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Lecture

Optimal Transport: Theory and Applications

In course

MATH-476: Optimal transport

The first part is devoted to Monge and Kantorovitch problems, discussing the existence and the properties of the optimal plan. The second part introduces the Wasserstein distance on measures and develops applications of optimal transport to PDEs, functiona ...

Description

This lecture covers the theory and applications of optimal transport, focusing on the concepts of infimal convolution, Kantorovich potentials, and the existence of optimal maps. The instructor discusses the nonexistence of maps in certain scenarios and the uniqueness of optimal plans. Various mathematical formulations and examples are presented to illustrate the principles of optimal transport.

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Instructors (2)

Maria Colombo

Xavier Fernandez-Real Girona

Official source

Related concepts (32)

Mathematical optimization

Mathematical optimization (alternatively spelled optimisation) or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfields: discrete optimization and continuous optimization. Optimization problems arise in all quantitative disciplines from computer science and engineering to operations research and economics, and the development of solution methods has been of interest in mathematics for centuries.

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In mathematics and economics, transportation theory or transport theory is a name given to the study of optimal transportation and allocation of resources. The problem was formalized by the French mathematician Gaspard Monge in 1781. In the 1920s A.N. Tolstoi was one of the first to study the transportation problem mathematically. In 1930, in the collection Transportation Planning Volume I for the National Commissariat of Transportation of the Soviet Union, he published a paper "Methods of Finding the Minimal Kilometrage in Cargo-transportation in space".

Map

A map is a symbolic depiction emphasizing relationships between elements of some space, such as objects, regions, or themes. Many maps are static, fixed to paper or some other durable medium, while others are dynamic or interactive. Although most commonly used to depict geography, maps may represent any space, real or fictional, without regard to context or scale, such as in brain mapping, DNA mapping, or computer network topology mapping.

Topographic map

In modern mapping, a topographic map or topographic sheet is a type of map characterized by large-scale detail and quantitative representation of relief features, usually using contour lines (connecting points of equal elevation), but historically using a variety of methods. Traditional definitions require a topographic map to show both natural and artificial features. A topographic survey is typically based upon a systematic observation and published as a map series, made up of two or more map sheets that combine to form the whole map.

Apple Maps

Apple Maps is a web mapping service developed by Apple Inc. The default map system of iOS, iPadOS, macOS, and watchOS, it provides directions and estimated times of arrival for driving, walking, cycling, and public transportation navigation. A "Flyover" mode shows certain densely populated urban centers and other places of interest in a 3D landscape composed of models of buildings and structures. First released in 2012, Apple Maps replaced Google Maps as the default map system on Apple devices.

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