Gradient flows with metric and differentiable structures, and applications to the Wasserstein space

Luigi Ambrosio; Nicola Gigli; Giuseppe Savaré

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (2004)

  • Volume: 15, Issue: 3-4, page 327-343
  • ISSN: 1120-6330

Abstract

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In this paper we summarize some of the main results of a forthcoming book on this topic, where we examine in detail the theory of curves of maximal slope in a general metric setting, following some ideas introduced in [11, 5], and study in detail the case of the Wasserstein space of probability measures. In the first part we derive new general conditions ensuring convergence of the implicit time discretization scheme to a curve of maximal slope, the uniqueness, and the error estimates. In the second part we study in detail the differentiable structure of the Wasserstein space, to which the metric theory applies, and use this structure to give also an equivalent concept of gradient flow. Our analysis includes measures in infinite-dimensional Hilbert spaces and it does not require any absolute continuity assumption on the measures involved.

How to cite

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Ambrosio, Luigi, Gigli, Nicola, and Savaré, Giuseppe. "Gradient flows with metric and differentiable structures, and applications to the Wasserstein space." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 15.3-4 (2004): 327-343. <http://eudml.org/doc/252389>.

@article{Ambrosio2004,
abstract = {In this paper we summarize some of the main results of a forthcoming book on this topic, where we examine in detail the theory of curves of maximal slope in a general metric setting, following some ideas introduced in [11, 5], and study in detail the case of the Wasserstein space of probability measures. In the first part we derive new general conditions ensuring convergence of the implicit time discretization scheme to a curve of maximal slope, the uniqueness, and the error estimates. In the second part we study in detail the differentiable structure of the Wasserstein space, to which the metric theory applies, and use this structure to give also an equivalent concept of gradient flow. Our analysis includes measures in infinite-dimensional Hilbert spaces and it does not require any absolute continuity assumption on the measures involved.},
author = {Ambrosio, Luigi, Gigli, Nicola, Savaré, Giuseppe},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Gradient flows; Wasserstein metric; Optimal transport},
language = {eng},
month = {12},
number = {3-4},
pages = {327-343},
publisher = {Accademia Nazionale dei Lincei},
title = {Gradient flows with metric and differentiable structures, and applications to the Wasserstein space},
url = {http://eudml.org/doc/252389},
volume = {15},
year = {2004},
}

TY - JOUR
AU - Ambrosio, Luigi
AU - Gigli, Nicola
AU - Savaré, Giuseppe
TI - Gradient flows with metric and differentiable structures, and applications to the Wasserstein space
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 2004/12//
PB - Accademia Nazionale dei Lincei
VL - 15
IS - 3-4
SP - 327
EP - 343
AB - In this paper we summarize some of the main results of a forthcoming book on this topic, where we examine in detail the theory of curves of maximal slope in a general metric setting, following some ideas introduced in [11, 5], and study in detail the case of the Wasserstein space of probability measures. In the first part we derive new general conditions ensuring convergence of the implicit time discretization scheme to a curve of maximal slope, the uniqueness, and the error estimates. In the second part we study in detail the differentiable structure of the Wasserstein space, to which the metric theory applies, and use this structure to give also an equivalent concept of gradient flow. Our analysis includes measures in infinite-dimensional Hilbert spaces and it does not require any absolute continuity assumption on the measures involved.
LA - eng
KW - Gradient flows; Wasserstein metric; Optimal transport
UR - http://eudml.org/doc/252389
ER -

References

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