Lecture

Optimal Transport: Regularity and Brenier's Theorem

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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 concept of optimal transport, focusing on the regularity of optimal maps and Brenier's theorem. It discusses the continuity of optimal maps and provides examples of when they are not continuous. The lecture also explores the notion of convexity in optimal transport and the implications of assuming the target is connected.

Instructors (2)

Maria Colombo

Teaching & PhD PhD Students Roberto Colombo, Antonio Tirotta, Carl Johan Peter Johansson, Giulia Mescolini Past EPFL PhD Students Luigi De Rosa, Silja Noëmi Aline Haffter, Massimo Sorella Courses Analysis IV - Lebesgue measure, Fourier analysis MATH-205 Learn the basis of Lebesgue integration and Fourier analysis Topics on the Euler and Navier-Stokes equations MATH-647 This topics course focuses on recent and classical fundamental results on the Euler and Navier-Stokes equa

Xavier Fernandez-Real Girona

Teaching & PhD PhD Students Roberto Colombo Courses Optimal transport MATH-476 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, functional/geometric inequalities, traffic models.

Official source

Related concepts (32)

Transportation theory (mathematics)

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".

Convex set

In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex region is a subset that intersects every line into a single line segment (possibly empty). For example, a solid cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is not convex. The boundary of a convex set is always a convex curve.

Wasserstein metric

In mathematics, the Wasserstein distance or Kantorovich–Rubinstein metric is a distance function defined between probability distributions on a given metric space . It is named after Leonid Vaseršteĭn. Intuitively, if each distribution is viewed as a unit amount of earth (soil) piled on , the metric is the minimum "cost" of turning one pile into the other, which is assumed to be the amount of earth that needs to be moved times the mean distance it has to be moved. This problem was first formalised by Gaspard Monge in 1781.

PowerVR

PowerVR is a division of Imagination Technologies (formerly VideoLogic) that develops hardware and software for 2D and 3D rendering, and for video encoding, decoding, associated and DirectX, OpenGL ES, OpenVG, and OpenCL acceleration. PowerVR also develops AI accelerators called Neural Network Accelerator (NNA). The PowerVR product line was originally introduced to compete in the desktop PC market for 3D hardware accelerators with a product with a better price–performance ratio than existing products like those from 3dfx Interactive.

Contradiction

In traditional logic, a contradiction occurs when a proposition conflicts either with itself or established fact. It is often used as a tool to detect disingenuous beliefs and bias. Illustrating a general tendency in applied logic, Aristotle's law of noncontradiction states that "It is impossible that the same thing can at the same time both belong and not belong to the same object and in the same respect.

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