This book concerns the theory of optimal transport (OT) and its applications to solving problems in geometric optics. It is a self-contained presentation including a detailed analysis of the Monge problem, the Monge-Kantorovich problem, the transshipment problem, and the network flow problem. A chapter on Monge-Ampère measures is included containing also exercises. A detailed analysis of the Wasserstein metric is also carried out. For the applications to optics, the book describes the necessary background concerning light refraction, solving both far-field and near-field refraction problems, and indicates lines of current research in this area.
Researchers in the fields of mathematical analysis, optimal transport, partial differential equations (PDEs), optimization, and optics will find this book valuable. It is also suitable for graduate students studying mathematics, physics, and engineering. The prerequisites for this book include a solid understanding of measure theory and integration, as well as basic knowledge of functional analysis.
Author(s): Cristian E. Gutiérrez
Series: SpringerBriefs on PDEs and Data Science
Edition: 1
Publisher: Springer Nature Singapore
Year: 2023
Language: English
Pages: 135
Tags: Monge-Kantorovich problem, Wasserstein distance, Kantorovich-Rubinstein duality, Monge-Ampère equation, Transhipment problem, Disintegration of measures, Sinkhorn algorithm, Snell’s law, Refractor problem, network flow problem
Preface
Acknowledgements
Contents
1 Introduction
1.1 The Transportation or Distribution Problem
1.2 Monge Problem
1.3 Kantorovitch Problem
1.4 Trans-Shipment Problem
1.5 Minimum Network Flow Problem
1.6 Conversion of the Network Flow Problem Into a Transportation Problem
1.7 Another Way to Convert the Network Flow Problem Into Optimal Transport
1.8 More on the Transshipment Problem
1.9 Linear Programming
2 The Normal Mapping or Subdifferential
2.1 Properties of the Normal Mapping
2.2 Weak Convergence of Monge-Ampère Measures
2.3 Exercises on the Subdifferential and Monge-Ampère Measures
3 Sinkhorn's Theorem and Application to the Distribution Problem
3.1 Application to the Distribution Problem
3.2 Sinkhorn's Algorithm
4 Monge-Kantorovich Distance
4.1 Disintegration of Measures
4.2 Wasserstein Distance
4.3 Topology Given by the Wasserstein Distance
5 Multivalued Measure Preserving Maps
6 Kantorovich Dual Problem
6.1 Kantorovich dual = Monge = Kantorovich
6.1.1 Invertibility of Optimal Maps
7 Brenier and Aleksandrov Solutions
8 Cyclical Monotonicity
9 Quadratic Cost
9.1 Cyclical Monotonicity of the Optimal Map: Heuristics
9.2 Pde for the Quadratic Cost
10 Brenier's Polar Factorization Theorem
11 Benamou and Brenier Formula
12 Snell's Law of Refraction
12.1 In Vector Form
12.1.1 κ<1
12.1.2 κ>1
12.1.3 κ=1
12.2 Derivation of the Snell Law
12.3 Surfaces with the Uniform Refracting Property: Far Field Case
12.3.1 Case κ=1
12.3.2 Case κ<1
12.3.3 Case κ>1
12.4 Uniform Refraction: Near Field Case
12.5 Case 0<κ<1
12.6 Case κ>1
13 Solution of the Far Field Refractor Problem κ<1
13.1 Refractor Measure and Weak Solutions
13.2 Existence and Uniqueness of Solutions to the RefractorProblem
13.3 Further Results
14 Proof of the Disintegration Theorem
Reference
