5,225 research outputs found
Numerical Computation of Weil-Peterson Geodesics in the Universal Teichm\"uller Space
We propose an optimization algorithm for computing geodesics on the universal
Teichm\"uller space T(1) in the Weil-Petersson () metric. Another
realization for T(1) is the space of planar shapes, modulo translation and
scale, and thus our algorithm addresses a fundamental problem in computer
vision: compute the distance between two given shapes. The identification of
smooth shapes with elements on T(1) allows us to represent a shape as a
diffeomorphism on . Then given two diffeomorphisms on (i.e., two
shapes we want connect with a flow), we formulate a discretized energy
and the resulting problem is a boundary-value minimization problem. We
numerically solve this problem, providing several examples of geodesic flow on
the space of shapes, and verifying mathematical properties of T(1). Our
algorithm is more general than the application here in the sense that it can be
used to compute geodesics on any other Riemannian manifold.Comment: 21 pages, 11 figure
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
Optimal transport with branching distance costs and the obstacle problem
We address the Monge problem in metric spaces with a geodesic distance: (X,
d) is a Polish space and dN is a geodesic Borel distance which makes (X,dN) a
possibly branching geodesic space. We show that under some assumptions on the
transference plan we can reduce the transport problem to transport problems
along family of geodesics. We introduce two assumptions on the transference
plan {\pi} which imply that the conditional probabilities of the first marginal
on each family of geodesics are continuous and that each family of geodesics is
a hourglass-like set. We show that this regularity is sufficient for the
construction of a transport map. We apply these results to the Monge problem in
d with smooth, convex and compact obstacle obtaining the existence of an
optimal map provided the first marginal is absolutely continuous w.r.t. the
d-dimensional Lebesgue measure.Comment: 27 pages, 1 figure; SIAM J. Math. Anal. 2012. arXiv admin note:
substantial text overlap with arXiv:1103.2796, arXiv:1103.279
Shape analysis on homogeneous spaces: a generalised SRVT framework
Shape analysis is ubiquitous in problems of pattern and object recognition
and has developed considerably in the last decade. The use of shapes is natural
in applications where one wants to compare curves independently of their
parametrisation. One computationally efficient approach to shape analysis is
based on the Square Root Velocity Transform (SRVT). In this paper we propose a
generalised SRVT framework for shapes on homogeneous manifolds. The method
opens up for a variety of possibilities based on different choices of Lie group
action and giving rise to different Riemannian metrics.Comment: 28 pages; 4 figures, 30 subfigures; notes for proceedings of the Abel
Symposium 2016: "Computation and Combinatorics in Dynamics, Stochastics and
Control". v3: amended the text to improve readability and clarify some
points; updated and added some references; added pseudocode for the dynamic
programming algorithm used. The main results remain unchange
Vortex sheets and diffeomorphism groupoids
In 1966 V.Arnold suggested a group-theoretic approach to ideal hydrodynamics
in which the motion of an inviscid incompressible fluid is described as the
geodesic flow of the right-invariant -metric on the group of
volume-preserving diffeomorphisms of the flow domain. Here we propose geodesic,
group-theoretic, and Hamiltonian frameworks to include fluid flows with vortex
sheets. It turns out that the corresponding dynamics is related to a certain
groupoid of pairs of volume-preserving diffeomorphisms with common interface.
We also develop a general framework for Euler-Arnold equations for geodesics on
groupoids equipped with one-sided invariant metrics.Comment: Final version accepted to Advances in Mathematics; 46 pages, 6
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