Optimal Transport and Applications to Geometric Optics
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Beschreibung
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 andintegration, as well as basic knowledge of functional analysis.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.
Explores the connection between optimal transport and geometric optics, focusing on the design of free-form lenses Offers self-contained presentation, with detailed explanations and full proofs of the theorems and results Leverages optimal transport principles for the design of non-rotationally symmetric lenses
Autor*in
Cristian E. Gutiérrez
Themen in »Optimal Transport and Applications to Geometric Optics«
Monge-Kantorovich problem Wasserstein distance Kantorovich-Rubinstein duality Monge-Ampère equation Optimal maps Transhipment problem Disintegration of measures Sinkhorn algorithm Brenier polar factorization Benamou and Brenier formula Snell’s law of refraction Refractor problem Near field refraction Far field refraction Legendre transform
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"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)
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