Optimal transportation describes the problem of finding the most efficient way to move distribution of materials
A common problem in these applications is to have two sets of data points
An illustration of such a matching, where both
As shown in (2) fluctuations are effective in low dimension, exhibiting a logarithmic correction when
Note that the optimal matching problem as described in (1) can be seen as a special case of optimal transportation problems where the original and final distributions are point measures, that is they are discrete measures. Indeed, defining
This allows to study (1) from an analytical point of view exploiting the rich structure of optimal transport combining tools from convex analysis and partial differential equations. In [7] a very precise ansatz regarding convergence of the re-scaled cost in dimension
So far, we have focused on the matching problem on large scales. Thus we are interested in matching infinite point clouds in the whole space
The proof of the non-existence of stationary, locally optimal matchings in 2d is based on a tool, which is called harmonic approximation. It says that optimal transportation maps are well-approximated by harmonic maps.
The harmonic approximation result was originally developed in [13], in order to study the regularity properties of solutions to the quadratic optimal transportation problem. These problems had been studied before mainly using techniques from nonlinear PDE theory developed by Caffarelli. The outcome of these studies was a partial
This approach relies primarily on the connection of the optimal transportation problem with a lineari\ed PDE problem. Note that we can view an optimal transportation map
It turns out that not only does the change of variables formula hold, but
The idea of the harmonic approximation is to perform a Taylor-expansion for around the identity. Since
If this argument is not just formal, but can be made rigorous, we should expect that
Regularity results for continuous densities were obtained in [12], while the boundary case was treated in [22]
The aforementioned regularity results can be extended to cost functions that locally look like the Euclidean cost function
A natural generalization of the optimal transport problem is the question of optimally coupling more than two measures, leading to the so-called multi-marginal optimal transport problem. Multi-marginal optimal transport, originally considered by Gangbo and Święch [11], has recently received attention due to its connection to barycenters of measures with respect to the Wasserstein geometry [1], which provides a natural way of interpolating many probability measures. However, the regularity properties of this interpolating density is far from being well-understood. If the cost function is given by pairwise Coulomb interaction, the multi-marginal optimal transport problem also arises as a semi-classical limit (in the strong coupling regime) of electronic density functional theory [8]. Due to the repulsive nature of the cost, the behavior of optimal transport in this case is rather different from the usually considered cost functions and many questions remain open.
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