977 research outputs found
Lecture Notes on Gradient Flows and Optimal Transport
We present a short overview on the strongest variational formulation for
gradient flows of geodesically -convex functionals in metric spaces,
with applications to diffusion equations in Wasserstein spaces of probability
measures. These notes are based on a series of lectures given by the second
author for the Summer School "Optimal transportation: Theory and applications"
in Grenoble during the week of June 22-26, 2009
Weighted Energy-Dissipation principle for gradient flows in metric spaces
This paper develops the so-called Weighted Energy-Dissipation (WED)
variational approach for the analysis of gradient flows in metric spaces. This
focuses on the minimization of the parameter-dependent global-in-time
functional of trajectories \mathcal{I}_\varepsilon[u] = \int_0^{\infty}
e^{-t/\varepsilon}\left( \frac12 |u'|^2(t) + \frac1{\varepsilon}\phi(u(t))
\right) \dd t, featuring the weighted sum of energetic and dissipative
terms. As the parameter is sent to~, the minimizers
of such functionals converge, up to subsequences, to curves of
maximal slope driven by the functional . This delivers a new and general
variational approximation procedure, hence a new existence proof, for metric
gradient flows. In addition, it provides a novel perspective towards
relaxation
A striking correspondence between the dynamics generated by the vector fields and by the scalar parabolic equations
The purpose of this paper is to enhance a correspondence between the dynamics
of the differential equations on and those
of the parabolic equations on a bounded
domain . We give details on the similarities of these dynamics in the
cases , and and in the corresponding cases ,
and dim() respectively. In addition to
the beauty of such a correspondence, this could serve as a guideline for future
research on the dynamics of parabolic equations
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