Edition by Anonim Pdf

Issues in Algebra, Geometry, and Topology / 2011 Edition is a ScholarlyEditions™ eBook that delivers timely, authoritative, and comprehensive information about Algebra, Geometry, and Topology. The editors have built Issues in Algebra, Geometry, and Topology: 2011 Edition on the vast information databases of ScholarlyNews.™ You can expect the information about Algebra, Geometry, and Topology in this eBook to be deeper than what you can access anywhere else, as well as consistently reliable, authoritative, informed, and relevant. The content of Issues in Algebra, Geometry, and Topology: 2011 Edition has been produced by the world’s leading scientists, engineers, analysts, research institutions, and companies. All of the content is from peer-reviewed sources, and all of it is written, assembled, and edited by the editors at ScholarlyEditions™ and available exclusively from us. You now have a source you can cite with authority, confidence, and credibility. More information is available at http://www.ScholarlyEditions.com/.

Lectures on Optimal Transport

Luigi Ambrosio,Elia Brué,Daniele Semola
Lectures on Optimal Transport Book PDF

Author : Luigi Ambrosio,Elia Brué,Daniele Semola
Publisher : Springer Nature
Page : 250 pages
File Size : 46,5 Mb
Release : 2021-07-22
Category : Mathematics
ISBN : 9783030721626

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Lectures on Optimal Transport by Luigi Ambrosio,Elia Brué,Daniele Semola Pdf

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.