273 research outputs found
On the Lagrangian branched transport model and the equivalence with its Eulerian formulation
First we present two classical models of Branched Transport: the Lagrangian
model introduced by Bernot, Caselles, Morel, Maddalena, Solimini, and the
Eulerian model introduced by Xia. An emphasis is put on the Lagrangian model,
for which we give a complete proof of existence of minimizers in a
--hopefully-- simplified manner. We also treat in detail some
-finiteness and rectifiability issues to yield rigorously the energy
formula connecting the irrigation cost I to the Gilbert Energy
E. Our main purpose is to use this energy formula and exploit a Smirnov
decomposition of vector flows, which was proved via the Dacorogna-Moser
approach by Santambrogio, to establish the equivalence between the Lagrangian
and Eulerian models
Full characterization of optimal transport plans for concave costs
This paper slightly improves a classical result by Gangbo and McCann (1996)
about the structure of optimal transport plans for costs that are concave
functions of the Euclidean distance. Since the main difficulty for proving the
existence of an optimal map comes from the possible singularity of the cost at
, everything is quite easy if the supports of the two measures are disjoint;
Gangbo and McCann proved the result under the assumption \mu(\spt(\nu))=0; in
this paper we replace this assumption with the fact that the two measures are
singular to each other. In this case it is possible to prove the existence of
an optimal transport map, provided the starting measure does not give
mass to small sets (i.e. rectifiable sets). When the measures are
not singular the optimal transport plan decomposes into two parts, one
concentrated on the diagonal and the other being a transport map between
mutually singular measures
Optimal transportation with an oscillation-type cost: the one-dimensional case
The main result of this paper is the existence of an optimal transport map
between two given measures and , for a cost which considers the
maximal oscillation of at scale , given by
. The minimization of this
criterion finds applications in the field of privacy-respectful data
transmission. The existence proof unfortunately only works in dimension one and
is based on some monotonicity considerations
The ELSA Database and What Can Be Done Regarding SERIES Networking Activities
This document critically presents the ELSA database for experimental results and compares its data structure with the one of the NEEScentral repository established in the US for storing the experimental results of 15 laboratories. This comparison leads to some proposed modifications of the ELSA data format that could be further used as a template for the SERIES/NA1 network activities. Some implementation directions are also given.JRC.G.5-European laboratory for structural assessmen
Large mass rigidity for a liquid drop model in 2D with kernels of finite moments
Motivated by Gamow's liquid drop model in the large mass regime, we consider
an isoperimetric problem in which the standard perimeter is replaced by
, with and a
nonlocal energy such that as vanishes.
We prove that unit area minimizers are disks for small enough.
More precisely, we first show that in dimension , minimizers are necessarily
convex, provided that is small enough. In turn, this implies that
minimizers have nearly circular boundaries, that is, their boundary is a small
Lipschitz perturbation of the circle. Then, using a Fuglede-type argument, we
prove that (in arbitrary dimension ) the unit ball in
is the unique unit-volume minimizer of the problem among centered nearly
spherical sets. As a consequence, up to translations, the unit disk is the
unique minimizer. This isoperimetric problem is equivalent to a generalization
of the liquid drop model for the atomic nucleus introduced by Gamow, where the
nonlocal repulsive potential is given by a radial, sufficiently integrable
kernel. In that formulation, our main result states that if the first moment of
the kernel is smaller than an explicit threshold, there exists a critical mass
such that for any , the disk is the unique minimizer of area
up to translations. This is in sharp contrast with the usual case of Riesz
kernels, where the problem does not admit minimizers above a critical mass.Comment: Final accepted versio
Convergence rate of entropy-regularized multi-marginal optimal transport costs
We investigate the convergence rate of multi-marginal optimal transport costs
that are regularized with the Boltzmann-Shannon entropy, as the noise parameter
tends to . We establish lower and upper bounds on the
difference with the unregularized cost of the form
for some explicit dimensional
constants depending on the marginals and on the ground cost, but not on the
optimal transport plans themselves. Upper bounds are obtained for Lipschitz
costs or locally semi-concave costs for a finer estimate, and lower bounds for
costs satisfying some signature condition on the mixed second
derivatives that may include degenerate costs, thus generalizing results
previously in the two marginals case and for non-degenerate costs. We obtain in
particular matching bounds in some typical situations where the optimal plan is
deterministic
A multi-material transport problem and its convex relaxation via rectifiable -currents
In this paper we study a variant of the branched transportation problem, that
we call multi-material transport problem. This is a transportation problem,
where distinct commodities are transported simultaneously along a network. The
cost of the transportation depends on the network used to move the masses, as
it is common in models studied in branched transportation. The main novelty is
that in our model the cost per unit length of the network does not depend only
on the total flow, but on the actual quantity of each commodity. This allows to
take into account different interactions between the transported goods. We
propose an Eulerian formulation of the discrete problem, describing the flow of
each commodity through every point of the network. We provide minimal
assumptions on the cost, under which existence of solutions can be proved.
Moreover, we prove that, under mild additional assumptions, the problem can be
rephrased as a mass minimization problem in a class of rectifiable currents
with coefficients in a group, allowing to introduce a notion of calibration.
The latter result is new even in the well studied framework of the
"single-material" branched transportation.Comment: Accepted: SIAM J. Math. Ana
Frequency and damping evolution during experimental seismic response of civil engineering structures
The results of the seismic tests on several reinforced-concrete shear walls and a four-storey frame are analysed in this paper. Each specimen was submitted to the action of a horizontal accelerogram, with successive growing amplitudes, using the pseudodynamic method. An analysis of the results allows knowing the evolution of the eigen frequency and damping ratio during the earthquakes thanks to an identification method working in the time domain. The method is formulated as a spatial model in which the stiffness and damping matrices are directly identified from the experimental displacements, velocities and restoring forces. The obtained matrices are then combined with the theoretical mass in order to obtain the eigen frequencies, damping ratios and modes. Those parameters have a great relevance for the design of this type of structures
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