MATHERIALS

MATHERIALS - 2024

2024Activity reportProject-TeamMATHERIALS

RNSR: 201421206U

Research center Inria Paris Centre
In partnership with:Ecole Nationale des Ponts et Chaussées
Team name: MATHematics for MatERIALS
In collaboration with:Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS)
Domain:Applied Mathematics, Computation and Simulation
Theme:Numerical schemes and simulations

Keywords

Computer Science and Digital Science

A6.1.1. Continuous Modeling (PDE, ODE)
A6.1.2. Stochastic Modeling
A6.1.4. Multiscale modeling
A6.1.5. Multiphysics modeling
A6.2.1. Numerical analysis of PDE and ODE
A6.2.2. Numerical probability
A6.2.3. Probabilistic methods
A6.2.4. Statistical methods
A6.2.7. High performance computing
A6.3.1. Inverse problems
A6.3.4. Model reduction
A6.4.1. Deterministic control

1 Team members, visitors, external collaborators

Research Scientists

Claude Le Bris [Team leader, ENPC, Senior Researcher, HDR]
Sébastien Boyaval [ENPC, Senior Researcher, HDR]
Eric Cancès [ENPC, Senior Researcher, HDR]
Virginie Ehrlacher [ENPC, Senior Researcher, HDR]
Frederic Legoll [ENPC, Senior Researcher, HDR]
Tony Lelièvre [ENPC, Senior Researcher, HDR]
Gabriel Stoltz [ENPC, Senior Researcher, HDR]
Urbain Vaes [INRIA, ISFP]

Faculty Members

Arnaud Guyader [SORBONNE UNIVERSITE, Professor Delegation, until Aug 2024]
Luca Nenna [UNIV PARIS SACLAY, Associate Professor Delegation, until Jun 2024]
Francis Nier [UNIV PARIS XIII, Professor Delegation, from Feb 2024]

Post-Doctoral Fellows

Thomas Borsoni [ENPC, Post-Doctoral Fellow, from Oct 2024]
Amandine Boucart [ENPC, Post-Doctoral Fellow, until Mar 2024]
Antonin Dellanoce [INRIA, Post-Doctoral Fellow]
Mathias Dus [ENPC, Post-Doctoral Fellow, until Sep 2024]
Laura Grazioli [ENPC, from Sep 2024]
Rodrigue Lelotte [ENPC, Post-Doctoral Fellow]
Annamaria Massimini [ENPC, Post-Doctoral Fellow, from Nov 2024]
Thomas Normand [INRIA, Post-Doctoral Fellow, from Oct 2024]
Etienne Polack [ENPC, Post-Doctoral Fellow, until Jul 2024]
Giulia Sambataro [ENPC, Post-Doctoral Fellow]
Hadrien Vroylandt [ENPC, Post-Doctoral Fellow, until Nov 2024]

PhD Students

Noe Blassel [ENPC]
Andrea Bordignon [ENPC, until Jul 2024]
Louis Carillo [ENPC, from Sep 2024]
Shiva Darshan [ENPC]
Théo Duez [CNRS, from Oct 2024]
François Escolan [ENPC, from Oct 2024]
Sofiane Ezzehi [ENPC, from Oct 2024]
Renato Freitas Spacek [INRIA, until Oct 2024]
Raphael Gastaldello [CNRS]
Clement Guillot [ENPC]
Alfred Kirsch [ENPC, until Nov 2024]
Alberic Lefort [ENPC]
Eloïse Letournel [ENPC, until Sep 2024]
Pierre Marmey [IFPEN]
Alicia Negre [INRIA]
Solal Perrin-Roussel [ENPC]
Simon Ruget [INRIA]
Régis Santet [ENPC, until Sep 2024]
Laurent Vidal [ENPC, until Apr 2024]

Interns and Apprentices

Theo Duez [INRIA, Intern, from Apr 2024 until Sep 2024]

Administrative Assistant

Julien Guieu [INRIA]

Visiting Scientist

Theron Guo [MIT, from Sep 2024 until Nov 2024]

2 Overall objectives

The MATHERIALS project-team was created jointly by the École des Ponts ParisTech (ENPC) and Inria in 2015. It is the follow-up and an extension of the former project-team MICMAC originally created in October 2002. It is hosted by the CERMICS laboratory (Centre d'Enseignement et de Recherches en Mathématiques et Calcul Scientifique) at École des Ponts. The permanent research scientists of the project-team have positions at CERMICS and at two other laboratories of École des Ponts: Institut Navier and Laboratoire Saint-Venant. The scientific focus of the project-team is to analyze and improve the numerical schemes used in the simulation of computational chemistry at the microscopic level and to create simulations coupling this microscopic scale with meso- or macroscopic scales (possibly using parallel algorithms). Over the years, the project-team has accumulated an increasingly solid expertise on such topics, which are traditionally not well known by the community in applied mathematics and scientific computing. One of the major achievements of the project-team is to have created a corpus of literature, authoring books and research monographs on the subject 3, 4, 5, 6, 8, 7, 9 that other scientists may consult in order to enter the field.

3 Research program

Our group, originally only involved in electronic structure computations, continues to focus on many numerical issues in quantum chemistry, but now expands its expertise to cover several related problems at larger scales, such as molecular dynamics problems and multiscale problems. The mathematical derivation of continuum energies from quantum chemistry models is one instance of a long-term theoretical endeavour.

4 Application domains

4.1 Electronic structure of large systems

Quantum Chemistry aims at understanding the properties of matter through the modelling of its behavior at a subatomic scale, where matter is described as an assembly of nuclei and electrons. At this scale, the equation that rules the interactions between these constitutive elements is the Schrödinger equation. It can be considered (except in few special cases notably those involving relativistic phenomena or nuclear reactions) as a universal model for at least three reasons. First it contains all the physical information of the system under consideration so that any of the properties of this system can in theory be deduced from the Schrödinger equation associated to it. Second, the Schrödinger equation does not involve any empirical parameters, except some fundamental constants of Physics (the Planck constant, the mass and charge of the electron, ...); it can thus be written for any kind of molecular system provided its chemical composition, in terms of natures of nuclei and number of electrons, is known. Third, this model enjoys remarkable predictive capabilities, as confirmed by comparisons with a large amount of experimental data of various types. On the other hand, using this high quality model requires working with space and time scales which are both very tiny: the typical size of the electronic cloud of an isolated atom is the Angström ( $10^{- 10}$ meters), and the size of the nucleus embedded in it is $10^{- 15}$ meters; the typical vibration period of a molecular bond is the femtosecond ( $10^{- 15}$ seconds), and the characteristic relaxation time for an electron is $10^{- 18}$ seconds. Consequently, Quantum Chemistry calculations concern very short time (say $10^{- 12}$ seconds) behaviors of very small size (say $10^{- 27}$ m $^{3}$ ) systems. The underlying question is therefore whether information on phenomena at these scales is useful in understanding or, better, predicting macroscopic properties of matter. It is certainly not true that all macroscopic properties can be simply upscaled from the consideration of the short time behavior of a tiny sample of matter. Many of them derive from ensemble or bulk effects, that are far from being easy to understand and to model. Striking examples are found in solid state materials or biological systems. Cleavage, the ability of minerals to naturally split along crystal surfaces (e.g. mica yields to thin flakes), is an ensemble effect. Protein folding is also an ensemble effect that originates from the presence of the surrounding medium; it is responsible for peculiar properties (e.g. unexpected acidity of some reactive site enhanced by special interactions) upon which vital processes are based. However, it is undoubtedly true that many macroscopic phenomena originate from elementary processes which take place at the atomic scale. Let us mention for instance the fact that the elastic constants of a perfect crystal or the color of a chemical compound (which is related to the wavelengths absorbed or emitted during optic transitions between electronic levels) can be evaluated by atomic scale calculations. In the same fashion, the lubricative properties of graphite are essentially due to a phenomenon which can be entirely modeled at the atomic scale. It is therefore reasonable to simulate the behavior of matter at the atomic scale in order to understand what is going on at the macroscopic one. The journey is however a long one. Starting from the basic principles of Quantum Mechanics to model the matter at the subatomic scale, one finally uses statistical mechanics to reach the macroscopic scale. It is often necessary to rely on intermediate steps to deal with phenomena which take place on various mesoscales. It may then be possible to couple one description of the system with some others within the so-called multiscale models. The sequel indicates how this journey can be completed focusing on the first smallest scales (the subatomic one), rather than on the larger ones. It has already been mentioned that at the subatomic scale, the behavior of nuclei and electrons is governed by the Schrödinger equation, either in its time-dependent form or in its time-independent form. Let us only mention at this point that

both equations involve the quantum Hamiltonian of the molecular system under consideration; from a mathematical viewpoint, it is a self-adjoint operator on some Hilbert space; both the Hilbert space and the Hamiltonian operator depend on the nature of the system;
also present into these equations is the wavefunction of the system; it completely describes its state; its $L^{2}$ norm is set to one.

The time-dependent equation is a first-order linear evolution equation, whereas the time-independent equation is a linear eigenvalue equation. For the reader more familiar with numerical analysis than with quantum mechanics, the linear nature of the problems stated above may look auspicious. What makes the numerical simulation of these equations extremely difficult is essentially the huge size of the Hilbert space: indeed, this space is roughly some symmetry-constrained subspace of $L^{2} (ℝ^{d})$ , with $d = 3 (M + N)$ , $M$ and $N$ respectively denoting the number of nuclei and the number of electrons the system is made of. The parameter $d$ is already 39 for a single water molecule and rapidly reaches $10^{6}$ for polymers or biological molecules. In addition, a consequence of the universality of the model is that one has to deal at the same time with several energy scales. In molecular systems, the basic elementary interaction between nuclei and electrons (the two-body Coulomb interaction) appears in various complex physical and chemical phenomena whose characteristic energies cover several orders of magnitude: the binding energy of core electrons in heavy atoms is $10^{4}$ times as large as a typical covalent bond energy, which is itself around 20 times as large as the energy of a hydrogen bond. High precision or at least controlled error cancellations are thus required to reach chemical accuracy when starting from the Schrödinger equation. Clever approximations of the Schrödinger problems are therefore needed. The main two approximation strategies, namely the Born-Oppenheimer-Hartree-Fock and the Born-Oppenheimer-Kohn-Sham strategies, end up with large systems of coupled nonlinear partial differential equations, each of these equations being posed on $L^{2} (ℝ^{3})$ . The size of the underlying functional space is thus reduced at the cost of a dramatic increase of the mathematical complexity of the problem: nonlinearity. The mathematical and numerical analysis of the resulting models has been the major concern of the project-team for a long time. In the recent years, while part of the activity still follows this path, the focus has progressively shifted to problems at other scales.

As the size of the systems one wants to study increases, more efficient numerical techniques need to be resorted to. In computational chemistry, the typical scaling law for the complexity of computations with respect to the size of the system under study is $N^{3}$ , $N$ being for instance the number of electrons. The Holy Grail in this respect is to reach a linear scaling, so as to make possible simulations of systems of practical interest in biology or materials science. Efforts in this direction must address a large variety of questions such as

how can one improve the nonlinear iterations that are the basis of any ab initio models for computational chemistry?
how can one more efficiently solve the inner loop which most often consists in the solution procedure for the linear problem (with frozen nonlinearity)?
how can one design a sufficiently small variational space, whose dimension is kept limited while the size of the system increases?

An alternative strategy to reduce the complexity of ab initio computations is to try to couple different models at different scales. Such a mixed strategy can be either a sequential one or a parallel one, in the sense that

in the former, the results of the model at the lower scale are simply used to evaluate some parameters that are inserted in the model for the larger scale: one example is the parameterized classical molecular dynamics, which makes use of force fields that are fitted to calculations at the quantum level;
while in the latter, the model at the lower scale is concurrently coupled to the model at the larger scale: an instance of such a strategy is the so called QM/MM coupling (standing for Quantum Mechanics/Molecular Mechanics coupling) where some part of the system (typically the reactive site of a protein) is modeled with quantum models, that therefore accounts for the change in the electronic structure and for the modification of chemical bonds, while the rest of the system (typically the inert part of a protein) is coarse grained and more crudely modeled by classical mechanics.

The coupling of different scales can even go up to the macroscopic scale, with methods that couple a microscopic representation of matter, or at least a mesoscopic one, with the equations of continuum mechanics at the macroscopic level.

4.2 Computational Statistical Mechanics

The orders of magnitude used in the microscopic representation of matter are far from the orders of magnitude of the macroscopic quantities we are used to: the number of particles under consideration in a macroscopic sample of material is of the order of the Avogadro number $𝒩_{A} \sim 6 \times 10^{23}$ , the typical distances are expressed in Å ( $10^{- 10}$ m), the energies are of the order of $k_{B} T ≃ 4 \times 10^{- 21}$ J at room temperature, and the typical times are of the order of $10^{- 15}$ s.

To give some insight into such a large number of particles contained in a macroscopic sample, it is helpful to compute the number of moles of water on earth. Recall that one mole of water corresponds to 18 mL, so that a standard glass of water contains roughly 10 moles, and a typical bathtub contains $10^{5}$ mol. On the other hand, there are approximately $10^{18}$ m $^{3}$ of water in the oceans, i.e. $7 \times 10^{22}$ mol, a number comparable to the Avogadro number. This means that inferring the macroscopic behavior of physical systems described at the microscopic level by the dynamics of several millions of particles only is like inferring the ocean's dynamics from hydrodynamics in a bathtub...

For practical numerical computations of matter at the microscopic level, following the dynamics of every atom would require simulating $𝒩_{A}$ atoms and performing $O (10^{15})$ time integration steps, which is of course impossible! These numbers should be compared with the current orders of magnitude of the problems that can be tackled with classical molecular simulation, where several millions of atoms only can be followed over time scales of the order of a few microseconds.

Describing the macroscopic behavior of matter knowing its microscopic description therefore seems out of reach. Statistical physics allows us to bridge the gap between microscopic and macroscopic descriptions of matter, at least on a conceptual level. The question is whether the estimated quantities for a system of $N$ particles correctly approximate the macroscopic property, formally obtained in the thermodynamic limit $N \to + \infty$ (the density being kept fixed). In some cases, in particular for simple homogeneous systems, the macroscopic behavior is well approximated from small-scale simulations. However, the convergence of the estimated quantities as a function of the number of particles involved in the simulation should be checked in all cases.

Despite its intrinsic limitations on spatial and timescales, molecular simulation has been used and developed over the past 50 years, and its number of users keeps increasing. As we understand it, it has two major aims nowadays.

First, it can be used as a numerical microscope, which allows us to perform “computer” experiments. This was the initial motivation for simulations at the microscopic level: physical theories were tested on computers. This use of molecular simulation is particularly clear in its historic development, which was triggered and sustained by the physics of simple liquids. Indeed, there was no good analytical theory for these systems, and the observation of computer trajectories was very helpful to guide the physicists' intuition about what was happening in the system, for instance the mechanisms leading to molecular diffusion. In particular, the pioneering works on Monte Carlo methods by Metropolis et al., and the first molecular dynamics simulation of Alder and Wainwright were performed because of such motivations. Today, understanding the behavior of matter at the microscopic level can still be difficult from an experimental viewpoint (because of the high resolution required, both in time and in space), or because we simply do not know what to look for! Numerical simulations are then a valuable tool to test some ideas or obtain some data to process and analyze in order to help assessing experimental setups. This is particularly true for current nanoscale systems.

Another major aim of molecular simulation, maybe even more important than the previous one, is to compute macroscopic quantities or thermodynamic properties, typically through averages of some functionals of the system. In this case, molecular simulation is a way to obtain quantitative information on a system, instead of resorting to approximate theories, constructed for simplified models, and giving only qualitative answers. Sometimes, these properties are accessible through experiments, but in some cases only numerical computations are possible since experiments may be unfeasible or too costly (for instance, when high pressure or large temperature regimes are considered, or when studying materials not yet synthesized). More generally, molecular simulation is a tool to explore the links between the microscopic and macroscopic properties of a material, allowing one to address modelling questions such as “Which microscopic ingredients are necessary (and which are not) to observe a given macroscopic behavior?”

4.3 Homogenization and related problems

Over the years, the project-team has developed an increasing expertise on multiscale modeling for materials science at the continuum scale. The presence of numerous length scales in material science problems indeed represents a challenge for numerical simulation, especially when some randomness is assumed on the materials. It can take various forms, and includes defects in crystals, thermal fluctuations, and impurities or heterogeneities in continuous media. Standard methods available in the literature to handle such problems often lead to very costly computations. Our goal is to develop numerical methods that are more affordable. Because we cannot embrace all difficulties at once, we focus on a simple case, where the fine scale and the coarse-scale models can be written similarly, in the form of a simple elliptic partial differential equation in divergence form. The fine scale model includes heterogeneities at a small scale, a situation which is formalized by the fact that the coefficients in the fine scale model vary on a small length scale. After homogenization, this model yields an effective, macroscopic model, which includes no small scale (the coefficients of the coarse scale equations are thus simply constant, or vary on a coarse length scale). In many cases, a sound theoretical groundwork exists for such homogenization results. The difficulty stems from the fact that the models generally lead to prohibitively costly computations (this is for instance the case for random stationary settings). Our aim is to focus on different settings, all relevant from an applied viewpoint, and leading to practically affordable computational approaches. It is well-known that the case of ordered (that is, in this context, periodic) systems is now well-understood, both from a theoretical and a numerical standpoint. Our aim is to turn to cases, more relevant in practice, where some disorder is present in the microstructure of the material, to take into account defects in crystals, impurities in continuous media... This disorder may be mathematically modeled in various ways.

Such endeavors raise several questions. The first one, theoretical in nature, is to extend the classical theory of homogenization (well developed e.g. in the periodic setting) to such disordered settings. Next, after homogenization, we expect to obtain an effective, macroscopic model, which includes no small scale. A second question is to introduce affordable numerical methods to compute the homogenized coefficients. An alternative approach, more numerical in nature, is to directly attack the oscillatory problem by using discretization approaches tailored to the multiscale nature of the problem (the construction of which is often inspired by theoretical homogenization results). For a comprehensive account of many of the research efforts of the team on these topics, we refer to 1, 2.

5 New software, platforms, open data

5.1 New software

5.1.1 DFTK

Keywords:
Molecular simulation, Quantum chemistry, Materials
Functional Description:
DFTK, short for the density-functional toolkit, is a Julia library implementing plane-wave density functional theory for the simulation of the electronic structure of molecules and materials. It aims at providing a simple platform for experimentation and algorithm development for scientists of different backgrounds.
Release Contributions:
In 2024 DFTK continued to be actively developed, and it received several contributions from members of MATHERIALS. The library has been used for several publications both inside and outside the project-team.
URL:
http://dftk.org
Contact:
Antoine Levitt

6 New results

6.1 Electronic structure calculations and related quantum-scale problems

Participants: Andrea Bordignon, Eric Cancès, Mathias Dus, Virginie Ehrlacher, Clément Guillot, Alfred Kirsch, Claude Le Bris, Eloïse Letournel, Solal Perrin-Roussel, Etienne Polack, Laurent Vidal.

6.1.1 Density functional theory

The team continued its long-standing project to study density functional theory from an applied mathematics perspective.

In collaboration with Geneviève Dusson (CNRS and University of Franche-Comté), Gaspard Kemlin (Université de Picardie Jules Verne), Rafael Antonio Lainez Reyes (University of Stuttgart, Germany), Benjamin Stamm (University of Stuttgart, Germany), A. Bordignon and E. Cancès derived fully guaranteed error bounds for the energy of convex nonlinear mean-field models 40. These results apply in particular to Kohn-Sham equations with convex density functionals, which includes the reduced Hartree–Fock (rHF) model, as well as the Kohn-Sham model with exact exchange-density functional (which is unfortunately not explicit and therefore not usable in practice). They then decomposed the obtained bounds into two parts, one depending on the chosen discretization and one depending on the number of iterations performed in the self-consistent algorithm used to solve the nonlinear eigenvalue problem, paving the way for adaptive refinement strategies. The accuracy of the bounds is demonstrated on a series of test cases, including a Silicon crystal and an Hydrogen Fluoride molecule simulated with the rHF model and discretized with planewaves. They also showed that, although not anymore guaranteed, the error bounds remain very accurate for a Silicon crystal simulated with the Kohn–Sham model using nonconvex exchange- correlation functionals of practical interest.

Etienne Polack and Laurent Vidal developed new functionalities in DFTK, a Julia library implementing plane-wave density functional theory for the simulation of the electronic structure of molecules and materials, whose development was launched in 2019 within the Project-Team MATHERIALS (main developers: Michael Herbst, now at EPFL, and Antoine Levitt, now at Université Paris-Saclay).

6.1.2 Analysis of molecular electronic densities

In collaboration with Yingxing Cheng (University of Stuttgart, Germany), Alston J. Misquitta (Queen Mary University of London, UK), and Benjamin Stamm (University of Stuttgart, Germany), E. Cancès and V. Ehrlacher analyzed various Iterative Stockholder Analysis (ISA) methods for molecular density partitioning 49, focusing on the numerical performance of the recently proposed Linear approximation of Iterative Stockholder Analysis model (LISA) 73. They first provide a systematic derivation of various iterative solvers to find the unique LISA solution. In a subsequent systematic numerical study, they evaluate their performance on 48 organic and inorganic, neutral and charged molecules and also compare LISA to two other well-known ISA variants: the Gaussian Iterative Stockholder Analysis (GISA) and Minimum Basis Iterative Stockholder analysis (MBIS). The study reveals that LISA-family methods can offer a numerically more efficient approach with better accuracy compared to the two comparative methods. Moreover, the well-known issue with the MBIS method, where atomic charges obtained for negatively charged molecules are anomalously negative, is not observed in LISA-family methods. Despite the fact that LISA occasionally exhibits elevated entropy as a consequence of the absence of more diffuse basis functions, this issue can be readily mitigated by incorporating additional or integrating supplementary basis functions within the LISA framework. This research provides the foundation for future studies on the efficiency and chemical accuracy of molecular density partitioning schemes.

6.1.3 Strongly-correlated systems

The treatment of strongly correlated quantum systems is a long-standing challenge in computational chemistry and physics.

In collaboration with Filippo Lipparini and Tommaso Nottoli (University of Pisa, Italy), E. Cancès and L. Vidal explored Riemannian optimization methods for Restricted-Open-shell Hartree-Fock (ROHF) and Complete Active Space Self-Consistent Field (CASSCF) methods 31. After showing that ROHF and CASSCF can be reformulated as optimization problems on so-called flag manifolds, they reviewed Riemannian optimization basics and their application to these specific problems. They compared these methods to traditional ones and find robust convergence properties without fine-tuning of numerical parameters. This study suggests Riemannian optimization as a valuable addition to orbital optimization for ROHF and CASSCF, warranting further investigation.

The application of high-accuracy first-principle methods such as CASSCF that are able to capture the electronic correlation effects at chemical accuracy is commonly stymied by a steep computational scaling with respect to system size. A potential remedy is provided by quantum embedding theories, which can be somehow interpreted as domain decomposition methods for the quantum many-body problem in the Fock space. Such approaches include the dynamical mean-field theory (DMFT) and the density matrix embedding theory (DMET).

In 44, E. Cancès, A. Kirsch and S. Perrin-Roussel provide a mathematical analysis of the Dynamical Mean-Field Theory, a celebrated representative of a class of approximations in quantum mechanics known as embedding methods. They start by a pedagogical and self-contained mathematical formulation of the Dynamical Mean-Field Theory equations for the finite Hubbard model. After recalling the definition and properties of one-body time-ordered Green's functions and self-energies, and the mathematical structure of the Hubbard and Anderson impurity models, they describe a specific impurity solver, namely the Iterated Perturbation Theory solver, which can be conveniently formulated using Matsubara's Green's functions. Within this framework, they prove under certain assumptions that the Dynamical Mean-Field Theory equations admit a solution for any set of physical parameters. Moreover, they establish some properties of the solution(s).

In 54, M. Dus, C. Guillot and V. Ehrlacher, in collaboration with Geneviève Dusson (Besançon) and Joel Pascal Soffo (Nantes), compare the efficiency of various numerical methods for strongly correlated systems, including the Density Matrix Normalization Group (DMRG) method (using tensor trains) and neural network approaches which have been recently proposed in the literature. The latter include the FermiNet and the PauliNet architectures. This is the first time that a careful study compares the efficiency of these two families of approaches on the same chemical systems.

This work was the motivation for 55, where M. Dus and V. Ehrlacher analyze the gradient descent algorithm used in neural network approaches for the resolution of high-dimensional Schrödinger eigenvalue problems. More precisely, considering two-layer neural networks in the mean-field limit, they proved, inspired by earlier contributions by Chizat and Bach, that this gradient descent can be interpreted as a constrained gradient curve in the space of probability measures defined on the set of parameters defining the neural network, of which they prove the existence. They also prove that, if the solution of this constrained gradient curve converges to some limit measure in the long time limit and if the interaction potential is sufficiently smooth, then the Lebesgue density of this measure is necessarily a solution to the original Schrödinger eigenvalue problem at hand.

6.1.4 Optimal transport and quantum chemistry

Recent research efforts have been carried out in the team on the development of efficient numerical methods for quantum chemistry using optimal transport theory.

In 50, together with Maxime Daléry and Geneviève Dusson, V. Ehrlacher developped new definitions of marginal-preserving Wasserstein-like barycenters between Gaussian mixture distributions. This new construction is based on fine properties of the Wasserstein-Bure metric between symmetric definite positive matrices. This new concept will be particularly useful for the construction of new reduced-order models to accelerate parametric electronic structure calculations.

6.1.5 Evolution of quantum system

In 52, C. Guillot and V. Ehrlacher, in collaboration with Mi-Song Dupuy (Sorbonne Université, Paris) derived a new global space-time variational formulation to express the solution of time-dependent Schrödinger equations. This new formulation is very promising in the sense that it can give rise to new efficient numerical methods for the approximation of solutions of these equations in low-complexity manifolds (like tensor formats or Gaussian mixtures) in high-dimensional contexts. This new variational principle leads to approximate low-complexity approximations that are defined globally in time, in contrast to the classical Dirac-Frenkel variational principle which only ensures local existence in time of its corresponding approximations.

6.2 Computational statistical physics

Participants: Noé Blassel, Louis Carillo, Shiva Darshan, Raphaël Gastaldello, Arnaud Guyader, Tony Lelièvre, Régis Santet, Renato Spacek, Gabriel Stoltz, Urbain Vaes.

The aim of computational statistical physics is to compute macroscopic properties of materials starting from a microscopic description, using concepts of statistical physics (thermodynamic ensembles and molecular dynamics). The contributions of the team can be divided into four main topics: (i) the improvement of techniques to sample the configuration space; (ii) the development of simulation methods to efficiently simulate nonequilibrium systems; (iii) the study of dynamical properties and rare event sampling; (iv) the use of particle methods for sampling and optimization.

6.2.1 Sampling of the configuration space

There is still a need to improve techniques to sample the configuration space, and to understand their performance. This includes the development of sampling techniques, and their numerical analysis.

Concerning the development of sampling techniques, T. Lelièvre, R. Santet and G. Stoltz, together with Grigorios A. Pavliotis (Imperial College London, Royaume-Uni) and Geneviève Robin (CNRS et Université d'Evry, France), considered in 63 the improvement of sampling efficiency of overdamped Langevin dynamics obtained when varying the diffusion coefficient. As there are in fact infinitely many overdamped Langevin dynamics which are reversible with respect to the target probability measure at hand, this suggests to optimize the diffusion coefficient in order to increase the convergence rate of the dynamics, as measured by the spectral gap of the generator associated with the stochastic differential equation. They analytically studied this problem, obtaining in particular necessary conditions on the optimal diffusion coefficient. They also derived an explicit expression of the optimal diffusion in some appropriate homogenized limit. Numerical results for low dimensional systems, both relying on discretizations of the spectral gap problem and Monte Carlo simulations of the stochastic dynamics, demonstrate the increased quality of the sampling arising from an appropriate choice of the diffusion coefficient.

In order to leverage the results obtained in the low dimensional setting in actual high dimensional scenarios, T. Lelièvre, R. Santet and G. Stoltz proposed in 65 a class of diffusion matrices, based on one-dimensional collective variables (CVs), which helps dynamics explore the latent space defined by the CV. The form of the diffusion matrix is such that the effective dynamics, which are approximations of the processes as observed on the latent space, are governed by the optimal effective diffusion coefficient in a homogenized limit, which possesses an analytical expression. They describe how this class of diffusion matrices can be constructed and learned during the simulation. They also provide implementations of the Metropolis–Adjusted Langevin Algorithm and Riemann Manifold (Generalized) Hamiltonian Monte Carlo algorithms, and discuss numerical optimizations in the case when the CV depends only on a few number of components of the position of the system. The efficiency gains of using this class of diffusion is illustrated by computing mean transition durations between two configurations of a dimer in a solvent.

G. Stoltz participated in 46 to a review article led by Francois Bottin (CEA/DAM, France), on the so-called "Machine Learning Assisted Canonical Sampling" method. This techniques allows to train simple empirical force fields on ab-initio data, in order to reproduce thermodynamic properties at finite temperature. It iterates between exploration phases where new configurations are efficiently sampled and generated, using the current version of the simple empirical potential at hand, and a training phase where the empirical potential is updated with new ab-initio data. Thermodynamic consistency is ensured via some nonlinear reweighting procedure.

On the numerical analysis side, G. Stoltz, together with Evan Camrud (Colorado State University, Etats-Unis), Alain Durmus (Ecole polytechnique, France) and Pierre Monmarché (Sorbonne-Université, France), provided in 75 a convergence analysis for generalized Hamiltonian Monte Carlo samplers, a family of Markov Chain Monte Carlo methods based on leapfrog integration of Hamiltonian dynamics and kinetic Langevin diffusion, that encompasses the unadjusted Hamiltonian Monte Carlo method. Assuming that the target distribution to sample satisfies a log-Sobolev inequality and mild conditions on the corresponding potential function, they establish quantitative bounds on the relative entropy of the iterates defined by the algorithm. Our approach is based on a perturbative and discrete version of the modified entropy method developed to establish hypocoercivity for the continuous-time kinetic Langevin process. Complexity bounds follow as a corollary of their results.

6.2.2 Mathematical understanding and efficient simulation of nonequilibrium systems

Many systems in computational statistical physics are not at equilibrium. This is in particular the case when one wants to compute transport coefficients, which determine the response of the system to some external perturbation. For instance, the thermal conductivity relates an applied temperature difference to an energy current through Fourier's law, while the mobility coefficient relates an applied external constant force to the average velocity of the particles in the system. The main limitations of usual methods to compute transport coefficients is the large variance of the estimators, which motivates searching for dedicated variance reduction strategies. Let us next describe the efforts of the team done in the previous year.

In 29, R. Spacek and G. Stoltz studied with Pierre Monmarché (Sorbonne-Université, France) a transient method where the system is started off equilibrium and its relaxation towards the equilibrium state is monitored through some time-integral. An appropriate choice of the initial perturbation allows to recover the transport coefficient of interest. The efficiency of the numerical method can be substantially improved through some synchronous coupling where the difference between the response of a system at equilibrium and the one of system whose initial distribution is slightly perturbed.

In 67, R. Spacek, G. Stoltz and U. Vaes proposed with Grigorios Pavliotis (Imperial College London, United Kingdom) a method utilizing physics-informed neural networks (PINNs) to solve Poisson equations that serve as control variates in the computation of transport coefficients via fluctuation formulas, such as the Green–Kubo and generalized Einstein-like formulas. By leveraging approximate solutions to the Poisson equation constructed through neural networks, their approach allows to significantly reduce the variance of the estimator at hand. They provide an extensive numerical analysis of the estimators and detail a methodology for training neural networks to solve these Poisson equations. The approximate solutions are then incorporated into Monte Carlo simulations as effective control variates, demonstrating the suitability of the method for moderately high-dimensional problems where fully deterministic solutions are computationally infeasible.

In 51, S. Darshan and G. Stoltz, together with Andreas Eberle (University of Bonn, Germany), present and analyze a control variate strategy based on couplings to reduce the variance of estimators of transport coefficients in the NEMD context. More precisely, they study the bias and variance of a sticky-coupling and a synchronous-coupling based estimator as the forcing magnitude parameter $η$ goes to zero. For diffusions with elliptic additive noise, they show that, when the drift is contractive outside a compact set, the bias of a sticky-coupling based estimator is bounded as $η \to 0$ and its variance scales as $1 / η$ , compared to the standard estimator whose bias and variance behave scale as $1 / η$ and $1 / η^{2}$ , respectively. The setting they consider includes overdamped Langevin dynamics with many physically relevant non-convex potentials. Their theoretical results are illustrated by numerical examples, including overdamped Langevin dynamics with a highly non-convex Lennard-Jones potential to demonstrate both the failure of synchronous coupling and the effectiveness of sticky coupling in the not globally contractive setting.

6.2.3 Sampling dynamical properties and rare events

Sampling transitions from one metastable state to another is a difficult task. The work of the team here consists in analyzing and developing new numerical methods to this end, and provide the associated mathematical analysis.

On the analysis side, T. Lelièvre, M. Rachid and G. Stoltz studied in 64 the narrow escape problem in the disk, which consists of identifying the first exit time and first exit point distribution of a Brownian particle from the ball in dimension 2, with reflecting boundary conditions except on small disjoint windows through which it can escape. This problem is motivated by practical questions arising in various scientific fields (in particular cellular biology and molecular dynamics). They applied the quasi-stationary distribution approach to metastability, which requires to study the eigenvalue problem for the Laplacian operator with Dirichlet boundary conditions on the small absorbing part of the boundary, and Neumann boundary conditions on the remaining reflecting part. They obtained rigorous asymptotic estimates of the first eigenvalue and of the normal derivative of the associated eigenfunction in the limit of infinitely small exit regions, which yield asymptotic estimates of the first exit time and first exit point distribution starting from the quasi-stationary distribution within the disk.

T. Lelièvre together with N. Champagnat (Inria Nancy, France), M. Ramil (Inria Rennes, France) and D. Villemonais (University of Strasbourg, France) considered in 47 the convergence of the Langevin dynamics conditioned to be trapped in a domain towards the quasi-stationary distribution. The result is obtained for coefficients with low regularity. For the solutions to such equations, we prove a Harnack inequality which then allows us to prove, under a Lyapunov condition, the existence and uniqueness (in a suitable class of measures) of a quasi-stationary distribution in cylindrical domains of the phase space. We finally exhibit two settings in which the Lyapunov condition holds: general kinetic SDEs in domains which are bounded in position, and Langevin processes with a non-conservative force and a suitable growth condition on the force.

In 58, T. Lelièvre together with Lucas Journel (Sorbonne Université, France) and Julien Reygner (Ecole des Ponts, France) studied the Fleming-Viot interacting particle process in the regime of a fast killing mechanism. The Fleming-Viot process is used in accelerated molecular dynamics algorithms to analyze the convergence to the quasi-stationary distribution of a stochastic process trapped in a domain. This analysis studies a pathological situation, when the domain is not metastable so that exits occur very frequently.

On the numerical side, T. Lelièvre and G. Stoltz, together with Christoph Schönle (Ecole polytechnique, France) and Marylou Gabrié (ENS Paris, France) considered in 69 the problem of sampling a high dimensional multimodal target probability measure, assuming that a good proposal kernel to move only a subset of the degrees of freedoms (also known as collective variables) is known a priori. This proposal kernel can for example be built using normalizing flows. They showed how to extend the move from the collective variable space to the full space and how to implement an accept-reject step in order to get a reversible chain with respect to a target probability measure. The accept-reject step does not require to know the marginal of the original measure in the collective variable (namely to know the free energy). The obtained algorithm admits several variants, some of them being very close to methods which have been proposed previously in the literature. They showed how the obtained acceptance ratio can be expressed in terms of the work which appears in the Jarzynski–Crooks equality, at least for some variants. Numerical illustrations demonstrate the efficiency of the approach on various simple test cases, and allow us to compare the variants of the algorithm.

6.2.4 Interacting particle methods for sampling

A number of stochastic numerical methods for optimization and sampling are based on interacting stochastic dynamics. Methods of this type are convenient because they can usually be implemented in parallel, and they may possess desirable properties that single-replica methods do not enjoy, such as faster convergence or invariance under affine transformations.

A well-known numerical method based on an interacting particle system is the ensemble Kalman filter, a methodology for incorporating noisy data into complex dynamical models to enhance predictive capability. This filter is widely adopted in the geophysical sciences, underpinning weather forecasting for example, and is starting to be used throughout the sciences and engineering. For high dimensional filtering problems, the ensemble Kalman filter has a robustness that is not shared by the particle filter; in particular it does not suffer from weight collapse. However, there is no theory which quantifies its accuracy as an approximation of the true filtering distribution, except in the Gaussian setting.

Study of the statistical accuracy of the ensemble Kalman filter is inherently technical, as it involves the evolution of probability measures according to a nonlinear and nonautonomous dynamical system. In 45, U. Vaes together with José A. Carrillo (University of Oxford, United Kingdom), Franca Hoffmann (Caltech, USA) and Andrew M. Stuart (Caltech, USA) provide an accessible overview of previous work in which they took first steps to analyze the accuracy of the filter beyond the linear Gaussian setting 17.

Building on this work, in 42 U. Vaes together with Edoardo Calvello (Caltech, USA), Pierre Monmarché (Sorbonne Université) and Andrew M. Stuart (Caltech, USA) conduct an analysis of the accuracy of the ensemble Kalman filter in the setting of near-linear dynamical models and observation operators, a concrete setting in which the filtering distribution is provably close to Gaussian. The analysis combines stability estimates for the mean field ensemble Kalman filter with a classical propagation of chaos estimate for the filter. It demonstrates that, in the near-linear and many-particle regime, the ensemble Kalman filter accurately approximates the true filtering distribution.

In 30, U. Vaes generalizes prior results concerning the mean field limit of the so-called ensemble Kalman sampler, a sampling method closely related to the ensemble Kalman filter and based on a collection of Langevin dynamics interacting via the ensemble covariance. The article provides a sharp, local-in-time propagation of chaos result for the sampler under mild assumptions on the logarithm of the target probability distribution. The proof employs a classical synchronous coupling approach and uses a stopping time argument to handle the merely local Lipschitz continuity of the interaction term. In addition, the work establishes new results on the well-posedness nonlinear process arising in the mean field limit.

In 10, U. Vaes together with Rafael Bailo (TU Delft), Alethea Barbaro (TU Eindhoven), Susana Gomes (University of Warwick), Konstantin Riedl (University of Oxford), Tim Roith (DESY – Deutsches Elektronen-Synchrotron), and Claudia Totzeck (University of Wuppertal), describe and motivate their work on developing robust and efficient implementations of consensus-based methods for optimization and sampling. This effort originated at the 2023 workshop “Purpose-driven particle systems” organized in March 2023 at the Lorentz center in Leiden. Led by Rafael Bailo (on the Julia side) Tim Roith (on the Python side), it resulted in the development of free software libraries written in Julia and Python (see the Github repository).

6.3 Homogenization

Participants: Amandine Boucart, Claude Le Bris, Albéric Lefort, Frédéric Legoll, Simon Ruget.

6.3.1 Homogenization theory

In collaboration with Yves Achdou (Université Paris-Cité), C. Le Bris has pursued the study of some homogenization problems for a class of stationary Hamilton-Jacobi equations in which the Hamiltonian is obtained by perturbing an otherwise periodic Hamiltonian. Homogenization then leads to an effective Hamilton-Jacobi equation supplemented with effective Dirichlet boundary conditions. After the case of a perturbation at the origin, the case of a perturbation near a half-line of the state space has been considered in 38. In all these cases, the limiting problem belongs to the class of stratified problems introduced by Alberto Bressan and Yunho Hong and later studied by Guy Barles and Emmanuel Chasseigne.

On the other hand, and this time in a series of works in collaboration with Andrea Braides and Giani Dal Maso (SISSA Trieste, Italy), C. Le Bris has used the tools of Gamma convergence to study several homogenization problems he has considered in the past using the tools of PDE theory, notably in collaboration with Xavier Blanc (Université Paris Cité) and Pierre-Louis Lions (Collège de France). In a first work 14, the stability of some classes of integrals, with respect to homogenization, was examined. Stability theorems in homogenization which comprise the case of perturbations with zero average on the whole space were then deduced. The results were also extended to the stochastic case, and several other cases. In a second work 41, the stability with respect to homogenization of classes of integrals arising in the control-theoretic interpretation of some Hamilton–Jacobi equations were investigated. The study revisits and, depending on the different assumptions, complements results obtained by Pierre-Louis Lions and various collaborators using PDE techniques.

A third research effort in the area of homogenization theory has been conducted by C. Le Bris in collaboration with Kento Kaneko and Anthony T. Patera (MIT), see 60, 61. The general context of the study is heat transfer, and more specifically the dunking problem: a solid body, at uniform temperature and possibly with heterogeneous material composition, is placed in an environment characterized by far field temperature and spatially uniform time independent heat transfer coefficient. The problem is described by a heat equation with Robin boundary conditions. The crucial parameter is the Biot number — a nondimensional heat transfer (Robin) coefficient. Various approximations were introduced, justified theoretically and tested numerically. Error estimates for these approximations were also provided and companion numerical solutions of the heat equation confirm the effectiveness of the error estimates for small Biot number.

6.3.2 Inverse multiscale problems

In the context of the PhD of S. Ruget, C. Le Bris and F. Legoll have pursued their work on the question of how to determine the homogenized coefficient of a multiscale problem without explicitly performing a homogenization approach. This work is a follow-up on earlier works over the years in collaboration with Kun Li, Simon Lemaire and Olga Gorynina. Current efforts are focused on investigating the robustness of the approach with respect to the available data. In particular, an attractive situation is to assume to only have access to coarse information on the solution, such as the global energy stored in the physical system for any given load (such inverse problems with partial information available are indeed relevant to engineering situations). While the reconstruction of the microstructure is known to be an ill-posed problem, the reconstruction of effective parameters based only on this aggregated information is possible. A manuscript collecting various theoretical and numerical results is currently in preparation.

6.3.3 Multiscale Finite Element approaches

From a numerical perspective, the Multiscale Finite Element Method (MsFEM) is a classical strategy to address the situation where the homogenized problem is not known (e.g. in difficult nonlinear cases), or when the scale of the heterogeneities, although small, is not considered to be zero (and hence the homogenized problem cannot be considered as a sufficiently accurate approximation). The MsFEM approach uses a Galerkin approximation of the problem on a pre-computed basis, obtained by solving local problems mimicking the problem at hand at the scale of mesh elements.

Together with Rutger Biezemans (now at CEA) and Alexei Lozinski (Université de Franche-Comté), C. Le Bris and F. Legoll have completed the writing of a manuscript (now published as 12) adressing the question of how to design accurate MsFEM approaches for the case of multiscale advection-diffusion problems, in the advection-dominated regime. Thin boundary layers are present in the exact solution, and numerical approaches should be carefully adapted to this situation, e.g. using stabilization. How stabilization and the multiscale nature of the problem interplay with one another is a challenging question, and several MsFEM variants have been compared. It is shown in 12 how MsFEM with weak continuity conditions of Crouzeix-Raviart type can be stabilized by adding specific bubble functions, satisfying the same type of weak boundary conditions.

In the context of the PhD of A. Lefort, C. Le Bris and F. Legoll have undertaken the study of a multiscale, reaction-diffusion equation. This problem is different from the equations previously studied by the team by the fact that it includes a large reaction term which competes with the diffusive term. From a numerical perspective, two difficulties are present in the time-dependent version of the problem. First, the coefficients of the equation (and therefore the solution) oscillate at a small spatial scale. In addition, the problem in time is stiff. In order to not address all difficulties at the same time, the associated eigenvalue problem has been considered. A promising MsFEM-type approach, which uses an oversampling procedure based on filtering ideas investigated by the team several years ago, has been introduced. Current investigations are targeted toward the extension of this MsFEM approach to vector-valued reaction-diffusion eigenvalue problemss, a very relevant case from the application viewpoint.

In parallel to the exploration of advection-diffusion equations and reaction-diffusion equations, another direction of research is focused on hyperbolic multiscale conservation laws. The homogenized limit of a large class of such conservation laws has recently been established in the literature. As a preliminary work on this topic, A. Boucart (who left the team this year at the end of her post-doc position), C. Le Bris and F. Legoll have put in action the classical homogenization approach, and numerically demonstrated that the two-scale approximation provided by homogenization theory is indeed an accurate approximation of the reference solution. Another question, currently under investigation, is to perform homogenization on the numerical scheme used to discretize the PDE (in contrast to performing homogenization on the PDE itself). Several alternatives in that direction are being compared.

6.4 Various topics

6.4.1 Complex fluids

Participants: Sébastien Boyaval.

In 2024, S. Boyaval has pursued his reformulation of viscoelastic systems for non-Newtonian fluids using symmetric-hyperbolic balance laws consistent with polyconvex elastodynamics. In 72, he proved with Na Wang and Yuxi Hu that for one-dimensional flows, the symmetric-hyperbolic conservation laws are asymptotically consistent with the standard viscous compressible Navier-Stokes equaions, when the viscoelastic stress relaxes infinitely fast.

In 71, S. Boyaval and his co-authors Jean-Paul Travert, Cédric Goeury, Vito Bacchi and Fabrice Zaoui study numerically, for various metrics, the sensitivity of the distance between a flood map inferred from a satellite, and a distribution of numerical flood maps, with respect to various parameters of the procedure. It results in a selection of only a few metrics actually useful for data assimilation.

6.4.2 Model-order reduction methods

Participants: Sébastien Boyaval, Virginie Ehrlacher, Tony Lelièvre, Giulia Sambattaro.

The objective of a model-order reduction method is the following: it may sometimes be very expensive from a computational point of view to simulate the properties of a complex system described by a complicated model, typically a set of PDEs. This cost may become prohibitive in situations where the solution of the model has to be computed for a very large number of values of the parameters involved in the model. Such a parametric study is nevertheless necessary in several contexts, for instance when the value of these parameters has to be calibrated so that numerical simulations give approximations of the solutions that are as close as possible to some measured data. A reduced-order model method then consists in constructing, from a few complex simulations that were performed for a small number of well-chosen values of the parameters, a so-called reduced model, much cheaper and quicker to solve from a numerical point of view, and which enables to obtain an accurate approximation of the solution of the model for any other values of the parameters.

In 70, S. Boyaval and his co-authors Guillaume Enchéry, Jana Tarhini, Quang Huy Tran, build a fast and reliable surrogate model of compressible porous-media flows with uncertain permeabilities, useful in the context of underground storage in deep aquifers (of $C O_{2}$ typically), using the most up-to-date Reduced-Basis techniques and a new a posteriori error estimator based on a fixed (i.e. parameter-independent) norm.

In 59, together with Alexandre Ern (SERENA) and collaborators from SAFRANTech (Abbas Kabalan, Fabien Casenave and Felipe Bordeu), V. Ehrlacher developed a new algorithm to compute morphings between different domains with a view to using them to construct new efficient nonlinear reduced-order models to accelerate parametric studies in Computational Fluid Dynamics. Abbas Kabalan and Fabien Casenave received the first prize of the NeurIPS 2024 ML4CFD challenge thanks to the methodology developed in this work.

6.4.3 Cross-diffusion systems

Participants: Jean Cauvin-Vila, Virginie Ehrlacher.

Cross-diffusion systems are nonlinear degenerate parabolic systems that naturally arise in diffusion models of multi-species mixtures in a wide variety of applications: tumor growth, population dynamics, materials science etc. In materials science they typically model the evolution of local densities or volumic fractions of chemical species within a mixture.

In 15, Clément Cancès, Jean Cauvin-Vila, Claire Chainais-Hillairet and V. Ehrlacher developed a new structure-preserving numerical scheme for a model of a physical vapor deposition process used for the fabrication of thin film layers. The model involves two different types of cross-diffusion systems coupled by an evolving interface. The moving interface is addressed with a cut-cell approach, where the mesh is locally deformed around the interface. The scheme is shown to preserve the structure of the continuous system, namely: mass conservation, nonnegativity, volume-filling constraints and decay of the free energy. Moreover, existence of structure-preserving solutions of the discrete scheme has been proved. New results have also been obtained on the characterization of non-trivial steady states for the continuous system and on necessary and sufficient conditions under which the latter can be proved to exist.

6.4.4 Numerical approaches for SPDEs

Participants: Claude Le Bris.

In collaboration with Ana Djurdjevac (FU Berlin) and Endre Suli (Oxford University), C. Le Bris has considered some dedicated numerical approaches for a large class of parabolic SPDEs with multiplicative noise. The specificity of the class considered is that it preserves positivity at the continuous level. Inspired by well-established techniques for the deterministic case, a FEM discretization was introduced that both is accurate and, unconditionally in the discretization parameters, also preserves positivity. Besides the numerical analysis, some numerical experiments were provided, which illustrate the added value with respect to other approaches in the literature. This work will soon be submitted for publication.

6.4.5 Modelisation and simulation of structure fatigue

Participants: Frederic Legoll.

In collaboration with Francois-Baptiste Cartiaux (OSMOS Group) and Julien Reygner (ENPC), and in the context of the PhD of Alex Libal, F. Legoll has investigated the probabilistic nature of fatigue life in structures subjected to cyclic loading with variable amplitude. In the article 18, the methodology introduced by the same authors in a previous work is applied to estimate the survival probability of an industrial structure using experimental data. The study considers both the randomness in the initial state of the structure and in the amplitude of loading cycles, which both have an impact on the randomness of the life before failure of the structure. The results indicate that the variability of loading cycles can be captured through the concept of deterministic equivalent damage, providing a computationally efficient method for assessing the fatigue life of the structure. The proposed approach is also compared with some standard approaches of the fatigue community.

7 Bilateral contracts and grants with industry

Participants: Claude Le Bris, Frédéric Legoll, Tony Leliève, Gabriel Stoltz.

Many research activities of the project-team are conducted in close collaboration with private or public companies: CEA, EDF, IFPEN, Sanofi, OSMOS Group, SAFRANTech. The project-team is also supported by the Office of Naval Research and the European Office of Aerospace Research and Development, for multiscale simulations of random materials. All these contracts are operated at and administrated by the École des Ponts, except the contracts with IFPEN, which are administrated by Inria.

8 Partnerships and cooperations

8.1 International initiatives

T. Lelièvre, G. Stoltz and F. Legoll participate in the Laboratoire International Associé (LIA) CNRS / University of Illinois at Urbana-Champaign on complex biological systems and their simulation by high performance computers. This LIA involves French research teams from Université de Nancy, Institut de Biologie Structurale (Grenoble) and Institut de Biologie Physico-Chimique (Paris). The LIA has been last renewed on January 1st, 2018.

Eric Cancès is one of the PIs of the Simons Targeted Grant “Moiré materials magic” (September 2021 - August 2026). His co-PIs are Allan MacDonald (UT Austin, coordinating PI), Svetlana Jitomirskaya (UC Berkeley), Efthimios Kaxiras (Harvard), Lin Lin (UC Berkeley), Mitchell Luskin (University of Minnesota), Angel Rubio (Max-Planck Institut), Maciej Zworski (UC Berkeley).

8.2 International research visitors

8.2.1 Visits of international scientists

Hugo Touchette

Status
Professor
Institution of origin:
Stellenbosch University
Country:
South Africa
Dates:
June 2024
Context of the visit:
Work on problems related to large deviations in computational statistical physics
Mobility program/type of mobility:
Invited professor fellowship from the COMUE Paris-Est

8.3 European initiatives

8.3.1 H2020 projects

EMC2

Participants: Noé Blassel, Eric Cancès, Shiva Darshan, Alfred Kirsch, Eloïse Letournel, Solal Perrin-Roussel, Régis Santet, Renato Spacek, Gabriel Stoltz, Laurent Vidal, Urbain Vaes.

EMC2 project on cordis.europa.eu

Title:
Extreme-scale Mathematically-based Computational Chemistry
Duration:
From September 1, 2019 to February 28, 2026
Partners:
- Institut National de Recherche en Informatique et Automatique (INRIA), France
- École Polytechnique Fédérale de Lausanne (EPFL), Switzerland
- École Nationale des Ponts et Chaussées (ENPC), France
- Centre National de la Recherche Scientifique (CNRS), France
- Sorbonne Université, France
Inria contact:
Laura GRIGORI (Alpines)
Coordinators:
Eric Cancès (ENPC), Laura Grigori (Inria Alpines), Yvon Maday (Sorbonne Université), J.-P. Piquemal (Sorbonne Université)
Summary:

Molecular simulation has become an instrumental tool in chemistry, condensed matter physics, molecular biology, materials science, and nanosciences. It will allow to propose de novo design of e.g. new drugs or materials provided that the efficiency of underlying software is accelerated by several orders of magnitude.

The ambition of the EMC2 project is to achieve scientific breakthroughs in this field by gathering the expertise of a multidisciplinary community at the interfaces of four disciplines: mathematics, chemistry, physics, and computer science. It is motivated by the twofold observation that, i) building upon our collaborative work, we have recently been able to gain efficiency factors of up to 3 orders of magnitude for polarizable molecular dynamics in solution of multi-million atom systems, but this is not enough since ii) even larger or more complex systems of major practical interest (such as solvated biosystems or molecules with strongly-correlated electrons) are currently mostly intractable in reasonable clock time. The only way to further improve the efficiency of the solvers, while preserving accuracy, is to develop physically and chemically sound models, mathematically certified and numerically efficient algorithms, and implement them in a robust and scalable way on various architectures (from standard academic or industrial clusters to emerging heterogeneous and exascale architectures).

EMC2 has no equivalent in the world: there is nowhere such a critical number of interdisciplinary researchers already collaborating with the required track records to address this challenge. Under the leadership of the 4 PIs, supported by highly recognized teams from three major institutions in the Paris area, EMC2 will develop disruptive methodological approaches and publicly available simulation tools, and apply them to challenging molecular systems. The project will strongly strengthen the local teams and their synergy enabling decisive progress in the field.

TIME-X

Participants: Frédéric Legoll, Tony Lelièvre.

TIME-X project on openaire.eu

Title:
TIME parallelisation: for eXascale computing and beyond
Duration:
From April 1, 2021 to March 31, 2024
Partners:
- KU Leuven, Belgium
- École Nationale des Ponts et Chaussées (ENPC), France
- Sorbonne Université, France
- University of Wuppertal, Germany
- Forschungszentrum Jülich, Germany
- Universita della Svizzera Italiana, Switzerland
- University of Geneva, Switzerland
- TU Darmstadt, Germany
- TU Munich, Germany
- Hamburg University of Technology, Germany
Coordinators:
Yvon Maday (Sorbonne Université) and Giovanni Samaey (KU Leuven)
Summary:

Recent successes have established the potential of parallel-in-time integration as a powerful algorithmic paradigm to unlock the performance of Exascale systems. However, these successes have mainly been achieved in a rather academic setting, without an overarching understanding. TIME-X will take the next leap in the development and deployment of this promising new approach for massively parallel HPC simulation, enabling efficient parallel-in-time integration for real-life applications. We will:

(i) provide software for parallel-in-time integration on current and future Exascale HPC architectures, delivering substantial improvements in parallel scaling;

(ii) develop novel algorithmic concepts for parallel-in-time integration, deepening our mathematical understanding of their convergence behaviour and including advances in multi-scale methodology;

(iii) demonstrate the impact of parallel-in-time integration, showcasing the potential on problems that, to date, cannot be tackled with full parallel efficiency in three diverse and challenging application fields with high societal impact: weather and climate, medicine and fusion.

To realise these ambitious, yet achievable goals, the inherently inter-disciplinary TIME-X Consortium unites top researchers from numerical analysis and applied mathematics, computer science and the selected application domains. Europe is leading research in parallel-in-time integration. TIME-X unites all relevant actors at the European level for the first time in a joint strategic research effort. A strategic investment from the European Commission would enable taking the necessary next step: advancing parallel-in-time integration from an academic/mathematical methodology into a widely available technology with a convincing proof of concept, maintaining European leadership in this rapidly advancing field and paving the way for industrial adoption.

HighLEAP

Participants: Virginie Ehrlacher, Clément Guillot, Mathias Dus, Rodrigue Lelotte, Giulia Sambataro.

EMC2 project on cordis.europa.eu

Title:
High-dimensional mathematical methods for LargE Agent and Particle Systems
Duration:
From December 1, 2023 to November 30, 2028
Partners:
- École Nationale des Ponts et Chaussées (ENPC), France
Coordinators:
Virginie Ehrlacher (ENPC)
Summary:
Interacting particle or agent-based systems are ubiquitous in science. They arise in an extremely wide variety of applications including materials science, biology, economics and social sciences. Several mathematical models exist to account for the evolution of such systems at different scales, among which stand optimal transport problems, Fokker-Planck equations, mean-field games systems or stochastic differential equations. However, all of them suffer from severe limitations when it comes to the simulation of high-dimensional problems, the high-dimensionality character coming either from the large number of particles or agents in the system, the high amount of features of each agent or particle, or the huge quantity of parameters entering the model. The objective of this project is to provide a new mathematical framework for the development and analysis of efficient and accurate numerical methods for the simulation of high-dimensional particle or agent systems, stemming from applications in materials science and stochastic game theory.

The main challenges which will be addressed in this project are:
- sparse optimization problems for multi-marginal optimal transport problems, using moment constraints;
- numerical resolution of high-dimensional partial differential equations, with randomized iterative algorithms;
- efficient approximation of parametric stochastic differential equations, by means of reduced-order modeling approaches.
The potential impacts of the project are huge: making possible such extreme-scale simulations will enable to gain precious insights on the predictive power of agent- or particle-based models, with applications in various fields, such as quantum chemistry, molecular dynamics, crowd motion or urban traffic.

8.4 National initiatives

The project-team is involved in several ANR projects:

F. Legoll is a member of the ANR Anohona (2024-2028), on Advanced nonlinear homogenization for structural analysis. PI: N. Lahellec (LMA Marseille).
G. Stoltz is the PI of the ANR project SINEQ (2022-2025), whose aim is to improve the mathematical understanding and numerical simulation of nonequilibrium stochastic dynamics, in particular their linear response properties. This project involves researchers from CEREMADE, Université Paris-Dauphine and the SIMSART project-team of Inria Rennes.
U. Vaes is the PI of the ANR project ISPO (2024-2025). The main objectives of this project are to improve, implement and mathematically analyze sampling and optimisation methods based on interacting particle systems.

The project-team is a partner of the DIM QuanTiP. It is also involved in the Projet CNRS Recherche à risque et à impact (RI)2 “Nouvelles approches mathématiques pour des systèmes quantiques en interaction (MAQUI)”, lead PI: M. Lewin (CEREMADE, CNRS and University Paris-Dauphine PSL), co-PI: E. Cancès, J. Toulouse (LCT, SU).

The project-team is also involved in PEPR projects:

T. Lelièvre is responsible of the node "Ecole des Ponts" of the project MAMABIO of PEPR B-BEST (Biomass, Biotechnologies & Environmentally Sustainable Technologies for chemicals and fuels; 2023-2028), to which G. Stoltz also participates.
E. Cancès, C. Le Bris , T. Lelièvre and G. Stoltz are part of the node "MATHERIALS" of the project EpiQ of PEPR Quantique, which is part of Plan France 2030.

Members of the project-team are participating in the following GdR or RT:

AMORE (Advanced Model Order REduction),
DYNQUA (time evolution of quantum systems),
MathGeoPhy (MAthematics for GeoPhysics), now RT Terre et Energies,
MANU (MAthematics for NUclear applications), now RT Terre et Energies,
GDM (Geometry and Mechanics),
IAMAT (Artificial Intelligence for MATerials),
MASCOT-NUM (stochastic methods for the analysis of numerical codes),
MEPHY (multiphase flows),
NBODY (electronic structure),
REST (theoretical spectroscopy).

The project-team is involved in the Labex Bezout (2011-2024).

9 Dissemination

9.1 Promoting scientific activities

S. Boyaval

is the director of Laboratoire d’Hydraulique Saint-Venant (Ecole des Ponts ParisTech - EDF R&D - CEREMA), since September 2021;
is currently a member of the RA1 (scientific committee) and CODIR+ (executive committee) of E4C.

E. Cancès

is a member of the editorial boards of Mathematical Modelling and Numerical Analysis $(2006-)$ , SIAM Multiscale Modeling and Simulation $(2012-)$ , the Journal of Computational Mathematics (2017-), and the Journal of Computational Physics (2023-);
is a member of the Scientific Committee of the MFO (Mathematisches Forschungsinstitut Oberwolfach);
is a member of the Scientific Committees of the GdR DynQua (quantum dynamics), NBody ( $N$ -body quantum problem in chemistry and physics), and Rest (Theoretical spectroscopy);
has co-organized (with E. Fromager, E. Giner, P.-F. Loos, and J. Toulouse) a summer school on Mathematics for theoretical chemistry and physics, Paris, May 28-31, 2024;
has co-organized (with L. Grigori, Y. Maday and J.-P. Piquemal) a workshop on Mathematical and numerical methods for electronic structure calculation and molecular dynamics, July 1-5, 2024, Roscoff, France
has co-organized (with S. Jitomirskaya, M. Luskin and A. MacDonald) a workshop on 2D and moiré materials, July 8-12, 2024, Roscoff, France
has co-organized (with G. Friesecke, Y. Maday, R. Schneider, B. Stamm, H. Yserentant and A. Zhou) the MANUEL conference, September 16-20, 2024, Stuttgart, Germany
has co-organized (with M. Esteban, G. Galli, L. Lin, A. Rodriguez and A. Tkatchenko) an IPAM workshop on "Advancing quantum mechanics with mathematics and statistics", December 9-13, 2024, UCLA, Lake Arrowhead, USA

V. Ehrlacher

is a member of the editorial boards of Mathematical Modelling and Numerical Analysis $(2024-)$ , Acta Applicandae Mathematicae $(2024-)$ and Mathematics of Computation $(2024-)$ ;
is head of the Modelisation, Analysis and Simulation team of the applied mathematics department (CERMICS) at Ecole des Ponts (since Sep. 2024);
is member of the board of the SMAI-SIGMA group;
is co-chair of the European Mathematical Society Topic Activity Group on "Scientific Machine Learning";
is an expert for the Scientific Committee of IFPEN (since 2024);
is a member of the Programme INRIA Quadrant (PIQ) (since 2024);
is a member of the “Conseil d'Administration” of Ecole des Ponts;
is a member of the “Conseil d'Administration” of the COMUE Paris-Est;
was a member of the scientific committee of the CANUM 2024 conference;
was a member of the evaluation committee of the Weierstrass Institute for Applied Analysis and Stochastics (WIAS), Berlin, Germany;
is a member of the scientific committee of the SMAI 2025 conference;
has been a member of the SMAI-GAMNI PhD prize selection committee.

C. Le Bris

is co-editor in chief (2024-) of Journal de Mathématiques Pures et Appliquées;
is a member of the editorial boards of Annales mathématiques du Québec (2013-), Archive for Rational Mechanics and Analysis (2004-), Calcolo (2019-), Communications in Partial Differential Equations (2022-), COCV (Control, Optimization and Calculus of Variations) (2003-), Mathematics in Action (2008-), Networks and Heterogeneous Media (2007-), Nonlinearity (2005-), Pure and Applied Analysis (2018-);
is a member of the editorial boards of the monograph series Mathématiques & Applications, Springer (2008-), Modelling, Simulations and Applications, Springer (2009-), Springer Monographs in Mathematics, Springer (2016-);
is the president (2016-2024) of the scientific advisory board of the Institut des Sciences du calcul et des données, Sorbonne Université, and a member (2020-) of the Scientific Advisory Committee of the Institute for Mathematical and Statistical Innovation, University of Chicago;
is a member (2019-) of the scientific advisory board of Framatome;
is the Vice-director of the French Education Committee for the China-France Mathematics Talents Class, Université Paris Cité, 2024-2029.

F. Legoll

is a member of the editorial boards of SIAM MMS (2012-), ESAIM: Proceedings and Surveys (2012-) and the Journal of Machine Learning for Modeling and Computing (2024-);
has co-organized (with G. Samaey) the minisymposium "Multilevel and asymptotic-preserving methods for uncertainty quantification in multiscale systems" within the SIAM UQ 2024 conference (Trieste), February 27 - March 1, 2024;
has co-organized (with L. Balazi and P. Omnes) the minisymposium "Méthodes multi-échelles pour les EDP" within the CANUM 2024 conference (Ile de Ré), May 27-31, 2024.

T. Lelièvre

is a member of the editorial boards of SIAM/ASA Journal of Uncertainty Quantification (2017-), IMA: Journal of Numerical Analysis (2018-), Communications in Mathematical Sciences (2019-), Journal of Computational Physics (2019-), ESAIM:M2AN (2020-), and Foundations of Computational Mathematics (2022-);
is an expert for the Scientific Committee of IFPEN (since 2022);
is the Chair of the External Advisory Board, Mathematical Theory of Radiation Transport: Nuclear Technology Frontiers (MaThRad) (since 2023);
is a member of the "Bureau de Comités des équipes-Projets d'Inria Paris" (since 2022; until Aug. 2024);
is an external member of the Conseil Scientifique et de Prospective of the Institut de Mathématiques de Toulouse (since 2023).

G. Stoltz

is the head of the applied mathematics department (CERMICS) at Ecole des Ponts (since Sep. 2024);
is a member of the editorial board of Journal of Computational Dynamics;
is a member of the Executive Board of GdR IAMAT (Artificial Intelligence and Materials Science);
is a member of the "Conseil d'Enseignement et de Recherche" of Ecole des Ponts and of the Faculty Board of EELISA (European Engineering Learning Innovation Science Alliance);
co-organized the 3-month research program “Data-Driven Materials Informatics” at IMSI, University of Chicago (March-May 2024);
was a member of the HCERES evaluation jury of SAMM (Université Paris Panthéon; January 2024).

9.2 Teaching - Supervision - Juries

9.2.1 Teaching

The members of the project-team have taught the following courses.

At École des Ponts 1st year (equivalent to L3):

Analyse et équations aux dérivées partielles, 30h (T. Borsoni, M. Dus, V. Ehrlacher, R. Gastadello, A. Massimini, A. Lefort, R. Lelotte, E. Loko, G. Sambataro, S. Ruget, S. Ezzehi, C. Guillot, T. Duez)
Bases scientifiques pour transition énergétique, 10h30 (L. Carillo)
Équations aux dérivées partielles: approches variationnelles, 15h (S. Darshan, F. Legoll, S. Ruget, R. Santet)
Initiation au travail en projet, 15h (R. Santet, R. Spacek, L. Vidal)
Introduction à l'optimisation (M. Dus)
Mise à niveau en mathématiques, 16h (E. Cancès)
Mécanique des milieux continus fluides, 25h (S. Boyaval)
Mécanique quantique, 15h (E. Cancès, A. Kirsch, A. Negre)
Pratique du calcul scientifique, 18h (N. Blassel, U. Vaes)
Probabilités, 24h (N. Blassel)
Programmation en Python, 24h (L. Carillo, G. Sambataro)

At École des Ponts 2nd year (equivalent to M1):

Contrôle de systèmes dynamiques et équations aux dérivées partielles, 18h (E. Cancès)
Problèmes d'évolution, 10h (F. Legoll)
Techniques de developpement logiciel, 32h30 (É. Polack)

At the M2 “Mathématiques de la modélisation” of Sorbonne Université:

Équations aux dérivées partielles et modélisation, 24h (F. Legoll)
Introduction to computational statistical physics, 20h (G. Stoltz)
Méthodes de tenseurs par la résolution d'équations aux dérivées partielles en grande dimension, 20h (V. Ehrlacher)
Théorie spectrale et méthodes variationnelles, 10h (E. Cancès)

At other institutions:

Aléatoire (MAP361), 40h, Ecole Polytechnique (V. Ehrlacher)
Gestion des incertitudes et analyse de risque (MAP568), 20h, Ecole Polytechnique (T. Lelièvre)
Introduction to Machine Learning, 64h, Institut polytechnique Paris, M1 Applied mathematics and statistics (G. Stoltz)
Modal de Mathématiques Appliquées (MAP473D), 15h, Ecole Polytechnique (T. Lelièvre)
Modélisation de phénomènes aléatoires (MAP432), 40h, Ecole Polytechnique (V. Ehrlacher, T. Lelièvre)
Optimisation et Transport Optimal, 38h, ENS Paris (S. Perrin-Roussel)
Numerical Analysis, 56h, NYU Paris (U. Vaes)
Théorie spectrale et mécanique quantique, 26h, ENS Paris (S. Perrin-Roussel)

9.2.2 Supervision

The following PhD theses supervised by members of the project-team have been defended:

Elisa Beteille, thèse CIFRE EDF, Propagation of Urban Flood waves, supervised by S. Boyaval (and F. Larrarte from UGE), defended in November 74
Alfred Kirsch, funding Simons foundation, Mathematical and numerical analysis of embedding methods in quantum mechanics, École des Ponts, co-supervised by E. Cancès and D. Gontier (Paris-Dauphine CEREMADE), defended in November
Eloïse Letournel, funding DIM Math Innov (Inria), Analyse théorique et numérique de modèles statiques et dynamiques en calcul de structure électronique, École des Ponts, supervised by A. Levitt, defended in November
Régis Santet, funding Ecole des Ponts, Foundations and optimization of Langevin samplers, Ecole des Ponts, co-supervised by T. Lelièvre and G. Stoltz, defended in December
Lev-Arcady Sellem, funding Advanced ERC Q-Feedback (PI: P. Rouchon), Codes bosoniques et réservoirs quantiques: dompter l’environnement, thèse Mines Paris-Tech, co-supervised by C. Le Bris and P. Rouchon (Inria QUANTIC), defended in March 37
Renato Spacek, funding FSMP CoFund, Efficient computation of transport coefficients in molecular dynamics, ED 386 Sorbonne-Université, co-supervised by G. Stoltz and P. Monmarché (Sorbonne Université), defended in December
Jana Tarhini, funded by Inria-IFPEN, Fast simulation of CO2/H2 storage in geological bassins, supervised by S. Boyaval (and G. Enchéry, H. Tran from IFPEN), defended in November 76
Laurent Vidal, funding ERC Synergy EMC2, Model reduction in physics and quantum chemistry, supervised by E. Cancès and A. Levitt (Université Paris-Saclay), defended in June

The following PhD theses supervised by members of the project-team are ongoing:

Noé Blassel, funding ERC Synergy EMC2, Approximation of the quasi-stationnary distribution, Ecole des Ponts, since October 2022, co-supervised by T. Lelièvre and G. Stoltz
Thomas Brunel, funded by ANR Neptune, Paddle sports physics: Velocity–stroke rate and active drags, supervised by S. Boyaval (and R. Carmigniani from ENPC)
Louis Carillo, funded by a CDSN fellowship with additional funding from ENPC, Mathematical analysis and numerical methods for metastable systems in statistical physics, since September 2024, co-supervised by T. Lelièvre and U. Vaes
Charlotte Chapellier, funding CIFRE Sanofi, Generative methods for drug design, since October 2023, co-supervised by T. Lelièvre and G. Stoltz
Shiva Darshan, funding ANR SINEQ, Linear response of constrained stochastic dynamics, since October 2022, co-supervised by G. Stoltz and S. Olla (Université Paris-Dauphine)
Théo Duez, funding CNRS, Contributions to the development of new approximations and numerical methods for Time-Dependent Density-Functional Theory (TDDFT) for molecules and materials, since October 2024, co-supervised by E. Cancès and M. Lewin (CEREMADE, CNRS and Université Paris-Dauphine PSL)
François Escolan, funding ERC HighLEAP, Stochastic particle methods for optimal transport, since November 2024, co-supervised by V. Ehrlacher, J. Reygner (CERMICS) and A. Alfonsi (MATHRISK).
Sofiane Ezzehi, funding Région Île-de-France and IFPEN, Nonlinear reduced-order modeling techniques for underground CO2 storage applications, since November 2024, co-supervised with G. Enchéry (IFPEN).
Raphaël Gastaldello, funding CNRS, Variance reduction methods for the computation of transport coefficients, since December 2023, co-supervised by G. Stoltz and U. Vaes
Clément Guillot, funding ENPC, Space-time variational principles for the Schrödinger equation in large dimension, since November 2023, co-supervised by V. Ehrlacher and M. Dupuy (Sorbonne Université)
Abbas Kabalan, thèse CIFRE SAFRANTech, Reduced-order models for problems with non-parametric geometrical variations, since November 2022, co-supervised by V. Ehrlacher and F. Casenave (SAFRANTech)
Albéric Lefort, funding CERMICS-ENPC, Multiscale numerical methods for reaction-diffusion equations and related problems, Ecole des Ponts, since November 2022, co-supervised by F. Legoll and C. Le Bris
Pierre Marmey, funding IFPEN, Evaluation of reaction constants using approaches coupling machine learning and quantum chemistry, since October 2023, co-supervised by T. Lelièvre and P. Raybaud (IFPEN), together with G. Stoltz and M. Corral-Valero (IFPEN)
Alicia Negre, funding Inria, Quantum computing for quantum embedding methods, since October 2023, co-supervised by E. Cancès and T. Ayral (Eviden)
Solal Perrin-Roussel, funding ENPC and ERC Synergy EMC2, Mathematical analysis and numerical simulation of electronic transport in moiré materials, co-supervised by É. Cances and by D. Gontier (CEREMADE, Université Paris-Dauphine PSL)
Jean Ruel, funding ENS-Saclay, Certified and robust reduced models for the simulation of elongated structures, Ecole des Ponts, since October 2023, co-supervised by F. Legoll, L. Chamoin (ENS Paris-Saclay) and A. Lebée (Ecole des Ponts)
Simon Ruget, funding Inria, Coarse approximation for a Schrödinger problem with highly oscillatory coefficients, Ecole des Ponts, since October 2022, co-supervised by F. Legoll and C. Le Bris
Jean-Paul Travert, funding CIFRE EDF, Data assimilation for flood predictions, since November 2022, supervised by S. Boyaval (and C. Goeury from EDF)

9.2.3 Juries

Project-team members have participated in the following PhD juries:

S. Boyaval, PhD of Nathan Shourick ("Modélisation mathématique et numérique du mouvement collectif dans les épithéliums"), defended at Université Grenoble Alpes in October (examiner)
S. Boyaval, PhD of Agustin Somacal ("Model reduction for forward simulation and inverse problems: towards non-linear approaches"), defended at Sorbonne Université in April (referee)
E. Cancès, PhD of Siwar Baddredine ("Symétries et structures de rang faible des matrices et tenseurs pour des problèmes en chimie quantique"), defended at SU in March (examiner)
E. Cancès, PhD of Eloïse Letournel ("Analyse théorique et numérique de modèles statiques et dynamiques en calcul de structure électronique"), defended at ENPC in November (examiner)
V. Ehrlacher, PhD of Ioanna-Maria Lygatsika ("Numerical methods for Gaussian discretizations in electronic structure theory problems"), defended at Sorbonne Université in March (referee)
V. Ehrlacher, PhD of Eki Agouzal ("Réduction de modèles en mécanique non-linéaire quasi-statique pour l’estimation de l’état par recalage en assimilation de données : application aux enceintes de confinement"), defended at INRIA Bordeaux in April (referee)
V. Ehrlacher, PhD of Charles Elbar ("Étude mathématique d’équations de type Cahn-Hilliard dégénérées"), defended at Sorbonne Université in May (president)
V. Ehrlacher, PhD of Tobias Blickhan ("Reduced basis methods for problems with moving features"), defended at TU Munich, Germany in July (referee)
V. Ehrlacher, PhD of Willy Haïk ("Hybrid variational data assimilation strategy for real-time monitoring of complex dynamical systems "), defended at Sorbonne Université, in September (referee)
V. Ehrlacher, PhD of Davide Murari ("Neural Networks, Differential Equations, and Structure Preservation"), defended at University of Trondheim, Norway, in October (referee)
V. Ehrlacher, PhD of Michel Fabrice Serret ("Analyse d’algorithmes quantiques variationnels pour la résolution d’équations différentielles en présence de bruit quantique : application à l’équation de Gross-Pitaevskii stationnaire"), defended at Sorbonne Université, in November (president)
V. Ehrlacher, PhD of Alfred Kirsch ("Analyse mathématique et numérique de méthodes de plongement en mécanique quantique"), defended at Ecole Nationale des Ponts et Chaussées in December (president)
F. Legoll, PhD of Loïc Balazi ("Multi-scale Finite Element Method for incompressible flows in heterogeneous media: Implementation and Convergence analysis"), defended at CEA Saclay in September (referee)
T. Lelièvre, PhD of Lucas Journel ("Long-time behavior of some Markov processes, and application to stochastic algorithms "), defended at Sorbonne Université in June (examiner)
T. Lelièvre, PhD of Davide Carbone ("Generative Models as Out-of-equilibrium Particle Systems: the case of Energy-Based Models"), defended at Politecnico di Torino in July (examiner)
T. Lelièvre, PhD of Valentin Milia ("Couplage de modèles de chimie quantique et d'algorithmes haute performance pour l'exploration globale du paysage énergétique de systèmes atomiques et moléculaires"), defended at Université de Toulouse in September (referee)
T. Lelièvre, PhD of Giovanni Michele Cherchi ("Generative Methods for Polymer Dynamics & Structure"), defended at Université Clermont Auvergne in October (president)
T. Lelièvre, PhD of Luke Daniel Shaw ("Geometric Numerical Integration for Hamiltonian Monte Carlo and Extrapolation), defended at Universitat Jaume I in November (referee)
T. Lelièvre, PhD of Xin Huang ("Numerical approximation of quantum canonical statistical observables with mean-field molecular dynamics and machine learning"), defended at KTH in November (opponent)
T. Lelièvre, PhD of Lune Maillard ("Bayesian methods for studying Nuclear Quantum Effects in Material Science"), defended at Sorbonne Université in Novembre (referee).
G. Stoltz, PhD of Iain Souttar ("Multiscale methods for Stochastic Differential Equations and applications"), defended at Heriot-Watt University in February (referee)

Project-team members have participated in the following habilitation juries:

E. Cancès, HdR of Luigi Genovese ("Les opportunités du formalisme ondelettes dans le calcul de structure électronique à haute performance"), defended at Université Grenoble Alpes in May (referee)
E. Cancès, HdR of Thomas Ayral ("Classical and quantum algorithms for many-body problems"), defended at Ecole Polytechnique IPP in November (referee)
V. Ehrlacher, HdR of Tommaso Taddei ("Contributions à la réduction de modèles de systèmes paramétriques en mécanique non linéaire"), defended at INRIA Bordeaux in April (referee)
T. Lelièvre, HdR of Nicolas Martzel ("Nouvelles méthodologies de montée en échelle, du microscopique au mésoscopique"), defended at Université Clermont-Auvergne in January (president)
T. Lelièvre, HdR of Manon Michel ("Analytical and computational developments around long-time behaviors of stochastic processes"), defended at Université Clermont-Auvergne in June (referee)
T. Lelièvre, HdR of Boris Nectoux ("Autour de l’analyse mathématique des réseaux de neurones et de la métastabilité en dynamique moléculaire"), defended at Université Clermont-Auvergne in November (examiner)

Project-team members have participated in the following selection committees:

V. Ehrlacher, professor position at ENSTA
V. Ehrlacher, MCF position at Université Paris-Dauphine
T. Lelièvre, professor position at Université Grenoble Alpes
G. Stoltz, research and teaching position at Ecole des Ponts
G. Stoltz, professor position at Université d'Orléans

9.3 Conference participation

Members of the project-team have delivered lectures in the following seminars, workshops and conferences:

N. Blassel, “Data-Driven Materials Informatics” IMSI long program, Chicago, March and April
N. Blassel, Platform for Advanced Scientific Computing (PASC), Zurich, June
N. Blassel, Journées de Probabilités, Bordeaux, June
N. Blassel, “Kinetic equations, Mathematical physics and Probability“, Bilbao, June
N. Blassel, MOANSI (GAMM) annual meeting, Berlin, November
T. Borsoni, Conference “Numerical Aspects of Hyperbolic Balance Laws and Related Problems”, Ferrara, Italy, December
E. Cancès, Model Systems in Quantum Mechanics (MSQM), Toulouse, January
E. Cancès, BRIN worskhop on Mathematical Models of Electronic Transport and Phases in Low-Dimensional Materials, University of Maryland, USA, March
E. Cancès, GAMM Annual Meeting (plenary conference), Magdeburg, Germany, March
E. Cancès, BIRS worskhop on Modern Methods for Differential Equations of Quantum Mechanics, Banff, Canada, April
E. Cancès, SIAM congress on linear algebra, May, Paris
E. Cancès, ICERM workshop on the Spectral Analysis of Schrödinger Operators, Providence, USA, August
A. Della Noce, Journées de Biostatistique, Paris, November
V. Ehrlacher, Seminar of the LMPS, ENS Paris-Saclay, January
V. Ehrlacher, Seminar "Numerical Analysis of Galerkin ROMs", online, February
V. Ehrlacher, Séminaire Saint-Gobain Recherche, Aubervilliers, February
V. Ehrlacher, Workshop "Mathématiques pour la neutronique", Sorbonne Université, Paris, February
V. Ehrlacher, Séminaire parisien d'Optimisation, Institut Henri Poincaré, Paris, March
V. Ehrlacher, Workshop "Optimal Transport: from theory to applications", Berlin, Germany, March
V. Ehrlacher, Workshop "Exploiting Algebraic and Geometric Structure in Time Integration methods" (plenary talk), Pisa, Italy, April
V. Ehrlacher, Académie des Sciences, Section SMI, May
V. Ehrlacher, 20 ans du Groupe Calcul, Sorbonne Université, Paris, June
V. Ehrlacher, Workshop "New trends in the numerical analysis of PDEs", Lille, June
V. Ehrlacher, SciML 2024 workshop, Strasbourg, July
V. Ehrlacher, Workshop "Multivariate approximation, discretization, and sampling recovery", Isaac Newton Institute, Cambridge, UK, July
V. Ehrlacher, MANUEL 2024 conference, Stuttgart, Germany, September
V. Ehrlacher, Journée Simulation Moléculaire, SAFRANTech, Saclay, October
V. Ehrlacher, SMAI-SIGMA workshop (plenary talk), CIRM, Luminy, October
V. Ehrlacher, FVOT conference, Orsay, November
V. Ehrlacher, Conference on "Uncertainty Quantification for High-Dimensional Problems" (plenary talk), Amsterdam CWI, Netherlands, November
C. Guillot, SIAM Conference on Mathematical Aspects of Materials Science, Pittsburgh, USA
A. Kirsch, Model Systems in Quantum Mechanics (MSQM), Toulouse, January
A. Kirsch, GAMM Annual Meeting, Magdeburg, Germany, March
A. Kirsch, Young Researchers Symposium, International Congress of Mathematical Physics, Strasbourg, June
A. Kirsch, Conference EMC2@Roscoff 2024, Roscoff, July
C. Le Bris, Conference ”Frontiers of the Calculus of Variations: A celebration of the mathematics of Gianni Dal Maso” (50 min plenary lecture), Samos, Greece, September
C. Le Bris, Conference ”Modelling, partial differential equations analysis and computational mathematics in material sciences” (50 min plenary lecture), Prague, Czech Republic, September
C. Le Bris, Joint Chinese-French Conference on Partial Differential Equations and Its Applications, Hong-Kong, November
F. Legoll, 28th Conference on Domain Decomposition (DD28), KAUST, Saudi Arabia, January
F. Legoll, SIAM conference on Uncertainty Quantification, Trieste, Italy, February
F. Legoll, European Mechanics of Materials Conference (EMMC), Madrid, Spain, May
F. Legoll, ECCOMAS congress, Lisbon, Portugal, June
F. Legoll, 9th International Conference on Computational Methods in Applied Mathematics (CMAM), Bonn, Germany, June
F. Legoll, WCCM congress, Vancouver, Canada, July
F. Legoll, HIM workshop on "Multiscale Problems: Algorithms, Numerical Analysis and Computation", Bonn, Germany, August
T. Lelièvre, ANR QuAMProcs, Nantes, January
T. Lelièvre, 6ème séminaire de l'IRL CRM, Montréal, Canada, February
T. Lelièvre, IMSI workshop, Chicago, USA, April
T. Lelièvre, Workshop "Constrained Dynamics, Stochastic Numerical Methods and the Modeling of Complex Systems", Oberwolfach, Germany, May
T. Lelièvre, Seminar One World Stochastic Numerics and Inverse Problems, online, June
T. Lelièvre, Conference EMC2@Roscoff, Roscoff, July
T. Lelièvre, Workshop Stochastic processes under constraints, Bielefeld, Germany (online), August
T. Lelièvre, MathRad workshop, Warwick, England, September
T. Lelièvre, CECAM workshop, Darmstadt, Germany, September
T. Lelièvre, MADSTAT seminar – Toulouse School of Economics, November
T. Lelièvre, Journée IFPEN-Inria, Paris, December
R. Lelotte, Conference "Numerical methods for optimal transport problems, mean field games, and multi-agent dynamics", Valparaíso, Chile, January
R. Lelotte, CANUM congress, Ile de Ré, May
R. Lelotte, Conference MeRiOT, Lake Como, Italy, September
R. Lelotte, SMAI-SIGMA meeting, CIRM, Luminy, October
A. Lefort, GAMM Seminar on Microstructures, Bochum, Germany, January
A. Lefort, CANUM congress, Ile de Ré, May
A. Lefort, ECCOMAS congress, Lisbon, Portugal, June
S. Perrin-Roussel, 2D Materials Workshop, Roscoff, July
S. Perrin-Roussel, DMA Analysis Workshop, Paris, October
S. Ruget, GAMM Seminar on Microstructures, Bochum, Germany, January
S. Ruget, CANUM congress, Ile de Ré, May
S. Ruget, ECCOMAS congress, Lisbon, Portugal, June
G. Sambataro, SIAM conference on Uncertainty Quantification, Trieste, Italy, February
G. Sambataro, YMMOR conference, Stuttgart, Germany, March
G. Sambataro, ECCOMAS congress, Lisbon, Portugal, June
G. Sambataro, WCCM, Vancouver, British Columbia, Canada, July
G. Stoltz, Applied and Computational Mathematics Seminar of Heriot-Watt and University of Edinburgh, United Kingdom, February
G. Stoltz, Mostly Monte Carlo seminar, Paris, May
G. Stoltz, Oberwolfach meeting "Constrained Dynamics, Stochastic Numerical Methods and the Modeling of Complex Systems", Germany, May
G. Stoltz, Workshop "Synergies Between Mathematics, Data Science, and Molecular Simulations in Materials Science", Birmingham, United Kingdom, July
G. Stoltz, Workshop "Uncertainty Quantification in Molecular Simulation", Magdeburg, Germany, August
G. Stoltz, Workshop "The many facets of kinetic theory", ICMS, Edinburgh, United Kingdom, September
G. Stoltz, Quantitative Methods for Finance seminar, University of Birmingham, United Kingdom, October
G. Stoltz, Workshop "Monte Carlo sampling: beyond the diffusive regime", Newton Institute, Cambridge, UK, November
U. Vaes, SIAM conference on Uncertainty Quantification, Trieste, Italy, February
U. Vaes, MINGuS BMS Workshop, Rennes, May
U. Vaes, ECCOMAS congress, Lisbon, Portugal, June
U. Vaes, 9th European Congress of Mathematics, Sevilla, Spain, June
U. Vaes, Journées MAS, Poitiers, August
U. Vaes, Discussions McKean–Vlasov, Jussieu, Paris, October
U. Vaes, Workshop “High-Dimensional Control Problems and Mean-Field Equations with Applications in Machine Learning”, Oberwolfach, Germany, December

Members of the project-team have delivered the following series of lectures:

E. Cancès, Mathematics for theoretical chemistry and physics, “Constrained optimization in quantum chemistry and physics” (6h), Paris, May
E. Cancès, International summer School in electronic structure Theory: electron correlation in Physics and Chemistry (ISTPC), “Mathematical aspects of electronic structure theory” (4h), Aussois, June
V. Ehrlacher, Winter-School on Mathematical Physics of the TRR 352, "Introduction to tensor methods in quantum chemistry" (8h), Kochel, Germany, March
T. Lelièvre, SFB 14841 Spring school, "The quasi-stationary distribution approach to metastability" (4h), Kloster Steinfeld, May

Members of the project-team have presented posters in the following seminars, workshops and international conferences:

L. Carillo, Workshop "Synergies Between Mathematics, Data Science, and Molecular Simulations in Materials Science", Birmingham, United Kingdom, July
L. Grazioli, Conference MANUEL, Stuttgart, September
C. Guillot, Conference MANUEL, Stuttgart, September
C. Guillot, Conference SIGMA, Luminy, October
S. Perrin-Roussel, ICMP 2024, July

Members of the project-team have participated (without giving talks nor presenting posters) in the following seminars, workshops and international conferences:

N. Blassel, Conference EMC2@Roscoff, Roscoff, July
L. Carillo, Workshop "Kinetic equation, mathematical physics and probability", Bilbao, Spain, June
L. Carillo, Workshop "Uncertainty quantification in molecular simulation", Magdeburg, Germany, August
A. Della Noce, MINGuS BMS Workshop, Rennes, May
T. Duez, Conference MANUEL, Stuttgart, September
T. Duez, Minischool GDR NBODY, Paris, May
C. Guillot, Minischool GDR NBODY, Paris, May
C. Guillot, TRR 352 Winter-School, Kochel, March
E. Letournel, Minischool GDR NBODY, Paris, June
E. Letournel, Workshop "Model Systems in Quantum Mechanics", Toulouse, January
S. Perrin-Roussel, Workshop "Model Systems in Quantum Mechanics", Toulouse, January
A. Negre, Minischool GDR NBODY, Paris, June
A. Negre, Conference EMC2@Roscoff, Roscoff, July

9.4 Popularization

V. Ehrlacher gave the scientific popularization conference at the prize ceremony of Olympiades, Lycée Louis-le-Grand.
V. Ehrlacher gave a SMAI/CNAM scientific popularization conference.
V. Ehrlacher did a CHICHE session and participated to the project "Calculottées" with Collège Eugène Delacroix, Roissy-en-Brie.
V. Ehrlacher presented the activities of the MATHERIALS team to INRIA interns (élèves de 3e)
V. Ehrlacher did 3 CHICHE sessions in Collège Pierre Mendès France, Lycée Montaigne and Fénelon Sup.
G. Stoltz did 7 sessions of CHICHE at lycées Maurice Ravel and Hélène Boucher (March: 2; April: 2; May: 1; December: 2)
G. Stoltz did 4 sessions of presentation of research in mathematics for primary school students in May 2024 at Ecole Joliot–Curie, Champs-sur-Marne

10 Scientific production

10.1 Major publications

1 bookX.Xavier Blanc and C.Claude Le Bris. Homogénéisation en milieu périodique... ou non.88Mathématiques et ApplicationsSpringer International Publishing2022HAL DOI back to text
2 bookX.Xavier Blanc and C.Claude Le Bris. Homogenization Theory for Multiscale Problems.21MS&ASpringer Nature Switzerland2023HAL DOI back to text
3 miscE.Eric Cancès, M.Mireille Defranceschi, W.Werner Kutzelnigg, C.Claude Le Bris and Y.Yvon Maday. Computational Quantum Chemistry: A Primer.2003back to text
4 bookE.Eric Cancès, C.Claude Le Bris and Y.Yvon Maday. Mathematical Methods in Quantum Chemistry. An Introduction. (Méthodes mathématiques en chimie quantique. Une introduction.).Mathématiques et Applications (Berlin) 53. Berlin: Springer. xvi, 409~p. 2006back to text
5 bookI.Isabelle Catto, C.Claude Le Bris and P.-L.Pierre-Louis Lions. The Mathematical Theory of Thermodynamic Limits: Thomas-Fermi Type Models.Oxford Mathematical Monographs. Oxford: Clarendon Press. xiii, 277~p.1998back to text
6 bookJ.-F.J.-F. Gerbeau, C.Claude Le Bris and T.Tony Lelièvre. Mathematical Methods for the Magnetohydrodynamics of Liquid Metals.Numerical Mathematics and Scientific Computation. Oxford: Oxford University Press., 324~p.2006back to text
7 bookC.Claude Le Bris and P.-L.Pierre-Louis Lions. Parabolic Equations with Irregular Data and Related Issues: Applications to Stochastic Differential Equations.4De Gruyter Series in Applied and Numerical Mathematics2019back to text
8 bookC.Claude Le Bris. Multi-scale Analysis. Modeling and Simulation. (Systèmes multi-échelles. Modélisation et simulation.).Mathématiques et Applications (Berlin) 47. Berlin: Springer. xi, 212~p.2005back to text
9 bookT.Tony Lelièvre, M.Mathias Rousset and G.Gabriel Stoltz. Free Energy Computations: A Mathematical Perspective.Imperial College Press, 458~p.2010back to text

10.2 Publications of the year

International journals

10 articleR.Rafael Bailo, A.Alethea Barbaro, S.Susana Gomes, K.Konstantin Riedl, T.Tim Roith, C.Claudia Totzeck and U.Urbain Vaes. CBX: Python and Julia packages for consensus-based interacting particle methods.Journal of Open Source Software998March 2024, 6611HAL DOI back to text
11 articleE.Elisa Beteille, F.Frederique Larrarte, S.Sebastien Boyaval, E.Eric Demay and M.-H.Minh-Hoang Le. Dam-break flow over various obstacles configurations.Journal of Hydraulic Research632March 2025, 156-170HAL DOI
12 articleR. A.Rutger A. Biezemans, C.Claude Le Bris, F.Frédéric Legoll and A.Alexei Lozinski. MsFEM for advection-dominated problems in heterogeneous media: Stabilization via nonconforming variants.Computer Methods in Applied Mechanics and Engineering4332025, 117496HAL DOI back to text back to text
13 articleN.Noé Blassel and G.Gabriel Stoltz. Fixing the flux: A dual approach to computing transport coefficients.Journal of Statistical Physics191January 2024, 17HAL DOI
14 articleA.Andrea Braides, G.Gianni Dal Maso and C.Claude Le Bris. A closure theorem for -convergence and $H$ -convergence with applications to non-periodic homogenization.Annales de l'Institut Henri Poincaré C, Analyse non linéaireNovember 2024HAL DOI back to text
15 articleC.Clément Cancès, J.Jean Cauvin-Vila, C.Claire Chainais-Hillairet and V.Virginie Ehrlacher. Cross-diffusion systems coupled via a moving interface.Interfaces and Free Boundaries : Mathematical Analysis, Computation and Applications2025. In press. HAL DOI back to text
16 articleC.Clément Cancès, V.Virginie Ehrlacher and L.Laurent Monasse. Finite Volumes for the Stefan-Maxwell Cross-Diffusion System.IMA Journal of Numerical Analysis4422024, 1029–1060HAL DOI
17 articleJ. A.Jose Antonio Carrillo, F.Franca Hoffmann, A. M.Andrew M. Stuart and U.Urbain Vaes. The Ensemble Kalman Filter in the Near-Gaussian Setting.SIAM Journal on Numerical Analysis626November 2024, 2549-2587HAL DOI back to text
18 articleF.-B.Francois-Baptiste Cartiaux, F.Frédéric Legoll, A.Alex Libal and J.Julien Reygner. Survival probability of structures under fatigue: a data-based approach.Probabilistic Engineering Mechanics772024, 103657HAL DOI back to text
19 articleJ.Jean Cauvin-Vila, V.Virginie Ehrlacher, G.Greta Marino and J.-F.Jan-Frederik Pietschmann. Stationary solutions and large time asymptotics to a cross-diffusion-Cahn-Hilliard system.Nonlinear Analysis: Theory, Methods and Applications241April 2024, 113482HAL DOI
20 articleY.Yonah Conjungo Taumhas, G.Geneviève Dusson, V.Virginie Ehrlacher, T.Tony Lelièvre and F.François Madiot. Reduced basis method for non-symmetric eigenvalue problems: application to the multigroup neutron diffusion equations.ESAIM: Mathematical Modelling and Numerical Analysis585October 2024, 1959-1987HAL DOI
21 articleJ.Jad Dabaghi and V.Virginie Ehrlacher. Structure-preserving reduced order model for parametric cross-diffusion systems.ESAIM: Mathematical Modelling and Numerical Analysis2024HAL DOI
22 articleR.Rémi Goudey and C.Claude Le Bris. Linear elliptic homogenization for a class of highly oscillating non-periodic potentials.SIAM Journal on Mathematical Analysis562March 2024, 2738-2782HAL DOI
23 articleT.Tony Lelièvre and P.Panos Parpas. Using Witten Laplacians to locate index-1 saddle points.SIAM Journal on Scientific Computing462March 2024, A770-A797HAL DOI
24 articleT.Tony Lelièvre, T.Thomas Pigeon, G.Gabriel Stoltz and W.Wei Zhang. Analyzing multimodal probability measures with autoencoders.Journal of Physical Chemistry B12811March 2024, 2607-2631HAL DOI
25 articleT.Tony Lelièvre, M.Mouad Ramil and J.Julien Reygner. Estimation of statistics of transitions and Hill relation for Langevin dynamics.Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques603August 2024HAL DOI
26 articleI.Idrissa Niakh, G.Guillaume Drouet, V.Virginie Ehrlacher and A.Alexandre Ern. A reduced basis method for frictional contact problems formulated with Nitsche's method.SMAI Journal of Computational Mathematics10June 2024, 29-54HAL
27 articleR.Rémi Robin, P.Pierre Rouchon and L.-A.Lev-Arcady Sellem. Convergence of bipartite open quantum systems stabilized by reservoir engineering.Annales Henri PoincaréNovember 2024HAL DOI
28 articleL.-A.Lev-Arcady Sellem, A.Alain Sarlette, Z.Zaki Leghtas, M.Mazyar Mirrahimi, P.Pierre Rouchon and P.Philippe Campagne-Ibarcq. Dissipative Protection of a GKP Qubit in a High-Impedance Superconducting Circuit Driven by a Microwave Frequency Comb.Physical Review X1512025, 011011HAL DOI
29 articleR.Renato Spacek, P.Pierre Monmarché and G.Gabriel Stoltz. Transient subtraction: A control variate method for computing transport coefficients.Journal of Statistical Physics1924April 2025, 53HAL DOI back to text
30 articleU.Urbain Vaes. Sharp Propagation of Chaos for the Ensemble Langevin Sampler.Journal of the London Mathematical Society1105March 2024HAL DOI back to text
31 articleL.Laurent Vidal, T.Tommaso Nottoli, F.Filippo Lipparini and E.Eric Cancès. Geometric Optimization of Restricted-Open and Complete Active Space Self-Consistent Field Wave Functions.Journal of Physical Chemistry A12831July 2024, 6601-6612HAL DOI back to text

International peer-reviewed conferences

32 inproceedingsF.-B.François-Baptiste Cartiaux, F.Frédéric Legoll, A.Alex Libal, J.Julien Reygner and J.Jorge Semiao. Application of a new method for probabilistic fatigue analysis from strain measurements on bridges.e-Journal of Nondestructive Testing11th European Workshop on Structural Health Monitoring (EWSHM 2024)Potsdam, GermanyJuly 2024HAL DOI

National peer-reviewed Conferences

33 inproceedingsV.Virginie Ehrlacher. Méthodes de décomposition d'une densité électronique moléculaire en contributions atomiques.16ème Colloque National en Calcul de Structures (CSMA 2024)Hyères, France2024HAL

Scientific book chapters

34 inbookC.Christophe Chipot, P.Paraskevi Gkeka, T.Tony Lelièvre and G.Gabriel Stoltz. Equilibrium and Nonequilibrium Methods for Free-Energy Calculations With Molecular Dynamics.3Comprehensive Computational ChemistryElsevier; Elsevier2024, 384-400HAL DOI
35 inbookD.Danny Perez and T.Tony Lelièvre. Recent advances in Accelerated Molecular Dynamics Methods: Theory and Applications.3Comprehensive Computational ChemistryElsevier2024, 360-383HAL DOI

Edition (books, proceedings, special issue of a journal)

36 proceedingsError estimates and variance reduction for nonequilibrium stochastic dynamics.Monte Carlo and Quasi-Monte Carlo Methods. MCQMC 2022460Springer Proceedings in Mathematics & StatisticsSpringer International PublishingJuly 2024, 163-187HAL DOI

Doctoral dissertations and habilitation theses

37 thesisL.-A.Lev-Arcady Sellem. Bosonic qubits and quantum reservoirs : taming the environment.Université Paris sciences et lettresMarch 2024HAL back to text

Reports & preprints

38 miscY.Yves Achdou and C.Claude Le Bris. Homogenization of Hamilton-Jacobi equations with defects leading to stratified problems.December 2024HAL back to text
39 miscN.Noé Blassel, T.Tony Lelièvre and G.Gabriel Stoltz. Quantitative low-temperature spectral asymptotics for reversible diffusions in temperature-dependent domains.January 2025HAL
40 miscA.Andrea Bordignon, É.Éric Cancès, G.Geneviève Dusson, G.Gaspard Kemlin, R. A.Rafael Antonio Lainez Reyes and B.Benjamin Stamm. Fully guaranteed and computable error bounds on the energy for periodic Kohn-Sham equations with convex density functionals.September 2024HAL back to text
41 miscA.Andrea Braides, G. D.Gianni Dal Maso and C.Claude Le Bris. A variational approach to the stability in the homogenization of some Hamilton-Jacobi equations.November 2024HAL back to text
42 miscE.Edoardo Calvello, P.Pierre Monmarché, A. M.Andrew M. Stuart and U.Urbain Vaes. Accuracy of the Ensemble Kalman Filter in the Near-Linear Setting.September 2024HAL back to text
43 miscC.Clément Cancès, M.Maxime Herda and A.Annamaria Massimini. Convergence and long-time behavior of finite volumes for a generalized Poisson-Nernst-Planck system with cross-diffusion and size exclusion.November 2024HAL DOI
44 miscÉ.Éric Cancès, A.Alfred Kirsch and S.Solal Perrin-Roussel. A mathematical analysis of IPT-DMFT.June 2024HAL back to text
45 miscJ. A.José Antonio Carrillo, F.Franca Hoffmann, A. M.Andrew M. Stuart and U.Urbain Vaes. Statistical Accuracy of Approximate Filtering Methods.February 2024HAL back to text
46 miscA.Aloïs Castellano, R.Romuald Béjaud, P.Pauline Richard, O.Olivier Nadeau, C.Clément Duval, G.Grégory Geneste, G.Gabriel Antonius, J.Johann Bouchet, A.Antoine Levitt, G.Gabriel Stoltz and F.François Bottin. Machine learning assisted canonical sampling (MLACS).December 2024HAL back to text
47 miscN.Nicolas Champagnat, T.Tony Lelièvre, M.Mouad Ramil, J.Julien Reygner and D.Denis Villemonais. Quasi-stationary distribution for kinetic SDEs with low regularity coefficients.October 2024HAL back to text
48 miscF.Frédérique Charles, A.Annamaria Massimini and F.Francesco Salvarani. Diffusion asymptotics of a kinetic model for gas-particle mixtures with energy exchanges.March 2025HAL
49 miscY.Yingxing Cheng, E.Eric Cancès, V.Virginie Ehrlacher, A.Alston Misquitta and B.Benjamin Stamm. Multi-center decomposition of molecular densities: A numerical perspective.2024HAL DOI back to text
50 miscM.Maxime Dalery, G.Geneviève Dusson and V.Virginie Ehrlacher. Marginal-preserving modified Wasserstein barycenters for Gaussian distributions and Gaussian mixtures.September 2024HAL back to text
51 miscS.Shiva Darshan, A.Andreas Eberle and G.Gabriel Stoltz. Sticky coupling as a control variate for sensitivity analysis.September 2024HAL back to text
52 miscM.-S.Mi-Song Dupuy, V.Virginie Ehrlacher and C.Clément Guillot. A space-time variational formulation for the many-body electronic Schrödinger evolution equation.May 2024HAL back to text
53 miscM.-S.Mi-Song Dupuy, E.Eloïse Letournel and A.Antoine Levitt. Linear response and resonances in adiabatic time-dependent density functional theory.July 2024HAL
54 miscM.Mathias Dus, G.Geneviève Dusson, V.Virginie Ehrlacher, C.Clément Guillot and J. P.Joel Pascal Soffo Wambo. Comparison between tensor methods and neural networks in electronic structure calculations.December 2024HAL back to text
55 miscM.Mathias Dus and V.Virginie Ehrlacher. Two-layers neural networks for Schrödinger eigenvalue problems.August 2024HAL DOI back to text
56 miscV.Virginie Ehrlacher and L.Luca Nenna. A sparse approximation of the Lieb functional with moment constraints.June 2024HAL
57 miscN. J.Nicolai Jurek Gerber and U.Urbain Vaes. Uniform-in-time propagation of chaos for the Cucker-Smale model.January 2025HAL
58 miscL.Lucas Journel, T.Tony Lelièvre and J.Julien Reygner. Condensation in Fleming--Viot particle systems with fast selection mechanism.July 2024HAL back to text
59 miscA.Abbas Kabalan, F.Fabien Casenave, F.Felipe Bordeu, V.Virginie Ehrlacher and A.Alexandre Ern. Elasticity-based mesh morphing technique with application to reduced-order modeling.2024HAL DOI back to text
60 miscK.Kento Kaneko, C.Claude Le Bris and A. T.Anthony T. Patera. Certified Lumped Approximations for the Conduction Dunking Problem.December 2024HAL DOI back to text
61 miscK.Kento Kaneko, C.Claude Le Bris and A. T.Anthony T. Patera. Error Estimators for the Small-Biot Lumped Approximation for the Conduction Dunking Problem.December 2024HAL DOI back to text
62 miscT.Tony Lelièvre, X.Xuyang Lin and P.Pierre Monmarché. Convergence rates for an Adaptive Biasing Potential scheme from a Wasserstein optimization perspective.January 2025HAL
63 miscT.Tony Lelièvre, G. A.Grigorios A. Pavliotis, G.Geneviève Robin, R.Régis Santet and G.Gabriel Stoltz. Optimizing the diffusion coefficient of overdamped Langevin dynamics.May 2024HAL back to text
64 miscT.Tony Lelièvre, M.Mohamad Rachid and G.Gabriel Stoltz. A spectral approach to the narrow escape problem in the disk.April 2024HAL back to text
65 miscT.Tony Lelièvre, R.Régis Santet and G.Gabriel Stoltz. Improving sampling by modifying the effective diffusion.October 2024HAL back to text
66 miscA.Alicia Negre, F.Fabian Faulstich, R.Raehyun Kim, T.Thomas Ayral, L.Lin Lin and É.Éric Cancès. New perspectives on Density-Matrix Embedding Theory.March 2025HAL
67 miscG.Grigorios Pavliotis, R.Renato Spacek, G.Gabriel Stoltz and U.Urbain Vaes. Neural network approaches for variance reduction in fluctuation formulas.September 2024HAL back to text
68 miscR.Rémi Robin, P.Pierre Rouchon and L.-A.Lev-Arcady Sellem. Unconditionally stable time discretization of Lindblad master equations in infinite dimension using quantum channels.March 2025HAL
69 miscC.Christoph Schönle, M.Marylou Gabrié, T.Tony Lelièvre and G.Gabriel Stoltz. Sampling metastable systems using collective variables and Jarzynski-Crooks paths.May 2024HAL back to text
70 miscJ.Jana Tarhini, S.Sébastien Boyaval, G.Guillaume Enchéry and Q. H.Quang Huy Tran. Reduced Basis method for finite volume simulations of parabolic PDEs applied to porous media flows.June 2024HAL back to text
71 miscJ.-P.Jean-Paul Travert, S.Sébastien Boyaval, C.Cédric Goeury, V.Vito Bacchi and F.Fabrice Zaoui. Evaluation of Performance Measures for Qualifying Flood Models with Satellite Observations.2024HAL DOI back to text
72 miscN.Na Wang, S.Sébastien Boyaval and Y.Yuxi Hu. Global solutions and uniform convergence stability for compressible Navier-Stokes equations with oldroyd-type constitutive law.June 2024HAL back to text

10.3 Cited publications

73 articleR.Robert Benda, E.Eric Cancès, V.Virginie Ehrlacher and B.Benjamin Stamm. Multi-center decomposition of molecular densities: a mathematical perspective.The Journal of Chemical Physics156April 2022, 164107HAL DOI back to text
74 phdthesisE.Elisa Beteille. Experimental study of unsteady flows in urbanised areas.École des Ponts ParisTechNovember 2024HAL back to text
75 unpublishedE.Evan Camrud, A. O.Alain Oliviero Durmus, P.Pierre Monmarché and G.Gabriel Stoltz. Second order quantitative bounds for unadjusted generalized Hamiltonian Monte Carlo.May 2024, working paper or preprintHAL back to text
76 phdthesisJ.Jana Tarhini. Simulations accélérées par bases réduites pour le stockage souterrain : cas d'incertitudes géologiques en poro-mécanique monophasique compressible.École des Ponts ParisTechNovember 2024HAL back to text

MATHERIALS - 2024

MATHERIALS - 2024

2024Activity reportProject-TeamMATHERIALS

Keywords

Computer Science and Digital Science

Other Research Topics and Application Domains

1 Team members, visitors, external collaborators

Research Scientists

Faculty Members

Post-Doctoral Fellows

PhD Students

Interns and Apprentices

Administrative Assistant

Visiting Scientist

2 Overall objectives

3 Research program

4 Application domains

4.1 Electronic structure of large systems

4.2 Computational Statistical Mechanics

4.3 Homogenization and related problems

5 New software, platforms, open data

5.1 New software

5.1.1 DFTK

6 New results

6.1 Electronic structure calculations and related quantum-scale problems

6.1.1 Density functional theory

6.1.2 Analysis of molecular electronic densities

6.1.3 Strongly-correlated systems

6.1.4 Optimal transport and quantum chemistry

6.1.5 Evolution of quantum system

6.2 Computational statistical physics

6.2.1 Sampling of the configuration space

6.2.2 Mathematical understanding and efficient simulation of nonequilibrium systems

6.2.3 Sampling dynamical properties and rare events

6.2.4 Interacting particle methods for sampling

6.3 Homogenization

6.3.1 Homogenization theory

6.3.2 Inverse multiscale problems

6.3.3 Multiscale Finite Element approaches

6.4 Various topics

6.4.1 Complex fluids

6.4.2 Model-order reduction methods

6.4.3 Cross-diffusion systems

6.4.4 Numerical approaches for SPDEs

6.4.5 Modelisation and simulation of structure fatigue

7 Bilateral contracts and grants with industry

8 Partnerships and cooperations

8.1 International initiatives

8.2 International research visitors

8.2.1 Visits of international scientists

Hugo Touchette

8.3 European initiatives

8.3.1 H2020 projects

EMC2

TIME-X

HighLEAP

8.4 National initiatives

9 Dissemination

9.1 Promoting scientific activities

9.2 Teaching - Supervision - Juries

9.2.1 Teaching

9.2.2 Supervision

9.2.3 Juries

9.3 Conference participation

9.4 Popularization

10 Scientific production

10.1 Major publications

10.2 Publications of the year

International journals

International peer-reviewed conferences

National peer-reviewed Conferences

Scientific book chapters

Edition (books, proceedings, special issue of a journal)

Doctoral dissertations and habilitation theses

Reports & preprints

10.3 Cited publications