14 research outputs found
Fast transport optimization for Monge costs on the circle
Consider the problem of optimally matching two measures on the circle, or
equivalently two periodic measures on the real line, and suppose the cost of
matching two points satisfies the Monge condition. We introduce a notion of
locally optimal transport plan, motivated by the weak KAM (Aubry-Mather)
theory, and show that all locally optimal transport plans are conjugate to
shifts and that the cost of a locally optimal transport plan is a convex
function of a shift parameter. This theory is applied to a transportation
problem arising in image processing: for two sets of point masses on the
circle, both of which have the same total mass, find an optimal transport plan
with respect to a given cost function satisfying the Monge condition. In the
circular case the sorting strategy fails to provide a unique candidate solution
and a naive approach requires a quadratic number of operations. For the case of
real-valued point masses we present an O(N |log epsilon|) algorithm that
approximates the optimal cost within epsilon; when all masses are integer
multiples of 1/M, the algorithm gives an exact solution in O(N log M)
operations.Comment: Added affiliation for the third author in arXiv metadata; no change
in the source. AMS-LaTeX, 20 pages, 5 figures (pgf/TiKZ and embedded
PostScript). Article accepted to SIAM J. Applied Mat
Local matching indicators for transport with concave costs
In this note, we introduce a class of indicators that enable to compute
efficiently optimal transport plans associated to arbitrary distributions of
demands and supplies in in the case where the cost
function is concave. The computational cost of these indicators is small and
independent of . A hierarchical use of them enables to obtain an efficient
algorithm
Local matching indicators for concave transport costs
International audienceIn this note, we introduce a class of indicators that enable to compute efficiently optimal transport plans associated to arbitrary distributions of demands and supplies in in the case where the cost function is concave. The computational cost of these indicators is small and independent of . A hierarchical use of them enables to obtain an efficient algorithm
Ergodic Transport Theory and Piecewise Analytic Subactions for Analytic Dynamics
We consider a piecewise analytic real expanding map of
degree which preserves orientation, and a real analytic positive potential
. We assume the map and the potential have a complex
analytic extension to a neighborhood of the interval in the complex plane. We
also assume is well defined for this extension.
It is known in Complex Dynamics that under the above hypothesis, for the
given potential , where is a real constant, there
exists a real analytic eigenfunction defined on (with a
complex analytic extension) for the Ruelle operator of .
Under some assumptions we show that
converges and is a piecewise analytic calibrated subaction. Our theory can be
applied when . In that case we relate the involution
kernel to the so called scaling function.Comment: 6 figure
Local matching indicators for transport problems with concave costs
In this paper, we introduce a class of indicators that enable to compute
efficiently optimal transport plans associated to arbitrary distributions of N
demands and M supplies in R in the case where the cost function is concave. The
computational cost of these indicators is small and independent of N. A
hierarchical use of them enables to obtain an efficient algorithm