Historically, optimal transport was about moving a pile of mortar efficiently or transferring the output of an array of steel mines optimally. This type of problem has been found to arise in many different fields of mathematics, science, and engineering—from fluid dynamics to many-electron physics to artificial intelligence—and in the last three decades interest in the subject has exploded.

This accessible book begins with an elementary and self-contained chapter on optimal transport on finite state spaces that does not require measure theory or functional analysis. It builds up mathematical theory rigorously and from scratch, aided by intuitive arguments, informal discussion, and carefully selected applications. It is the first book to cover modern topics such as Wasserstein GANs and multimarginal problems and includes a discussion of numerical methods and basic MATLAB code for simulating optimal transport problems directly via linear programming or more efficiently via the Sinkhorn algorithm. Additionally, it provides classroom-tested exercises in every chapter.

This book is for advanced undergraduate students, beginning graduate students, and researchers in applied mathematics. It will also be of interest to students and researchers in physics, engineering, computer science, data science, and machine learning who want to become familiar with cornerstone concepts, results, and methods. 

Optimal Transport: A Comprehensive Introduction to Modeling, Analysis, Simulation, Applications is appropriate for a special topics course on optimal transport. It can also be used as a supplementary text for a general course on linear optimization, convex analysis, calculus of variations, or mathematical methods in data science.