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SUMMARY:Covariance modulated optimal transport: geometry and gradient flow
s
DTSTART:20240514T091500Z
DTEND:20240514T101500Z
DTSTAMP:20240720T212300Z
UID:indico-event-10138@indico.math.cnrs.fr
DESCRIPTION:Speakers: Daniel Matthes\n\nThis talk is about a novel variant
of optimal mass transport: particles move with a mobility that is modulat
ed by the covariance of the ensemble density. This gives rise to a modulat
ed Wasserstein metric which provides a rigorous gradient flow formulation
of the mean-field limit in for the ensemble Kalman sampling. In combinatio
n with the abstract machinery of metric gradient flows\, the new metric is
an effective tool to study the rate of convergence of these methods. I sh
all present several analytic results about the modulated Wasserstein metri
c. The first is the splitting representation\, which allows to write the m
odulated metric as the sum of two simpler metrics\, one measuring the dist
ance in terms of first and second moments\, the other one measuring in ter
ms of shapes. The second result is about geodesic convexity and the relate
d rates of convergence in gradient flows. Specifically\, we prove exponent
ial equilibration in linear Fokker-Planck equations with Gaussian steady s
tates at a rate that does not depend on the covariance of the Gaussian. Th
e third and only partial result is about geodesics\, that we prove to exis
t for sufficiently close densities\, or densities with multiple reflection
symmetries. We also characterize geodesics in terms of particle trajector
ies\, that are no longer straight line as in the genuine Wasserstein metri
c\, but follow more complicated curves that satisfy second order ordinary
differential equations.This is joint work with Andre Schlichting\, Matthia
s Erbar\, Franca Hoffmann and Martin Burger.\n\nhttps://indico.math.cnrs.f
r/event/10138/
LOCATION:Amphi Schwartz (IMT)
URL:https://indico.math.cnrs.fr/event/10138/
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