April 25, 2024

Jacob Francis: Entropic Free Marginal Optimal Transport Solutions for the Shallow-Water Semi-Geostrophic Equations

We shall solve the shallow-water semi-geostrophic (SWSG) equations through an optimal transport (OT) type problem with entropic regularisation. In its original form OT seeks to minimise the transport of mass between two distributions in space, provided a cost of transportation. Canonical examples come from logistics, such as finding the optimal route to distribute bread from bakeries to cafes. Since its initial formulation, OT theory has found many varying applications from signal processing and economics to meteorology and machine learning, notably through the famed Wasserstein distance. Its success is first due to Kantorovich’s dual formulation and recent advancements in novel algorithms and GPU computing. These advances combined allow regularised OT problems to be solved efficiently.

April 18, 2024

Charly Andral: Piecewise deterministic Markov processes for Monte Carlo integration

This talk will give an introduction to the general framework of piecewise deterministic Markov processes (PDMP) and how to use them for Monte Carlo integration. After the introduction of the main PDMPs samplers, the Zigzag sampler and the bouncy particle sampler, we will focus on a more computational aspect of the process, i.e. how to sample them exactly. We will discuss the different approaches of the last few years on this topic.

April 11, 2024

Antonio Ocello: A Stochastic Target Problem for Branching Diffusion Processes

We consider an optimal stochastic target problem for branching diffusion processes. This problem consists in finding the minimal condition for which a control allows the underlying branching process to reach a target set at a finite terminal time for each of its branches. This problem is motivated by an example from fintech where we look for the super-replication price of options on blockchain-based cryptocurrencies. We first state a dynamic programming principle for the value function of the stochastic target problem. Next, we show that the value function can be simplified into a novel function with the use of a finite-dimensional argument through a concept known as the branching property. Under wide conditions, this last function is shown to be the unique viscosity solution to an HJB variational inequality. This is a joint work with Idris Kharroubi.

April 4, 2024

Suney Toste Regalado: Modeling the development of adipose tissue as a function of the energy flow it receives

Throughout our lives, our bodies grow and increase in size according to the hormonal balance we have at any given time, and the amount of nutrients/energy we get from our diet. Even during embryonic development, when the flow of nutrients is limited by the nutrients provided by the mother, a large number of cells are created during this phase of development. This developmental phase is followed by a growth phase, in which, although new cells appear, the rate of appearance is slower than during the developmental phase. This growth phase is characterized by an increase in cell size. Moreover, it is well known that in the absence of energy, fat cells reduce their size, releasing lipids to give the surrounding tissue the energy it needs to maintain itself. And, when the tissue receives a very high flow of energy, the cells increase in size and even inseminate new cells to store as much lipid as possible. I wanted to find a model capable of describing these behaviors as a function of the energy received by the system. More specifically, I'm interested in finding parameter sets in the model that can reproduce different waves of insemination in cell number, allowing to explain different phases of body growth and being in agreement with bimodal distributions of adipocyte size in adult tissue.

March 28, 2024

Thaddeus Roussigné: Peierls-type instability in graphene

In this talk, we present the phenomenon of Peierls symmetry breaking in one-dimensional atomic chains, and its analogue for two-dimensional honeycomb lattices, known as the Kekulé distortion. We recall the well-known tight-binding model for non-interacting electrons hopping on a periodic lattice, and apply it to the case of a chain of carbon atoms (polyacetylene), with a quadratic elasticity term to account for displacement of individual atoms in the chain. It can then be shown that the translation-invariant configuration of hopping amplitudes is unstable : at minimal energy, the chain is dimerized, i.e. its bonds alternate in length – this occurs both in the infinite-volume limit and in finite cases for specific chain lengths. We investigate whether such breaking of translation invariance can occur when considering the material graphene, by applying the same model to a hexagonal lattice : in this case we show that, for a sufficiently small rigidity parameter, a Kekulé distortion of the lattice opens a gap in the band structure, thus lowering the system’s energy.

March 21, 2024

Henri Surugue: Do we care about poll manipulation in political elections ?

In this talk, we consider the problem of poll manipulation in political elections. More precisely, we use a game-theoretical approach to model the strategic behaviour of voters in elections, who manipulate according to the poll information available to them, and we study whether a polling institute can manipulate the information it communicates to the voters in order to influence the outcome of the election. We investigate this problem using a probabilistic tool to understand the frequency of such poll manipulations in large elections. We will start with the description of our model and the specific poll manipulation problem under consideration. We will then present the main results of this work for the unrestricted version of the problem, with no restriction on the communicated poll information, and a uniform statistical culture. Finally, we will extend this work by considering restrictions on the communicated poll information that cannot be too far from reality, more general behaviour for voters’ manipulation and other statistical distributions (consistent with empirical studies).

March 14, 2024

Luca Davron: Control of the induced earthquakes and Sobolev towers

In this presentation I will present two works in control theory that are still in progress and closely linked to each other. The first one is the study of the controllability of a (simplified) fluid-structure model which represents the dynamics of earthquakes that are triggered by human activities (induced earthquakes). This will be the occasion to present the so-called LTI formalism which will allow us to decide the controllability of the model. Doing so we will see that the system is not controllable and even worse: in a sens it is "very not controllable". More precisely, the associated observability inequality has an infinite order of defect. This triggers the following intuition: the lack of controllability does not arise because the control laws are chosen in class that is too small, and we shall obtain similar non-controllability results even after enlarging the class of admissible control laws. The second part of the presentation will be devoted to a construction allowing one to make more rigorous this intuition. This will be done by considering two Sobolev towers, one for the state space and one for the control laws, that allow for various regularities and are suited for the generalization of well-posedness and control by duality. If the time permits I will discuss the subtleties associated to irregular control laws, such as the time discontinuity of the state trajectories, which a priori makes impossible the formulation of a control problem. The first work is in collaboration with Pierre Lissy (Cermics) and Swann Marx (LS2N).

March 7, 2024

Daniele de Gennaro: Explicit diffusion/redistancing scheme for the mean curvature flow

After an introductory section, aimed at non-specialists, I will introduce  a fully discrete and explicit scheme to simulate the mean curvature flow of boundaries. The scheme is based on an elementary diffusion step and a more costly redistancing operation. I'll then present some ideas to show the convergence of the scheme. After an introductory section, aimed at non-specialists, I will introduce  a fully discrete and explicit scheme to simulate the mean curvature flow of boundaries. The scheme is based on an elementary diffusion step and a more costly redistancing operation. I'll then present some ideas to show the convergence of the scheme.

February 29, 2024

Joao Machado: Convergence of Nash equilibria to Cournot-Nash equilibria: tools from optimal transport and Gamma convergence

In this talk we address a central question in the Mean Field Games theory: does a sequence of Nash equilibria converge to an equilibrium to a game with infinitely many players? To study this question first we introduce the notion of Cournot-Nash equilibria, introduced in the 60s to study games with infinitely many players and recently revisted with the machinery of Optimal Transport. We review some results from Blanchet and Carlier about the potential structure of these games, meaning that equilibria can be obtained through the minimization of a potential functional, and make a connection with the recent literature of Lagrangian Mean Field Games. In the sequel, we see that an iid sample of an initial distribution of players to this continuous game induces a sequence of N-players games that inherits this potential structure.  We then show that the potential functionals for the N-players games converge in the sense of Gamma-convergence to the potential functional of the continuous game, with probability 1. From this we conclude that Nash equilibria from the N-players game converge to Cournot-Nash equilibria with full probability, giving a positive answer to this question for a wise class of models.

February 22, 2024

Raphaël Poirier: Stability in shape optimization with regularity theory

Shape optimization consists in the study of the minimization inf{J(Ω), Ω ∈ Sad} where Sad is a class of subsets of Rd and J : Sad → R ∪ {∞} is a functional.

There are many problems where one expects the Euclidean ball B ⊂ Rd to be the (unique, up to translation) minimizer of J under volume constraint, as in the celebrated isoperimetric problem which asserts that the ball is in fact the unique minimizer of the perimeter among sets of fixed volume. Under this assumption of minimality of the ball, one can wonder further about its stability: if J(Ω) ≈ J(B), can we say that Ω is close to being a ball in some appropriate sense? After setting up the context and giving some examples of stability of the ball, we present a regularity-based strategy that has been widely used in the literature over the past decade for obtaining such results.

February 15, 2024

Adrien Cances: Around optimal transport

I will begin with a general presentation of optimal transport (OT), the main aim of which is to transport a probability measure to another probability while minimizing the total transport cost. Among other things, I will present Monge's original formulation and Kantorovitch's (more complete) one, the notion of duality, the Wasserstein distance, and a particularly enlightening variational formulation of the latter. In the last part of the presentation, I will explain the overall goal of my thesis and the key ideas that motivated it.

February 8, 2024

Gabriela Bayolo: Test allocation based on risk of infection from first and second order contact tracing

Under limited available resources, strategies for mitigating the propagation of an epidemic such as random testing and contact tracing become inefficient. Here, we propose to accurately allocate the resources by computing over time an individual risk of infection based on the partial observation of the epidemic spreading on a contact network; this risk is defined as the probability of getting infected from any possible transmission chain up to length two, originating from recently detected individuals. To evaluate the performance of our method and the effects of some key parameters, we carry out comparative simulated experiments using data generated by an agent-based model.

February 1, 2024

Alexandre Verine: Optimal Budgeted Rejection Sampling for Generative Models

Rejection sampling methods have recently been proposed to improve the performance of discriminator-based generative models. However, these methods are only optimal under an unlimited sampling budget, and are usually applied to a generator trained independently of the rejection procedure. We first propose an Optimal Budgeted Rejection Sampling (OBRS) scheme that is provably optimal with respect to any f-divergence between the true distribution and the postrejection distribution, for a given sampling budget. Second, we propose an end-toend method that incorporates the sampling scheme into the training procedure to further enhance the model’s overall performance. Through experiments and supporting theory, we show that the proposed methods are effective in significantly improving the quality and diversity of the samples.

January 25, 2024

Guillaume Garnier: Nonparametric estimator of the distribution of fitness effects of new mutations

Mutations are an essential mechanism that plays a key role in the history of life; in particular, they explain the appearance of new hereditary traits within a population. These new traits can modify the selective or fitness value of an individual. In evolutionary biology, biologists are interested in distribution of fitness effects (DFE) of new mutations since it is a key element to understanding the evolutionary trajectory of a population. In this talk, we present a probabilistic model based on compound Poisson processes that describes the evolution of the fitness of a cell line over the time . We propose an nonparametric estimator of the DFE based on the noisy observation of a i.i.d. sample of cell line over discrete time. In particular, it is not possible to observe fitness jumps related to the occurrence of a new mutation. We use a Fourier approach to construct this estimator, to provide risk bounds and an adaptive procedure.

January 18, 2024

Cristóbal Loyola: On unique continuation for nonlinear waves in finite time

In this talk, we will address the topic of global unique continuation for the semilinear wave equation where the nonlinearity is assumed to be subcritical, defocusing, and analytic. First, we will motivate the topic by showing how a unique continuation property (UCP for short) in infinite time is used to obtain the energy decay of the damped semilinear wave equation when the damped zone satisfies the geometric control condition only. This UCP is obtained by using an asymptotic smoothing effect due to Hale and Raugel in the context of dynamical systems.Then, we will discuss how to adapt this smoothing effect technique to obtain a UCP for the semilinear wave equation but posed in a finite-time interval. A key point in this adaptation is to keep the assumption that our observation is made on a zone satisfying the geometric control condition. This will allow us to employ tools coming from the context of control theory. This result is part of an ongoing joint work with C. Laurent.

January 11, 2024

Robin Roussel: Shape optimization of harmonic helicity in toroidal domains

In this talk, we introduce the notion of harmonic helicity of a toroidal domain and discuss its shape optimization both from the theoretical and numerical point of view. Given a toroidal domain, we consider its associated harmonic field. The latter is the magnetic field obtained uniquely up to normalization when imposing zero normal trace and zero electrical current inside the domain. We then study the helicity of this field, which is a quantity of interest in magneto-hydrodynamics corresponding to the $L^2$ product of the field with its image by the Biot-Savart operator. To do so, we begin by discussing the appropriate functional framework and an equivalent PDE characterization. We then discuss shape optimization, and we identify the shape gradient of the harmonic helicity. Finally, we give a finite elements numerical scheme to approximate the harmonic helicity of a domain, as well as its shape gradient, before giving some numerical results.

Slides available here!

December 14, 2023

François Quentin: Around the additif problem

Consider two Hermitian matrices A and B. What can be said about the eigenvalues of their sum C = A+B with respect to the eigenvalues of A and those of B ? This is the Horn problem which emerged in the 60’s. Using the historical approach, we will present the various areas involved in the resolution of this problem, from linear inequalities on the eigenvalues to combinatorial criteria using Schubert Calculus.

December 7, 2023

Herbert Sussman: Targeted Bayesian Estimation of Marginal Structural Models

Two principle tasks of causal inference are defining and estimating the effect of a treatment on an outcome of interest. Such treatment effects are defined as a possibly functional summary of the data generating distribution, and are referred to as target parameters. Estimation of the target parameter can be difficult, especially when it is high-dimensional. Marginal Structural Models (MSMs) provide a way to summarize such target parameters in terms of a lower dimensional working model. We introduce the semi-parametric efficiency bound for estimating MSM parameters in a general setting. We then present a frequentist estimator that achieves this bound based on Targeted Minimum Loss-Based Estimation. Our results are derived in a general context, and can be easily adapted to specific data structures and target parameters. We describe a novel targeted Bayesian estimator and provide a Bernstein von-Mises type result analyzing its asymptotic behavior. 

November 30, 2023 

Othmane Zarhali : From rough to multifractal volatility: introduction to Log S-fbm model

This talk is going to be a broad introduction to the Log S-fbm model. First, we are going to recall some background on set indexed process by giving an alternative construction of the fractional Brownian motion through this perspective. We will follow up by evoking the insights behind rough and multifractal volatility in financial modelling. Then, a special focus will be given to the Log S-fbm model as a new formalism that reconciliates the rough and multifractal volatility worlds.

Slides available here!

November 23 2023

Grégoire Szymanski : The two square root laws of Market Impact and the role of sophisticated market participants

In this study, we revisit the Hawkes order flow model for market impact initially introduced by Jaisson in 2015, which assumes linear market impact for individual orders and a price process modeled as a martingale. Our approach extends this model by introducing sophisticated market participants capable of reshaping the volatility profile. The strength of our methodology lies in two primary aspects: firstly, it relies on minimal or no additional assumptions, and secondly, it yields closed-form expressions for market impact. Notably, our analysis leads to the recovery of two well-known square root laws. Specifically, for a fixed duration, market impact scales proportionally to $\sqrt{\gamma}$, where $\gamma$ denotes the participation rate. Additionally, for a given order size, market impact adheres to a power-law behavior with respect to the total duration.

November 16, 2023

Florin Suciu: Training of (two-layer) Neural Networks as a Mean Field Optimisation Problem

In this talk, we start by recalling how the non-convex optimisation problem over the neuron parameters of a two-layer neural network becomes convex when lifted to an infinite-dimensional space of measures. 

We proceed by analysing the existence and uniqueness of minimisers for the corresponding regularised energy functional. We recover the correspondence between the minimiser of the energy functional and the invariant measure of a suitable McKean-Vlasov SDE called mean field Langevin equation. 

Finally, we overview and discuss some recent (N-interacting-particles system) and new (Self-interacting process) methods to approximate the invariant measure of the McKean-Vlasov SDE introduced above.

November 9, 2023

Raphaël Maillet: Invariant density estimation for ergodic Markov processes.

In this talk, we consider the problem of estimating the invariant measure of a diffusion process. We begin with a concise review of density estimation techniques, emphasizing the kernel density estimation method for observing n iid variables derived from a common density. We study the optimal estimation rate for functions within an anisotropic Hölder class and discuss the adaptive selection of the estimation window. Then, we delve into the invariant measure estimation in the context of a continuous diffusion process that admits a unique invariant measure. This problem has been extensively studied, and we will present the seminal results of Dalalyan & Reiss with continous observations, which showcase non-standard estimation rates. Lastly, we explore the challenges posed by noisy data in the estimation process. This final section is a work in progress, conducted in collaboration with Grégoire Szymanski.

Slides available here!

November 2, 2023

Richard Medina: An introduction to the kinetic Fokker-Planck equation and its long-time behaviour. 

In this talk, we consider the problem of estimating the invariant measure of a diffusion process. We begin with a concise review of density estimatio

In this presentation we will introduce the theory related with the long-time behaviour of the kinetic Fokker Planck equation in bounded domains. First, we will provide a hypocoercivity approach to obtain the spectral gap of our operator in a weighted L^2 space. Furthermore we will try to extend this result to every L^p : we will proceed by obtaining bounds on the growth of the semigroup and then through the splitting of the operator into a (hypo)dissipative plus a regularizing part we will be able to obtain convergence by the use of the Duhamel formula.

October 26, 2023

Carmen Moschella: A general model for non-instantaneous collisions with alignment.

In this talk I am going to consider a Boltzmann type equation for the description of a collision dynamic which is not instantaneous.This new class of kinetic equations has been introduced by Kanzler, Schmeiser and Tora to model ensembles of living agents, where the changes of state are the result of complicated internal processes, and not simple mechanical interactions. We extend the latter work introducing a first order approximation to the instantaneous equation, where non binary collisions are included. This is motivated by the fact that during an extended collision period there is a positive probability that a colliding pair is joined by additional particles. The interaction kernel is of alignment type, where the states of the particles approach each

other. For this spacially homogeneous approximation, we check that the formal properties of the system are kept. Furthermore existence and uniqueness of solutions and the instantaneous limit are examined.

October 19, 2023

Brune Massoulié: Transience time of the facilitated exclusion process

The facilitated exclusion process (FEP) is a particle system, where particles evolve on a discrete lattice, making random jumps while obeying local constraints. Because of these constraints, the process will almost surely become blocked (frozen), or reach an absorbing set of configurations after a finite time. This process can thus be seen as a toy model for the liquid-solid transition. In our work, we study the timescale after which these events occur, by introducing a new representation of this process. 

October 12, 2023

Luca Ziviani: Introduction to the Dolbeault-Mouhot-Schmeiser method for Hypocoercivity.

In this presentation, we will introduce the Dolbeault-Mouhot-Schmeiser method, a significant tool in kinetic theory for understanding the long-term behavior of evolutive partial differential equations. This method, which made its debut in 2010, fits in the context of Hypocoercivity. It enables us to study the long-term dynamics of a differential equation, even in cases where coercivity is lacking.

Over the years, from its invention to the present day, this method has demonstrated remarkable flexibility and adaptability, extending its applicability to situations with weaker assumptions. For this reason it has gained much success among the kinetic community.With this seminar we will see how it works with examples and variations.