Optimal Transport and Applications to Geometric Optics

by Cristian E. Gutiérrez

Overview

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.

Product Details

ISBN-13:	9789819948666
Publisher:	Springer Nature Singapore
Publication date:	12/10/2023
Series:	SpringerBriefs on PDEs and Data Science
Edition description:	1st ed. 2023
Pages:	135
Product dimensions:	6.10(w) x 9.25(h) x (d)

About the Author

Dr. Cristian E. Gutierrez is a Professor at Temple University. His research areas include partial differential equations, geometric optics, optimal transport, and electromagnetism. He is a Fellow of the American Mathematical Society, a Member of the Academy of Sciences of Argentina, and a Member of the Academy of Sciences of the University of Bologna, Italy.

Table of Contents

Chapter 1 Introduction.- Chapter 2 The normal mapping or subdifferential.-Chapter 3 Sinkhorn’s theorem and application to the distribution problem.- Chapter4 Monge-Kantorovich distance.- Chapter 5 Multivalued measure preserving maps.- Chapter 6 Kantorovich Dual Problem.- Chapter 7 Brenier and Aleksandrov solutions.- Chapter 8 Cyclical monotonicity.- Chapter 9 Quadratic cost.- Chapter 10 Brenier ’s Polar Factorization Theorem.- Chapter 11 Benamou and Brenier formula.- Chapter 12 Snell’s law of refraction.- Chapter 13 Solution of the far field refractor problem < 1.- Chapter 14 Proof of the Disintegration Theorem.- Chapter 15 Acknowledgements.- Chapter 16 References.

Editorial Reviews

"This book gives a comprehensible presentation which covers several important topics on optimal transport in a concise yet detailed way. … Its succinct style of presentation makes this book suitable for an introductory graduate-level course on optimal transport, or, with appropriate selection of topics, for a summer school on optimal transport. Additionally, it serves as a valuable reference for researchers who plan to apply optimal transport theory in their work." (Henok Mawi, Mathematical Reviews, February, 2025)

From the Publisher