Gradient Flows

Gradient Flows
Title Gradient Flows PDF eBook
Author Luigi Ambrosio
Publisher Springer Science & Business Media
Pages 333
Release 2008-10-29
Genre Mathematics
ISBN 376438722X

Download Gradient Flows Book in PDF, Epub and Kindle

The book is devoted to the theory of gradient flows in the general framework of metric spaces, and in the more specific setting of the space of probability measures, which provide a surprising link between optimal transportation theory and many evolutionary PDE's related to (non)linear diffusion. Particular emphasis is given to the convergence of the implicit time discretization method and to the error estimates for this discretization, extending the well established theory in Hilbert spaces. The book is split in two main parts that can be read independently of each other.

Gradient Flows

Gradient Flows
Title Gradient Flows PDF eBook
Author Luigi Ambrosio
Publisher Springer Science & Business Media
Pages 330
Release 2006-03-30
Genre Mathematics
ISBN 3764373091

Download Gradient Flows Book in PDF, Epub and Kindle

This book is devoted to a theory of gradient ?ows in spaces which are not nec- sarily endowed with a natural linear or di?erentiable structure. It is made of two parts, the ?rst one concerning gradient ?ows in metric spaces and the second one 2 1 devoted to gradient ?ows in the L -Wasserstein space of probability measures on p a separable Hilbert space X (we consider the L -Wasserstein distance, p? (1,?), as well). The two parts have some connections, due to the fact that the Wasserstein space of probability measures provides an important model to which the “metric” theory applies, but the book is conceived in such a way that the two parts can be read independently, the ?rst one by the reader more interested to Non-Smooth Analysis and Analysis in Metric Spaces, and the second one by the reader more oriented to theapplications in Partial Di?erential Equations, Measure Theory and Probability.

Hamiltonian and Gradient Flows, Algorithms, and Control

Hamiltonian and Gradient Flows, Algorithms, and Control
Title Hamiltonian and Gradient Flows, Algorithms, and Control PDF eBook
Author Anthony Bloch
Publisher American Mathematical Soc.
Pages 172
Release
Genre Mathematics
ISBN 9780821871362

Download Hamiltonian and Gradient Flows, Algorithms, and Control Book in PDF, Epub and Kindle

This is the proceedings of a conference held at the Fields Insitute and designed to bring together traditionally disparate fields of mathematical research. On such key interraction occurs between dynamical systems and algorithms. This volume explores many such interractions as well as related work in optimal control and partial differential equations.

The Space of Spaces: Curvature Bounds and Gradient Flows on the Space of Metric Measure Spaces

The Space of Spaces: Curvature Bounds and Gradient Flows on the Space of Metric Measure Spaces
Title The Space of Spaces: Curvature Bounds and Gradient Flows on the Space of Metric Measure Spaces PDF eBook
Author Karl-Theodor Sturm
Publisher American Mathematical Society
Pages 124
Release 2023-11-27
Genre Mathematics
ISBN 1470466961

Download The Space of Spaces: Curvature Bounds and Gradient Flows on the Space of Metric Measure Spaces Book in PDF, Epub and Kindle

View the abstract.

The Ricci Flow in Riemannian Geometry

The Ricci Flow in Riemannian Geometry
Title The Ricci Flow in Riemannian Geometry PDF eBook
Author Ben Andrews
Publisher Springer Science & Business Media
Pages 306
Release 2011
Genre Mathematics
ISBN 3642162851

Download The Ricci Flow in Riemannian Geometry Book in PDF, Epub and Kindle

This book focuses on Hamilton's Ricci flow, beginning with a detailed discussion of the required aspects of differential geometry, progressing through existence and regularity theory, compactness theorems for Riemannian manifolds, and Perelman's noncollapsing results, and culminating in a detailed analysis of the evolution of curvature, where recent breakthroughs of Böhm and Wilking and Brendle and Schoen have led to a proof of the differentiable 1/4-pinching sphere theorem.

Lectures on Optimal Transport

Lectures on Optimal Transport
Title Lectures on Optimal Transport PDF eBook
Author Luigi Ambrosio
Publisher Springer Nature
Pages 250
Release 2021-07-22
Genre Mathematics
ISBN 3030721620

Download Lectures on Optimal Transport Book in PDF, Epub and Kindle

This textbook is addressed to PhD or senior undergraduate students in mathematics, with interests in analysis, calculus of variations, probability and optimal transport. It originated from the teaching experience of the first author in the Scuola Normale Superiore, where a course on optimal transport and its applications has been given many times during the last 20 years. The topics and the tools were chosen at a sufficiently general and advanced level so that the student or scholar interested in a more specific theme would gain from the book the necessary background to explore it. After a large and detailed introduction to classical theory, more specific attention is devoted to applications to geometric and functional inequalities and to partial differential equations.

An Invitation to Optimal Transport, Wasserstein Distances, and Gradient Flows

An Invitation to Optimal Transport, Wasserstein Distances, and Gradient Flows
Title An Invitation to Optimal Transport, Wasserstein Distances, and Gradient Flows PDF eBook
Author Alessio Figalli
Publisher European Mathematical Society
Pages 0
Release 2023-05-15
Genre Mathematics
ISBN 3985470502

Download An Invitation to Optimal Transport, Wasserstein Distances, and Gradient Flows Book in PDF, Epub and Kindle

This book provides a self-contained introduction to optimal transport, and it is intended as a starting point for any researcher who wants to enter into this beautiful subject. The presentation focuses on the essential topics of the theory: Kantorovich duality, existence and uniqueness of optimal transport maps, Wasserstein distances, the JKO scheme, Otto's calculus, and Wasserstein gradient flows. At the end, a presentation of some selected applications of optimal transport is given. Suitable for a course at the graduate level, the book also includes an appendix with a series of exercises along with their solutions. The second edition contains a number of additions, such as a new section on the Brunn–Minkowski inequality, new exercises, and various corrections throughout the text.