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.
Optimal transport is a beautiful theory with plenty of applications both inside and outside mathematics; even though it has its roots in the work of Monge at the end of the 18th century, it is still the object of an intensive stream of pure, applied, and computational research. Gero Friesecke brilliantly succeeds in giving a rich, modern, and mathematically rigorous overview of the field while sharing his deep intuitions with the reader." - Guillaume Carlier, Université Paris Dauphine
"I wish that this book had existed when I first learned (and taught) optimal transport!" - David Bourne, Heriot-Watt University
