Tag - Optimization

Julian Barreiro-Gomez: The role of stochastic differential games of mean-field type in smart cities applications

13th June, 2022

This brief talk aims to show how the stochastic differential games contribute to the optimal solution of large-scale engineering problems emerging in smart cities where several dynamical interactions occur, e.g., the water distribution system, the crowd management, the traffic flow, power systems, among many others. We show that the general simplest problem statement leads to a complex PIDE system involving a backward Hamilton-Jacobi-Bellman equation coupled with a forward Fokker-Plank-Kolmogorov equation. Then, we discuss how this complexity can be handled for specific cases pursuing to develop real implementation. As an example, we focus on the crowd evacuation problem. Finally, future directions we are currently working on involving machine learning and stability are presented.

Robert McCann: Optimal Transportation, Geometry and Dynamics

11th January, 2022

This is a 24-lecture course, with each lecture being around 80 minutes, given by Robert McCann. It gives an introduction to optimal transport.

This course is an introduction to the active research areas surrounding optimal transportation and its deep connections to problems in geometry, physics, nonlinear partial differential equations, and machine learning. The basic problem is to find the most efficient structure linking two or more continuous distributions of mass; think of pairing a cloud of electrons with a cloud of positrons so as to minimize average distance to annihilation.

Applications include existence, uniqueness, and regularity of surfaces with prescribed Gauss curvature (the underlying PDE is Monge-Ampère), geometric inequalities with sharp constants, image processing, optimal decision making, long time asymptotics of dissipative systems, and the geometry of fluid motion (Euler's equation and approximations appropriate to atmospheric, oceanic, damped and porous medium flows). The course builds on a background in analysis, including measure theory, but will develop elements as needed from the calculus of variations, game theory, differential equations, fluid mechanics, physics, economics, and geometry. A particular goal will be to expose the developing theories of curvature and dimension in metric-measure geometry, which provide a framework for adapting powerful ideas from Riemannian and Lorentzian geometry to non-smooth settings which arise both naturally in applications, and as limits of smooth problems.

Vesselin Drensky: From a Diophantine transport problem from 2016 and its possible solution from 1903 to classical problems in algebra

17th December, 2020

Motivated by a recent Diophantine transport problem about how to transport profitably a group of persons or objects, we survey classical facts about solving systems of linear Diophantine equations and inequalities in non-negative integers. We emphasize on the method of Elliott from 1903 and its further development by MacMahon in his 'Ω-Calculus' or Partition Analysis. Then we show how this approach can be used to solve problems in classical and non-commutative invariant theory and theory of algebras with polynomial identities.

Sanjeev Arora: Toward Theoretical Understanding of Deep Learning

12th September, 2018

The empirical success of deep learning drives much of the excitement about machine learning today. This success vastly outstrips our mathematical understanding. This lecture surveys progress in recent years toward developing a theory of deep learning. Works have started addressing issues such as speed of optimization, sample requirements for training, effect of architecture choices, and properties of deep generative models.

Sanjeev Arora: What is Machine Learning?

11th September, 2018

Machine learning is the sub-field of computer science concerned with creating programs and machines that can improve from experience and interaction. It relies upon mathematical optimization, statistics, and algorithm design. The talk will be an introduction to machine learning for a mathematical audience. We describe the mathematical formulations of basic types of learning such as supervised, unsupervised, interactive, etc., and the philosophical and scientific issues raised by them.