Displaying similar documents to “Gradient flow for the one-dimensional Mumford-Shah functional”

Non-local approximation of functionals: variational and evolution problems

Massimo Gobbino (2000)

Bollettino dell'Unione Matematica Italiana

Similarity:

Questa nota tratta dell'approssimazione di funzionali, usati in problemi con discontinuità libere, mediante famiglie di funzionali non locali in cui il gradiente è sostituito dal rapporto incrementale. Vengono inoltre presentate alcune applicazioni di questa teoria a problemi variazionali e di evoluzione.

The heat equation on manifolds as a gradient flow in the Wasserstein space

Matthias Erbar (2010)

Annales de l'I.H.P. Probabilités et statistiques

Similarity:

We study the gradient flow for the relative entropy functional on probability measures over a riemannian manifold. To this aim we present a notion of a riemannian structure on the Wasserstein space. If the Ricci curvature is bounded below we establish existence and contractivity of the gradient flow using a discrete approximation scheme. Furthermore we show that its trajectories coincide with solutions to the heat equation.

Wasserstein gradient flows from large deviations of many-particle limits

Manh Hong Duong, Vaios Laschos, Michiel Renger (2013)

ESAIM: Control, Optimisation and Calculus of Variations

Similarity:

We study the Fokker–Planck equation as the many-particle limit of a stochastic particle system on one hand and as a Wasserstein gradient flow on the other. We write the path-space rate functional, which characterises the large deviations from the expected trajectories, in such a way that the free energy appears explicitly. Next we use this formulation via the contraction principle to prove that the discrete time rate functional is asymptotically equivalent in the Gamma-convergence sense...

Gradient flows in Wasserstein spaces and applications to crowd movement

Filippo Santambrogio (2010-2011)

Séminaire Équations aux dérivées partielles

Similarity:

Starting from a motivation in the modeling of crowd movement, the paper presents the topics of gradient flows, first in ℝ n , then in metric spaces, and finally in the space of probability measures endowed with the Wasserstein distance (induced by the quadratic transport cost). Differently from the usual theory by Jordan-Kinderlehrer-Otto and Ambrosio-Gigli-Savaré, we propose an approach where the optimality conditions for the minimizers of the optimization problems that one solves at every...