Optimal transport :
Villani, Cédric 1973-
Optimal transport : old and new / Cédric Villani. - First Edition. - xxii, 973 pages : illustrations ; 24 cm. - Grundlehren der mathematischen Wissenschaften, 338. 0072-7830 ; . - Grundlehren der mathematischen Wissenschaften ; 338. .
Includes bibliographical references (pages 915-956) and index.
1 Couplings and changes of variables -- 2 Three examples of coupling techniques -- 3 The founding fathers of optimal transport -- Part. I Qualitative description of optimal transport -- 4 Basic properties -- 5 Cyclical monotonicity and Kantorovich duality -- 6 The Wasserstein distances -- 7 Displacement interpolation -- 8 The Monge-Mather shortening principle -- 9 Solution of the Monge problem I: Global approach -- 10 Solution of the Monge problem II: Local approach -- 11 The Jacobian equation -- 12 Smoothness -- 13 Qualitative picture -- Part II. Optimal transport and Riemannian geometry -- 14 Ricci curvature -- 15 Otto calculus -- 16 Displacement convexity I -- 17 Displacement convexity II -- 18 Volume control -- 19 Density control and local regularity -- 20 Infinitesimal displacement convexity -- 21 Isoperimetric-type inequalties -- 22 Concentration inequalities -- 23 Gradient flows I -- 24 Gradient flows II: Qualitative properties -- 25 Gradient flows III: Functional inequalities -- Part. III Synthetic treatment of Ricci curvature -- 26 Analytic and synthetic points of view -- 27 Convergence of metric-measure spaces -- 28 Stability of optimal transport -- 29 Weak Ricci curvature bounds I: Definition and stability -- 30 Weak Ricci curvature bounds II: Geometric and analytic properties -- Conclusions and open problems.
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.
9783540710493 (acidfree paper) 3540710493 (acidfree paper) 9783540710509 (e-ISBN)
2008932183
Mathematical optimization
Transportation problems (Programming)
Probabilities
Dynamics.
Optimización matemática
Problemas de transporte (Programación)
Probabilidades
Dinámica
519.6
Optimal transport : old and new / Cédric Villani. - First Edition. - xxii, 973 pages : illustrations ; 24 cm. - Grundlehren der mathematischen Wissenschaften, 338. 0072-7830 ; . - Grundlehren der mathematischen Wissenschaften ; 338. .
Includes bibliographical references (pages 915-956) and index.
1 Couplings and changes of variables -- 2 Three examples of coupling techniques -- 3 The founding fathers of optimal transport -- Part. I Qualitative description of optimal transport -- 4 Basic properties -- 5 Cyclical monotonicity and Kantorovich duality -- 6 The Wasserstein distances -- 7 Displacement interpolation -- 8 The Monge-Mather shortening principle -- 9 Solution of the Monge problem I: Global approach -- 10 Solution of the Monge problem II: Local approach -- 11 The Jacobian equation -- 12 Smoothness -- 13 Qualitative picture -- Part II. Optimal transport and Riemannian geometry -- 14 Ricci curvature -- 15 Otto calculus -- 16 Displacement convexity I -- 17 Displacement convexity II -- 18 Volume control -- 19 Density control and local regularity -- 20 Infinitesimal displacement convexity -- 21 Isoperimetric-type inequalties -- 22 Concentration inequalities -- 23 Gradient flows I -- 24 Gradient flows II: Qualitative properties -- 25 Gradient flows III: Functional inequalities -- Part. III Synthetic treatment of Ricci curvature -- 26 Analytic and synthetic points of view -- 27 Convergence of metric-measure spaces -- 28 Stability of optimal transport -- 29 Weak Ricci curvature bounds I: Definition and stability -- 30 Weak Ricci curvature bounds II: Geometric and analytic properties -- Conclusions and open problems.
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.
9783540710493 (acidfree paper) 3540710493 (acidfree paper) 9783540710509 (e-ISBN)
2008932183
Mathematical optimization
Transportation problems (Programming)
Probabilities
Dynamics.
Optimización matemática
Problemas de transporte (Programación)
Probabilidades
Dinámica
519.6