Webpage of R.L.

I am a postdoctoral fellow at École des ponts et chaussées (CERMICS lab), also affiliated with the MATHERIALS team at INRIA, under the supervision of Virginie Ehrlacher, Tony Lelièvre, and Julien Reygner. My research is supported by the HighLEAP ERC grant, led by Virginie Ehrlacher.

I completed my PhD at Université Paris-Dauphine under the supervision of Mathieu Lewin, where I was a member of the MDFT ERC grant team. Prior to that, I studied at École Polytechnique and École Normale Supérieure de Paris-Saclay.

You can contact me at firstname.name@enpc.fr.

Research interests

1. Theory and numerics of optimal transport, notably its multimarginal extension as it appears in density functional theory in quantum chemistry. Here is a partial overview of some contributions of mine on this subject:
   • In [1], I proved that the Kantorovich potential for the multimarginal optimal transport with Coulomb cost is a superharmonic function, which can therefore be expressed as the electrostatic potential generated by a charge density. I then utilise this parametrisation to numerically solve the multimarginal optimal transport problem by discretising this charge density and solving the Kantorovich dual using classical optimisation and Monte Carlo-type methods.
   • In [2], I proved a conjecture regarding the asymptotics of the Kantorovich potential for repulsive transportation costs as commonly encountered in physics. This conjecture can be interpreted as a ionization conjecture for the strictly-correlated electrons system.
   • At the present time, I am working on several questions on this topic, among which the regularity of the multimarginal optimal transport with general costs (and implicitely on the existence of Monge transport maps in the multimarginal setting), as well as on the moment-constrained optimal transport and other numerical methods to solve the optimal transport problem with a possibly important number of marginals.
2. Study of classical (and continuous) statistical physics systems such as Coulomb and Riesz gases, with a particular focus on the existence of the thermodynamic limit and phase transitions. Here is a contribution of mine on this subject:
   • In [3], I showed numerical evidences that the one-dimensional Riesz gases exhibit phase transitions with respect the temperature in the long-range interaction regime, more precisely for the non-singular Riesz pair potential with exponent −1 ≤ s ≤ 0.
3. Model order reduction techniques for high-dimensional stochastic differential equations and Wasserstein gradient flows on manifolds.
   • In the forthcoming paper [4] in collaboration with Virginie Ehrlacher, Tony Lelièvre and Julien Reygner, we explain how to build projection of stochastic differential equations onto submanifolds of ℝ^d and develop numerical schemes for these objects that remain exactly onto the submanifold.

Publications

Teaching material

I was a teaching assistant at Université Paris-Dauphine. Some personal material for the fourth-year course Analyse numérique: évolution of Gabriel Turinici are available, namely a practical work (.pdf, .ipynb) on the numerical integration of (stiff) ordinary differential equations. Although the material is fairly standard, I did manage to find an example of a differential equation y′ = f(t, y) for a non-Lipschitz dynamics f which nevertheless admits an unique solution, and for which standard numerical schemes fail to converge. Another practical work (.pdf, .ipynb) that deals with numerical integration of stochastic differential equations — with a somewhat long additional exercise that includes a remarkable result from random matrix theory and statistical physics.

For the course Analyse des équations aux dérivées partielles for first year students of Ecole des Ponts, here are some (sketchy) corrections.

Some slides

• I recently gave a presentation on The multimarginal optimal transport with Coulomb cost at the SAMM lab at Centre Pierre Mendès France (Sorbonne)
• I was invited at the occasion of the conference On system of interacting particles at the IHÉS (Bures-sur-Yvette), where I gave a talk on The Riesz gases and their phase diagram in dimension 1
• For the 16th conference of the GDR Dynqua, I was invited to gave a talk on Strictly-Correlated Electrons systems and their dissociation at infinity

Some vignettes

Accelerating the GenCol algorithm

I am currently working on ways to improve the GenCol algorithm, notably by the introduction of stochastic rules in order to update the sparse domain Ω of the reduced OT problem. In the above .gif, an accelerated version of the GenCol algorithm is depicted. There is N = 5 marginals with L = 15 points of discretisation for each marginal, for the Coulomb cost with (one-dimensional) uniform marginal. On the left, the value of the reduced OT? In the center and on the right, the (symmetrised) two-point density (resp. for the original and accelerated GenCol).

Phase transition in the 1D Riesz gases for s ∈ (−1, 0)

Here is a .gif that depicts the simulation of the 1D Riesz gases when the Riesz exponent s is within the range (−1, 0) – thus between the 1D Coulomb gas and the Dyson log-gas. On the left, the pair correlation g(r) is depicted at decreasing temperatures, and we can observe that at sufficiently low temperature it becomes a periodic function (as in a crystal) whereas at high temperature it is monotonic (as in a fluid). On the right, I plot the structure factor S(k) (essentially the Fourier transform of g(r). It is coherent with the behaviour seen previously on the pair correlation, as we can witness the appearance of a so-called Bragg peak(s) at low temperature, coherent with the formation of a crystalline phase.

Projection(s) of SDE onto submanifolds

On the left, we start from the SDE dX_t = b(X_t) + dW_t where W_t is a two-dimensional standard Brownian motion and b(x₁, x₂) = (x₂, −x₁), and we build a projection of this SDE using a principle à la Dirac–Frenkel, so that the projected SDE be such that its solution verifies 𝔼[X_t] ∈ 𝕊 where 𝕊 is the unit circle. On the right, we depict three trajectories for three different ways of projecting an SDE as introduced in [4]. In the present case, we project the SDE onto the sphere, meaning that we demand that almost-surely X_t ∈ 𝕊².

Some prehistoric works

Here are some prehistoric works that may be of (relatively minor) interests for students.

• My mémoire de M2 on the two-dimensional Uniform Electron Gas through the lens of (multimarginal) optimal transport. Of potential interest may be the general blueprint to prove Kantorovich duality, as well as a minor extension to the logarithmic cost of an important result in the study of the multimarginal optimal transport problem with Coulomb cost, namely that the electrons are at optimality far enough from each others.
• My mémoire de M1 where I study the spectral properties of the N-body electronic Hamiltonian, and where I extend a result of Lenzman & Lewin to the molecular setting that states that when the number of electrons is bigger that N ≥ 4Z + 1 where Z is the number of protons, the electronic Hamiltonian has no eigenvalues (to wit, the atom is not stable). Also, of potential interest may be a comprehensive introduction to the famous Mourre estimates.