Groupe de Travail des Thésard·e·s du LJLL

Exposés passés

21 mars 2023
Norbert Tognon (INRIA Paris / LJLL)
Paraopt Algorithm and Runge-Kutta Methods
In this talk, I present the convergence analysis of Paraopt algorithm applied to a linear optimal control problem. The Paraopt algorithm is a parallel in time method for solving a optimality system arising in partial differential equations (PDEs) constrained optimization. This solving required two time integration methods and depends in crucial way on a quadrature formula and a Runge-Kutta method. We use an operator norm analysis to show that the convergence rate depends on the discretization parameters and the smaller of the orders of the time integration methods. We illustrate this by some numerical examples.
8 mars 2023
Kacem Lefki (Université Gustave Eiffel, Cermics)
Atomic decomposition of a positive compact operator on $L^p$
The well known Perron-Frobenius theorem states that for any nonnegative square matrix, the spectral radius is an eigenvalue and has a nonnegative eigenvector. When the matrix is irreducible, there is uniqueness of this nonnegative eigenvector. This theorem has been generalized by Krein and Rutman to the infinite dimensional case of positive compact operators on $L^p$ spaces. In this talk, we present a generalization of the uniqueness in Krein-Rutman theorem where there is no irreducibility. We prove a characterization of nonnegative eigenfunction of a positive compact operator on $L^p$ based on "irreducible components" of the operators, and an application to the equilibria of a SIS epidemic model.
1 mars 2023
Ruiyang DAI (LJLL)
Discrete moments models for Vlasov equations with non constant strong magnetic limit
We describe the structure of an original application of the method of moments to the Vlasov-Poisson system with non constant strong magnetic field in three dimensions of space. Using basis functions which are aligned with the magnetic field, one obtains a Friedrichs system where the kernel of the singular part is made explicit. A projection of the original model on this kernel yields what we call the reduced model.
22 février 2023
Michel-Fabrice Serret (LJLL)
A brief introduction to quantum computing
In this talk, we delve into the basics of quantum computation and try to demystify concepts from quantum physics, such as superposition and entanglement, required to comprehend the mechanisms at play in quantum algorithms. We then take a closer look at the effects of noise on quantum systems and provide an intuitive description of the breakdown of quantum algorithms on current quantum hardware. Finally, we consider other types of quantum algorithms which might still be able to provide a quantum advantage on noisy hardware.
15 février 2023
Gong Chen (LJLL)
Construction of the methods of force field parameterization in molecular simulation by machine learning
This talk introduces the methods to deal with molecules by neural networks. Machine learning has long been crucial to drug discovery and materials science and it has achieved great improvements due to the development of natural language processing and graph neural networks. Based on different message passing algorithms, neural networks perform well on detecting chemical environments and predicting molecular properties. Force field refers to the functional form and parameter sets used to calculate the potential energy of a system of atoms in molecular mechanics. We expect to give appropriate force field parameters by neural networks.
In this talk, firstly we talk about the different neural networks to deal with molecules and explain the attention mechanism briefly. Next we propose a new message passing algorithm, one-directional graph attention networks (OD-GAT), which outperforms other graph neural networks. Finally, based on OD-GAT, we predict force field parameters by neural networks and propose a new function to evaluate the stretching energy.
1 février 2023
Annamaria Massimini (Universität Wien)
Analysis of a Poisson–Nernst–Planck–Fermi model for ion transport in biological channels and nanopores
In this talk, we analyze a Poisson-Nernst-Planck-Fermi model to describe the evolution of a mixture of finite size ions in liquid electrolytes, which move through biological membranes or nanopores. The ion concentrations solve a cross-diffusion system in a bounded domain with mixed Dirichlet-Neumann boundary conditions. A drift term due to the electric potential is also present in the equations. The latter is coupled to the concentrations through a Poisson-Fermi equation. The novelty and the advantage of this model is to take into account ion-ion correlations, which is really important in case of strong electrostatic coupling and high ion concentrations. The global-in-time existence of bounded weak solutions is proved, employing the boundedness-by-entropy method, extended to nonhomogeneous boundary conditions. Furthermore, the weak-strong uniqueness result is also presented.
25 janvier 2023
Charles Elbar (LJLL)
Degenerate Cahn-Hilliard equation: From nonlocal to local
I will discuss the derivation of the local Cahn-Hilliard equation starting from its nonlocal counterpart (arXiv:2208.08955). For that purpose, I will present a new method which includes the use of nonlocal Poincaré and compactness inequalities. When the parameter of the nonlocality is sent to 0, we recover the local Cahn-Hilliard equation.

The motivation for this work is threefold:
- Firstly, the interest for the nonlocal Cahn-Hilliard equation is an old problem that can be traced back to Giacomin and Lebowitz. These seminal works establish the derivation of the nonlocal Cahn-Hilliard equation departing from stochastic systems of particles. However, they left open the question of deriving the local Cahn-Hilliard equation from the nonlocal one.
- Second, the nonlocal Cahn-Hilliard equation can be seen as a porous medium equation with a smooth advection term that is well understood, conversely to the local degenerate Cahn-Hilliard equation.
- Finally, the nonlocal Cahn-Hilliard equation is in fact an aggregation-diffusion equation with a nonlocal term corresponding to the aggregation effect. Thus, in this paper, we show that if the nonlocal effect is appropriately scaled, one approaches Cahn-Hilliard equation. This limit was formally stated in few papers and it was open to provide a rigorous mathematical argument for this approximation.
18 janvier 2023
Yipeng Wang (LJLL)
A New Well-Posedeness Analysis for the Single Reference Coupled Cluster Method
In this talk, we provide a new well-posedeness analysis for the single reference coupled cluster method based on the invertibility of the CC derivative. Under the minimal assumption that the sought-after eigenpair is non-degenerate and the CC ansatz is valid, we prove that the continuous (infinite-dimensional) CC equations are always locally well-posed. Under the same minimal assumptions and provided that the discretisation is fine enough, we prove that the discrete Full-CC equations targeted at the ground state are locally well-posed, and we derive residual-based error estimates with guaranteed positive constants.
11 janvier 2023
Ignacio Madrid (CMAP)
Exponential ergodicity of a degenerate age-size piecewise deterministic process
We aim to study the steady-state cell size distribution of a population of E. coli cells, integrating information collected at the individual scale. To that extent, we propose a stochastic individual-based dynamic model which can be calibrated using temporal single-cell lineage data acquired via microfluidic techniques. This data also grants access to the age structure, which then can be used to to provide a more precise non-Markovian characterisation of the growing population. More generally, this model leads us to the study of the long-time behaviour of a non conservative piecewise deterministic measure-valued stochastic process with support on R^2. The process is driven by a deterministic flow between random jump times, with a transition kernel which has a degenerate form. I will give some ideas on how to obtain the exponential ergodicity of such a process from a probabilistic approach, using Harris’ Theorem and the spectral properties of the infinitesimal generator of the process. In particular, the construction of explicit trajectories which explore the space state with positive probability permits us to prove a so-called petite-set condition for the compact sets of the state space, which happens to be one sufficient condition for the exponential ergodicity. An application to an age-structured growth-fragmentation process modelling bacterial growth will be shown.
14 décembre 2022
Andrea Poiatti (Politecnico di Milano, Polimi)
Mathematical analysis of phase separation models
Phase separation in a binary liquid (e.g. oil and vinegar) is a phenomenon which can be described as
a competition between an entropy mixing effect and a demixing effect due to the internal energy.
Typical mathematical models are given by the so-called Cahn-Hilliard (CH) equation or by the
conserved Allen-Cahn (AC) equation with singular potential. The Cahn-Hilliard (CH) equation can
be considered as to be local or nonlocal, according to the type of interactions (short range or long
range) we take into account. These equations govern the evolution of the relative concentration of
one component (phase field) and conserve the total mass. Moreover, one can consider a
multicomponent mixture, so that more than two chemical species can come into play in phase
separation phenomena.
In this talk, I will first present the phase separation phenomenon through some of its recent and
unexpected applications in cell biology, showing its growing and growing importance in many
fields of mathematical modelling. Then I will introduce some of the aforementioned equations, like
CH and AC equations on bounded domains or on compact evolving surfaces, trying to give an
insight of the most recent results about the mathematical analysis concerning these equations.
In conclusion, I will introduce the instantaneous strict separation property, meaning that each
(weak) solution, which is not a pure phase initially, stays uniformly away from the pure phases from
any positive time on. I will show how this property is essential to obtain higher-order regularity as
well as to study the long-time behavior of solutions (convergence to equilibria, existence of finite
dimensional global attractors...) and I will present some recent advancements in showing the
validity of this property in case of 3D AC and nonlocal CH with singular potential.
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