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E-raamat: Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling

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Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and ModelingSuurem pilt
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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 researche

rs in both pure and applied mathematics interested in the problems and applications of optimal transport will find this to be an invaluable resource.

Preface.- Primal and Dual Problems.- One-Dimensional Issues.- L^1 and L^infinity Theory.- Minimal Flows.- Wasserstein Spaces.- Numerical Methods.- Functionals over Probabilities.- Gradient Flows.- Exercises.- References.- Index.

Arvustused

This book offers an excellent, exciting and enjoyable, tour through the theory of optimal transportation, with a very good choice of topics . It is well written and thorough and provides an excellent introduction to applied mathematicians . a carefully selected list of exercises, make it ideal either as a textbook for an advanced postgraduate of doctoral level course, or for independent study. (Athanasios Yannacopoulos, zbMATH 1401.49002, 2019)

This book is very well written, and the proofs are carefully chosen and adapted. It is suitable for the researcher or the student willing to enter this field as well as for the professor planning a course on this topic. Thanks to the discussions at the end of each chapter and to the rich bibliography it is also a very good reference book. (Luigi De Pascale, Mathematical Reviews, January, 2017)

Preface vii
Introduction to optimal transport xiv
Acknowledgements xix
Notation xxv
1 Primal and dual problems
1(58)
1.1 Kantorovich and Monge problems
1(8)
1.2 Duality
9(4)
1.3 The case c(x, y) = h(x -- y) for h strictly convex and the existence of an optimal T
13(7)
1.3.1 The quadratic case in Rd
16(2)
1.3.2 The quadratic case on the flat torus
18(2)
1.4 Counterexamples to existence
20(1)
1.5 Kantorovich as a relaxation of Monge
21(4)
1.6 c-concavity, duality and optimality
25(16)
1.6.1 Convex and c-concave functions
25(3)
1.6.2 c-Cyclical monotonicity and duality
28(7)
1.6.3 A direct proof of duality
35(2)
1.6.4 Sufficient conditions for optimality and stability
37(4)
1.7 Discussion
41(18)
1.7.1 Probabilistic interpretation
41(1)
1.7.2 Polar factorization
42(2)
1.7.3 Matching problems and economic interpretations
44(4)
1.7.4 Multi-marginal transport problems
48(3)
1.7.5 Martingale optimal transport and financial applications
51(3)
1.7.6 Monge-Ampere equations and regularity
54(5)
2 One-dimensional issues
59(28)
2.1 Monotone transport maps and plans in 1D
59(4)
2.2 The optimality of the monotone map
63(4)
2.3 The Knothe transport
67(5)
2.4 Knothe as a limit of Brenier maps
72(7)
2.5 Discussion
79(8)
2.5.1 Histogram equalization
79(2)
2.5.2 Monotone maps from 1D constructions
81(2)
2.5.3 Isoperimetric inequality via Knothe or Brenier maps
83(4)
3 L1 and L∞ theory
87(34)
3.1 The Monge case, with cost |x - y|
87(17)
3.1.1 Duality for distance costs
88(1)
3.1.2 Secondary variational problem
89(2)
3.1.3 Geometric properties of transport rays
91(8)
3.1.4 Existence and nonexistence of an optimal transport map
99(2)
3.1.5 Approximation of the monotone transport
101(3)
3.2 The supremal case, L∞
104(5)
3.3 Discussion
109(12)
3.3.1 Different norms and more general convex costs
109(4)
3.3.2 Concave costs (Lp, with 0 < p < 1)
113(8)
4 Minimal flows
121(56)
4.1 Eulerian and Lagrangian points of view
121(6)
4.1.1 Static and dynamical models
121(2)
4.1.2 The continuity equation
123(4)
4.2 Beckmann's problem
127(17)
4.2.1 Introduction, formal equivalences, and variants
127(2)
4.2.2 Producing a minimizer for the Beckmann Problem
129(5)
4.2.3 Traffic intensity and traffic flows for measures on curves
134(6)
4.2.4 Beckman problem in one dimension
140(2)
4.2.5 Characterization and uniqueness of the optimal w
142(2)
4.3 Summability of the transport density
144(7)
4.4 Discussion
151(26)
4.4.1 Congested transport
151(11)
4.4.2 Branched transport
162(15)
5 Wasserstein spaces
177(42)
5.1 Definition and triangle inequality
179(4)
5.2 Topology induced by Wp
183(4)
5.3 Curves in Wp and continuity equation
187(15)
5.3.1 The Benamou-Brenier functional Bp
189(3)
5.3.2 AC curves admit velocity fields
192(2)
5.3.3 Regularization of the continuity equation
194(3)
5.3.4 Velocity fields give Lipschitz behavior
197(1)
5.3.5 Derivative of Wpp along curves of measures
198(4)
5.4 Constant-speed geodesies in Wp
202(5)
5.5 Discussion
207(12)
5.5.1 The Woo distance
207(2)
5.5.2 Wasserstein and negative Sobolev distances
209(2)
5.5.3 Wasserstein and branched transport distances
211(3)
5.5.4 The sliced Wasserstein distance
214(1)
5.5.5 Barycenters in W2
215(4)
6 Numerical methods
219(30)
6.1 Benamou-Brenier
220(5)
6.2 Angenent-Hacker-Tannenbaum
225(7)
6.3 Numerical solution of Monge-Ampere
232(3)
6.4 Discussion
235(14)
6.4.1 Discrete numerical methods
235(7)
6.4.2 Semidiscrete numerical methods
242(7)
7 Functionals over probabilities
249(36)
7.1 Semi-continuity
250(10)
7.1.1 Potential, interaction, transport costs, dual norms
250(4)
7.1.2 Local functionals
254(6)
7.2 Convexity, first variations, and subdifferentials
260(9)
7.2.1 Dual norms
263(1)
7.2.2 Transport costs
264(3)
7.2.3 Optimality conditions
267(2)
7.3 Displacement convexity
269(7)
7.3.1 Displacement convexity of V and W
270(1)
7.3.2 Displacement convexity of F
271(4)
7.3.3 Convexity on generalized geodesies
275(1)
7.4 Discussion
276(9)
7.4.1 A case study: min F(q) + W22(q, v)
276(3)
7.4.2 Brunn-Minkowski inequality
279(1)
7.4.3 Urban equilibria
280(5)
8 Gradient flows
285(40)
8.1 Gradient flows in Rd and in metric spaces
285(5)
8.2 Gradient flows in W2, derivation of the PDE
290(3)
8.3 Analysis of the Fokker-Planck case
293(8)
8.4 Discussion
301(24)
8.4.1 EVI, uniqueness, and geodesic convexity
301(3)
8.4.2 Other gradient flow PDEs
304(7)
8.4.3 Dirichlet boundary conditions
311(2)
8.4.4 Evolution PDEs: not only gradient flows
313(12)
Exercises 325(14)
References 339(12)
Index 351
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