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In order to tackle this difficulty we introduced a new continuous framework based on a -convergence result obtained recently by F. Santambrogio which leads to a very efficient numerical procedure. Let us introduce first this kind of problem in a discrete setting. Consider a compact convex domain
and two measures which are sum of dirac masses :
where and
are positive numbers. We ask to
and
to have the same total mass, that is
. Following Xia, we define a transport path
from
to
as both a weighted directed graph $G$ which vertices contains the points
and
and a weight function
where
is the set of directed edges of
. Moreover we ask
to satisfy Kirchhoff's law that is for all vertex
of
we have:
where and
denote the starting and ending points of each directed edge
. To every transport path
we associate a cost of transportation defined by
for some fixed parameter .
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We recall below how this discrete problem can be translated in a continuous context and the main -convergence obtained by F. Santambrogio. Let
be a transport path associated to the two measures
and
. In order to introduce a relaxed formulation to the previous problem we associate to every
the vectorial measure
By definition, the measure is in the class of vectorial measures of the type
, where
and
are respectively a positive function and a unit vector field defined on
a set of finite
-Hausdorff measure. Moreover Kirchhoff's law in this continuous setting is equivalent to the divergence constraint
In a analogous way to (costalpha}) we define for every vectorial measure the cost functional:
Thus, our relaxed optimisation problem is to minimise under the previous (weak) divergence constraint. The
-convergence result we are going to present is based on functionals very similar to the ones of Modica and Mortola but of the form
defined on with
. The main difference with Modica and Mortola's functionnal is the fact that the double-well potential has been replaced by a concave power
which forces the modulus of the vector field
to tends to
or
. The idea is now to choose the parameters
,
and
to obtain a functional equivalent (asymptotically when
tends to
) to the cost (costcont}). Considering that the support of
is concentrated on a segment, an heuristic argument leads to the following choice of the parameters :
Let us call the functional associated to a set of parameters satisfying conditions (paramirrig}). F.
: Suppose
and
. Then
-converges to
with respect to the convergence of measures for some suitable constant
when
tends to
.
The previous -convergence result makes it possible to replace an hard discrete problem by a sequence of optimisation problems under linear constraints. In addition, we observed that for
the functional
is close from being convex. This observation was the starting point of our optimisation strategy. The main difference between this situation and the previous one is related to the divergence constraint. Due to the very simple structure of the constraints we had to deal with, it was straightforward to compute a projection on the linear constraints in the context.
We present below the first results obtained with our simple approach. The following figures are the results of four different experiments with two different values of the parameter . On the first rows of the figures, we represent two views of the graph of the given density
. The second rows represent two views of the graph of the norm of the optimal vector field for each value of
. As expected, ``Kirchhoff's law is approximatively satisfied'' by the support of the vector field which converges to a one dimensional set. Moreover we observe that two different values of
may lead to very different optimal structures.
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