Minimax estimation of smooth optimal transport maps

Brenier's theorem is a cornerstone of optimal transport that guarantees the existence of an optimal transport map $T$ between two probability distributions $P$ and $Q$ over $\mathbb{R}^d$ under certain regularity conditions. The main goal of this work is to establish the minimax estimation rates for such a transport map from data sampled from $P$ and $Q$ under additional smoothness assumptions on $T$. To achieve this goal, we develop an estimator based on the minimization of an empirical version of the semi-dual optimal transport problem, restricted to truncated wavelet expansions. This estimator is shown to achieve near minimax optimality using new stability arguments for the semi-dual and a complementary minimax lower bound. Furthermore, we provide numerical experiments on synthetic data supporting our theoretical findings and highlighting the practical benefits of smoothness regularization. These are the first minimax estimation rates for transport maps in general dimension.Comment: 53 pages, 6 figure

Formulation and convergence of the finite volume method for conservation laws on spacetimes with boundary

We study nonlinear hyperbolic conservation laws posed on a differential (n+1)-manifold with boundary referred to as a spacetime, and defined from a prescribed flux field of n-forms depending on a parameter (the unknown variable), a class of equations proposed by LeFloch and Okutmustur in 2008. Our main result is a proof of the convergence of the finite volume method for weak solutions satisfying suitable entropy inequalities. A main difference with previous work is that we allow for slices with a boundary and, in addition, introduce a new formulation of the finite volume method involving the notion of total flux functions. Under a natural global hyperbolicity condition on the flux field and the spacetime and by assuming that the spacetime admits a foliation by compact slices with boundary, we establish an existence and uniqueness theory for the initial and boundary value problem, and we prove a contraction property in a geometrically natural L1-type distance.Comment: 32 page

Dry microfoams: Formation and flow in a confined channel

We present an experimental investigation of the agglomeration of microbubbles into a 2D microfoam and its flow in a rectangular microchannel. Using a flow-focusing method, we produce the foam in situ on a microfluidic chip for a large range of liquid fractions, down to a few percent in liquid. We can monitor the transition from separated bubbles to the desired microfoam, in which bubbles are closely packed and separated by thin films. We find that bubble formation frequency is limited by the liquid flow rate, whatever the gas pressure. The formation frequency creates a modulation of the foam flow, rapidly damped along the channel. The average foam flow rate depends non-linearly on the applied gas pressure, displaying a threshold pressure due to capillarity. Strong discontinuities in the flow rate appear when the number of bubbles in the channel width changes, reflecting the discrete nature of the foam topology. We also produce an ultra flat foam, reducing the channel height from 250 $\mu$m to 8 $\mu$m, resulting in a height to diameter ration of 0.02; we notice a marked change in bubble shape during the flow.Comment: 7 pages; 7 figures; 1 tex file+ 22 eps-file

A periodic microfluidic bubbling oscillator: insight into the stability of two-phase microflows

This letter describes a periodically oscillating microfoam flow. For constant input parameters, both the produced bubble volume and the flow rate vary over a factor two. We explicit the link between foam topology alternance and flow rate changes, and construct a retroaction model where bubbles still present downstream determine the volume of new bubbles, in agreement with experiment. This gives insight into the various parameters important to maintain monodispersity and at the same time shows a method to achieve controlled polydispersity.Comment: 4 page

A numerical approach to study the impact of packing density on fluid flow distribution in hollow fiber module.

The aim of this study was to analyze the influence of hollow fiber module design, specially packing density, and filtration operating mode on the filtration performance. In order to perform this analysis, a model based on the finite element method was used to simulate numerically the flow and filtration velocity along the fiber. An annular region of fluid surrounding the fiber was considered in order to account for the packing density Φ of the module. The originality of this approach lies in the study of fiber density effect on the hydrodynamic conditions, both for inside/out (IO) and outside/in (OI) filtration modes. The numerical simulations of fluid flow have shown a modification of the axial filtration velocity profile with packing density. When the density of fibers was high, filtration took place preferentially in the bottom of the fiber. In contrast, when the packing density was low, permeate flow was higher at the top of the fiber, i.e. the filtration module. Two experimental hollow fiber modules with two packing densities were tested and showed good agreement with the numerical data. These results underline the variations of filtration velocity along the fiber that will allow some predictions on fouling deposit to be done