Gutierrez C. Optimal Transport and Applications to Geometric Optics 2023

Download this torrent!

Gutierrez C. Optimal Transport and Applications to Geometric Optics 2023

Category: Other
Total size: 1.64 MB
Added: 2025-03-10 23:38:59

Share ratio: 9 seeders, 2 leechers
Info Hash: A689991EABB77DFEDB56B4F3C634557E8C082122
Last updated: 12.1 hours ago

Textbook in PDF format 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. Preface Acknowledgements Introduction The Transportation or Distribution Problem Monge Problem Kantorovitch Problem Trans-Shipment Problem Minimum Network Flow Problem Conversion of the Network Flow Problem Into a Transportation Problem Another Way to Convert the Network Flow Problem Into Optimal Transport More on the Transshipment Problem Linear Programming The Normal Mapping or Subdifferential Properties of the Normal Mapping Weak Convergence of Monge-Ampère Measures Exercises on the Subdifferential and Monge-Ampère Measures Sinkhorn's Theorem and Application to the Distribution Problem Application to the Distribution Problem Sinkhorn's Algorithm Monge-Kantorovich Distance Disintegration of Measures Wasserstein Distance Topology Given by the Wasserstein Distance Multivalued Measure Preserving Maps Kantorovich Dual Problem Kantorovich dual = Monge = Kantorovich Invertibility of Optimal Maps Brenier and Aleksandrov Solutions Cyclical Monotonicity Quadratic Cost Cyclical Monotonicity of the Optimal Map: Heuristics Pde for the Quadratic Cost Brenier's Polar Factorization Theorem Benamou and Brenier Formula Snell's Law of Refraction In Vector Form κ1 κ=1 Derivation of the Snell Law Surfaces with the Uniform Refracting Property: Far Field Case Case κ=1 Case κ1 Uniform Refraction: Near Field Case Case 0