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 $L^2\left({ℝ}^d\right)$, 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 $L^2\left({ℝ}^3\right)$. 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 uid10Computational 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 $k_{\mathrm{B}}T\simeq 4\times {10}^{-21}$ 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 $\mathrm{O}\left({10}^{15}\right)$ 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.

uid10First, 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).

5.1 New software

6.1 Electronic structure calculations

6.2 Computational statistical physics

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 five main topics: (i) the development of methods for sampling the configuration space; (ii) the efficient computation of dynamical properties which requires to sample metastable trajectories; (iii) the simulation of nonequilibrium systems and the computation of transport coefficients; (iv) coarse-graining techniques to reduce the computational cost of molecular dynamics simulations and gain some insights on the models; (v) the use of particle methods for sampling and optimization.

6.3 Homogenization

6.4 Complex fluids

In 2021, S. Boyaval has pursued his research to improve mathematical models of non-Newtonian fluids for application to large time-space domains, when waves propagate at finite speed. The symmetric-hyperbolic system of conservation laws introduced in 2020 to unify the elastodynamics of hyperelastic materials with viscous fluid dynamics (through viscoelasticity of Maxwell-type) has been extended 30, in particular to cover the non-isothermal flows (highly relevant for application to polymer melts).

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 renewed for 4 years, starting January 1st, 2018.

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

8.2 European initiatives

The ERC Synergy Grant EMC2 (ERC Grant Agreement number 810367, PIs E. Cancès, L. Grigori, Y. Maday, J.-P. Piquemal) has started in September 2019.

The Euro HPC grant TIME-X (PIs: Y. Maday and G. Samaey), focusing on parallel-in-time computations and in which F. Legoll and T. Lelièvre participate, has started on 1st April 2021.

8.3 National initiatives

The project-team is involved in several ANR projects:

• S. Boyaval is the PI of the ANR JCJC project SEDIFLO (2016-2022) to investigate new numerical models of solid transport in rivers.
• V. Ehrlacher is the PI of the ANR project COMODO (2020-2024) which focuses on the development of efficient numerical methods to simulate cross-diffusion systems on moving domains, with application to the simulation of the fabrication process of thin film solar cells. It includes Inria project-teams from Lille and Sophia-Antipolis as well as research teams from Germany.
• V. Ehrlacher is a member of the ANR project ADAPT (2018-2022), PI: D. Lombardi, Inria COMMEDIA team-project. This project is concerned with the parallelization of tensor methods for high-dimensional problems.
• F. Legoll is a member of the ANR project CINE-PARA (2015-2021), PI: Y. Maday, Sorbonne Université. This project is concerned with parallel-in-time algorithms.
• T. Lelièvre is responsible of the node "Ecole des Ponts" of the ANR QuAMProcs (2019-2023), to which G. Stoltz also participates, PI: L. Michel, Université de Bordeaux.

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

• AMORE (Advanced Model Order REduction),
• CHOCOLAS (experimental and numerical study of shock waves),
• DYNQUA (time evolution of quantum systems),
• EGRIN (gravity flows),
• MANU (MAthematics for NUclear applications),
• 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 two Labex: the Labex Bezout (2011-) and the Labex MMCD (2012-).

C. Le Bris is a participant to the Inria Challenge EQIP (Engineering for Quantum Information Processors), in particular in collaboration with P. Rouchon (QUANTIC project-team).

9.1 Promoting Scientific Activities

R. Benda co-organizes the PhD students and postdocs seminar of CERMICS.

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 https://­www.­e4c.­ip-paris.­fr/

E. Cancès is a member of the editorial boards of Mathematical Modelling and Numerical Analysis $\text{(2006-}\text{)}$, SIAM Journal of Scientific Computing (2008-), SIAM Multiscale Modeling and Simulation $\text{(2012-)}$, and the Journal of Computational Mathematics (2017-).

V. Ehrlacher

• is a member of the “Conseil d'Enseignement et de Recherche” of Ecole des Ponts,
• is a member of the “Conseil d'Administration” of the COMUE Paris-Est,
• is a member of the Cordi-S selection committee of INRIA,
• has co-organized the CEMRACS 2021 (6 weeks summer school at CIRM, Luminy, on “Data assimilation and reduced-order modeling for high-dimensional problems”) and an Oberwolfach workshop together with Jean-David Benamou and Daniel Matthes on “Applications of optimal transportation in the natural sciences”.

C. Le Bris

• is a member of the editorial boards of Annales mathématiques du Québec (2013-), Archive for Rational Mechanics and Analysis (2004-), Calcolo (2019-), COCV (Control, Optimization and Calculus of Variations) (2003-), Mathematics in Action (2008-), Networks and Heterogeneous Media (2007-), Nonlinearity (2005-), Journal de Mathématiques Pures et Appliquées (2009-), 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-),
• has been a member of the “Conseil de la Faculté des sciences et ingénierie”, Sorbonne Université, until October 2021,
• is the president of the scientific advisory board of the Institut des Sciences du calcul et des données, Sorbonne Université, and a member of the Scientific Advisory Committee of the Institute for Mathematical and Statistical Innovation, University of Chicago,
• is a member of several scientific advisory boards in the industrial sector, in particular (since 2020) of the Energy Division of the Atomic Energy Council (CEA) and (since 2019) of Framatome senior management,
• holds a position of Visiting Professor at the University of Chicago, for one quarter a year,
• has co-organized (with Q. Du, I. Fonseca, R. James, J. Lu, D. Perez) the workshop Mathematical and Computational Materials Science at the Institute for Mathematical and Statistical Innovation, Chicago, February 15-19, 2021, and (with A.-L. Dalibard, P. Souganidis) the workshop Homogenization theory: periodic and beyond, Oberwolfach, 14-20 March 2021.

F. Legoll

• is a member of the editorial board of SIAM MMS (2012-) and of ESAIM: Proceedings and Surveys (2012-),
• was a member of the DFG review panel for a Collaborative Research Centre on ”Multiscale Simulation Methods for Soft-Matter Systems” (dec 2021).

T. Lelièvre

• is a member of the editorial boards of IMA: Journal of Numerical Analysis, SIAM/ASA Journal of Uncertainty Quantification, Communications in Mathematical Sciences, Journal of Computational Physics and ESAIM:M2AN,
• is a member of the “Conseil d'Administration” of École des Ponts,
• is the head of the applied mathematics department (CERMICS) at Ecole des Ponts,
• has organized the 13th International Workshop on Rare-Event Simulation - RESIM 2021, Les Cordeliers Paris, 18th-21st May 2021 (with G. Fort, E. Gobet and A. Guyader).

A. Levitt co-organizes the applied mathematics seminar of the CERMICS lab, and the internal seminar of the EMC2 project (Sorbonne Université).

G. Stoltz

• is a member of the scientific council of UNIT (Université Numérique Ingénierie et Technologie),
• 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 with X. Shang (University of Birmingham, United Kingdom) a mini-symposium at SIAM Materials Science'21 (17-28 May) on "Computational Statistical Physics and Related Fields".

9.2 Teaching - Supervision - Juries

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

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

• Analyse et calcul scientifique, 30h (R. Biezemans, V. Ehrlacher, R. Goudey, G. Kemlin, G. Stoltz)
• Équations aux dérivées partielles et éléments finis, 15h (R. Biezemans, O. Gorynina, R. Goudey, F. Legoll)
• Hydraulique numérique, 15h (S. Boyaval)
• Mécanique quantique, 15h (E. Cancès, A. Levitt)
• Méthodes numériques pour les problèmes en grande dimension, 17h30 (V. Ehrlacher)
• Optimisation, 15h (O. Gorynina, J. Cauvin-Vila)
• Outils mathématiques pour l’ingénieur (E. Cancès: 18h, V. Ehrlacher, J. Cauvin-Vila, R. Goudey, F. Legoll, A. Levitt: 9h)
• Pratique du calcul scientifique, 15h (A. Levitt)

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

• Analyse de Fourier, 15h (A. Levitt)
• Contrôle de systèmes dynamiques et équations aux dérivées partielles, 18h (E. Cancès)
• Problèmes d'évolution, 36h (F. Legoll, V. Ehrlacher)
• Projets Modéliser Programmer Simuler (V. Ehrlacher, T. Lelièvre, G. Stoltz)
• Projets Semaine IMI, 10h (O. Gorynina)
• Statistics and data sciences, 24h (G. Stoltz)

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

• Introduction to computational statistical physics, 20h (G. Stoltz)
• Méthodes numériques probabilistes, 24h (T. Lelièvre)
• Problèmes multiéchelles, aspects théoriques et numériques, 24h (F. Legoll)
• Théorie spectrale et méthodes variationnelles, 10h (E. Cancès)

At other institutions:

• Modal de Mathématiques Appliquées (MAP472A, MAP463D), 15h, Ecole polytechnique (T. Lelièvre)
• Aléatoire (MAP361), 40h, Ecole Polytechnique (T. Lelièvre)
• Modélisation de phénomènes aléatoires (MAP432), 20 h (T. Lelièvre)
• Probabilités de 1ère année, 27h, L3, Ecole des Mines (M. Ramil)
• Analyse fonctionnelle, 46h, L3 Sorbonne Université (J. Cauvin-Vila)

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

• Rafaël Coyaud, funding Ecole des Ponts, Deterministic and stochastic approaches for multimarginal optimal transport problems, since October 2017, co-supervised by V. Ehrlacher and A. Alfonsi (École des Ponts and Inria MATHRISK), defended in January
• Lise Maurin, funding Sorbonne Université, Adaptive biasing algorithms: mathematical analysis and applications in molecular dynamics, co-supervised by T. Lelièvre and J.P. Piquemal (SU), defended in December

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

• Zineb Belkacemi, thèse CIFRE SANOFI, Machine learning for model reduction in molecular dynamics, Université Paris-Est, since October 2018, supervised by T. Lelièvre and G. Stoltz
• Robert Benda, funding MTES, Multi-scale modeling of water quality nanosensors based on carbon nanotubes and conjugated polymers, Institut Polytechnique de Paris, since September 2018, supervised by E. Cancès and B. Lebental (Polytechnique)
• Rutger Biezemans, funding DIM Math Innov (Inria), Difficult multiscale problems and non-intrusive approaches, Ecole des Ponts, since October 2020, supervised by C. Le Bris and A. Lozinski (University of Besançon)
• Raed Blel, funding UM6P, Model-order reduction methods for stochastic problems, since October 2018, co-supervised by T. Lelièvre and V. Ehrlacher
• Jean Cauvin-Vila, funding Ecole des Ponts, Cross-diffusion systems on moving boundary domains, since October 2020, co-supervised by V. Ehrlacher and A. Hayat
• Yonah Conjugo-Taumhas, thèse CIFRE CEA, Reduced basis methods for non-symmetric eigenvalue problems, since October 2020, co-supervised by T. Lelièvre and V. Ehrlacher together with G. Dusson (CNRS Besançon) and F. Madiot (CEA)
• Maria Fuente-Ruiz, funding INRIA, Parallel algorithms for tensor methods, since September 2020, supervised by V. Ehrlacher and D. Lombardi (Inria COMMEDIA)
• Rémi Goudey, funding CDSN, Homogenization problems in the presence of defects, Université Paris-Est, since September 2019, supervised by C. Le Bris
• Gaspard Kemlin, Mathematical and numerical analysis for electronic structures, École des Ponts, since September 2019, supervised by E. Cancès and A. Levitt
• Eloïse Letournel, funding Ecole des Ponts, Finite size effects in electronic structure, École des Ponts, since September 2021, supervised by A. Levitt
• Alex Libal, funding Ecole des Ponts, Probabilistic approaches for materials fatigue, Ecole des Ponts, since September 2020, supervised by F. Legoll and J. Reygner (Ecole des Ponts)
• Idrissa Niakh, thèse CIFRE EDF, Reduced basis for variational inequalities, since October 2019, co-supervised by V. Ehrlacher and A. Ern (Inria SERENA)
• Thomas Pigeon, funding Inria, Combining machine learning and quantum computations to discover new catalytic mechanisms, Université de Lyon, since October 2020, supervised by P. Raybaud (IFPEN) and T. Lelièvre, together with G. Stoltz and M. Corral-Vallero (IFPEN)
• Régis Santet, funding Ecole des Ponts, Enhancing the sampling efficiency of reversible and non-reversible dynamics, Ecole des Ponts, since October 2021, supervised by T. Lelièvre and G. Stoltz
• Inass Sekkat, funding UM6P, Large scale Bayesian inference, Université Paris-Est, since March 2018, supervised by G. Stoltz
• Lev-Arcady Sellem, funding Advanced ERC Q-Feedback (PI: P. Rouchon), Mathematical approaches for simulation and control of open quantum systems, Ecole des Mines de Paris, since October 2020, co-supervised by C. Le Bris and P. Rouchon (Inria QUANTIC)
• Renato Spacek, funding FSMP CoFund, Efficient computation of linear response of nonequilibrium stochastic dynamics, ED 386 Sorbonne-Université, since November 2021, supervised by G. Stoltz and P. Monmarché (Sorbonne Université)
• Elisa Beteille, thèse CIFRE EDF, Propagation of Urban Flood waves, since November 2021, supervised by S. Boyaval
• Jana Tarhini, thèse IFPEN, Fast simulation of CO2/H2 storage in geological bassins, since November 2021, supervised by S. Boyaval

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

• S. Boyaval, PhD of Benjamin Streichenberger (“Approches multi-fidélités pour la simulation rapide d’écoulements d’air en milieu urbain”), defended at Université Gustave Eiffel in September
• V. Ehrlacher, PhD of Anne-Françoise Lempereur de Guerny ("De l’influence des conditions aux bords périodiques pour la simulation de quelques quantités d’intérêt pour la chimie séparative"), defended at Sorbonne Université in April (president)
• V. Ehrlacher, PhD of Thomas Daniel ("Apprentissage statistique pour la réduction de modèles non linéaire"), defended at Ecole des Mines de Paris in September (referee)
• V. Ehrlacher, PhD of Benoit Gaudeul ("Approximation numérique entropique pour des systèmes de diffusion croisée issus de la physique"), defended at Université de Lille in August
• V. Ehrlacher, PhD of Gabriele Todeschi ("Finite volume approximation of optimal transport and Wasserstein gradient flows"), defended at Université Paris-Dauphine in December
• G. Stoltz, PhD of Anne-Françoise Lempereur de Guerny ("De l’influence des conditions aux bords périodiques pour la simulation de quelques quantités d’intérêt pour la chimie séparative"), defended at Sorbonne Université in April (referee)
• G. Stoltz, PhD of Louis Thiry ("On the efficiency of local methods in image classification and energy regression in Physics"), defended at ENS Ulm in July (president)
• G. Stoltz, PhD of Lorenz Richter ("Solving high-dimensional PDEs, approximation of path space measures and importance sampling of diffusions"), defended at BTU Cottbus in December (referee)
• G. Stoltz, PhD of Robert Benda ("Multi-scale modeling of water quality nanosensors based on carbon nanotubes and conjugated polymers"), defended at Ecole polytechnique in December
• T. Lelièvre, PhD of Giacomo Garegnani ("Probabilistic and Bayesian methods for uncertainty quantification of deterministic and stochastic differential equations"), defended at EPFL in June (referee)
• T. Lelièvre, PhD of Arthur Macherey ("Approximation et réduction de modèle pour les équations aux dérivées partielles avec interprétation probabiliste"), defended at Centrale Nantes in June (president)
• T. Lelièvre, PhD of Adrien Laurent ("Algebraic Tools and Multiscale Methods for the Numerical Integration of Stochastic Evolutionary Problems"), defended at Université de Genève in July
• T. Lelièvre, PhD of Louis Thiry ("On the efficiency of local methods in image classification and energy regression in physics") defended at Ecole Normale Supérieure in July (referee)
• T. Lelièvre, PhD of Dario Lucente ("Predicting probabilities of climate extremes from observations and dynamics"), defended at Ecole Normale Supérieure de Lyon in November

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

• T. Lelièvre, HdR of Pierre Monmarché ("Degenerated Markov processes for sampling, optimization and modelling"), defended at Sorbonne Université in January
• T. Lelièvre, HdR of Charles-Edouard Bréhier ("Contributions to stochastic numerics: simulation of SPDEs and rare events") defended at Université de Lyon 1, in June
• T. Lelièvre, HdR of Julien Reygner ("Theoretical analysis and numerical methods for conservation laws, metastability and uncertainty propagation "), defended at Paris Est Sup in September

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

• C. Le Bris has chaired the committee for CEA-INSTN PhD fellowships in the Numerics program at CEA
• C. Le Bris was a member of the committee for CEA-FOCUS PhD fellowships in the Experimentation and Numerical Twins program at CEA.
• T. Lelièvre, Postes en mathématiques appliquées, Ecole Polytechnique.
• T. Lelièvre, Professeur en probabilités et statistiques, Université de Lorraine

9.3 Conference participation

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

• R. Biezemans, CERMICS Young Researchers Seminar, online, April
• R. Biezemans, Congrès SMAI 2021, la Grande-Motte, June
• R. Biezemans, 2nd French-German Workshop on Multiscale Problems, Besançon, September
• J. Cauvin-Vila, Journée école doctorale MSTIC, online, June.
• J. Cauvin-Vila, Congrès des jeunes chercheuses et chercheurs en mathématiques appliquées (CJC-MA), Ecole Polytechnique, October
• J. Dabaghi, WCCM conference, invited in a minisymposium, online, January
• J. Dabaghi, European Finite Element Fair conference, online, September
• J. Dabaghi, SIMRACE conference, Rueil-Malmaison, December
• J. Dabaghi, CEMRACS research project with Christoph Strossner and supervised by Virginie Ehrlacher, Luminy, July-August
• J. Dabaghi, research stay at ACSIOM, invited by M. Daniele Di Pietro, Université de Montpellier, November
• V. Ehrlacher, University of Strathclyde applied mathematics seminar, online, October
• V. Ehrlacher, Rencontres CEA-LJLL, December
• V. Ehrlacher, Plenary talk at the ICOSAHOM 2020 conference, online, July
• V. Ehrlacher, Journées IHP ”Réduction de modèles et traitement géométrique des données”, November
• V. Ehrlacher, University of Warwick applied mathematics seminar, online, June
• V. Ehrlacher, IPAM Workshop “Tensor Methods and their Applications in the Physical and Data Sciences”, online, June
• L. Garrigue, Seminar of the laboratory of mathematics of the university of Bourgogne, Dijon, March
• L. Garrigue, Seminar of mathematical physics of the Centre for theoretical physics of Toulon, Toulon, March
• L. Garrigue, Seminar on multiscale problems in mathematical physics, Angers, June
• L. Garrigue, International congress on mathematical physics, Geneva, August
• L. Garrigue, MOANSI meeting 2021, online, September
• O. Gorynina, Congrès SMAI 2021, la Grande-Motte, June
• R. Goudey, Oberwolfach workshop on "Homogenization Theory: Periodic and Beyond", online, March
• R. Goudey, Congrès SMAI 2021, la Grande-Motte, June
• G. Kemlin, MOANSI meeting 2021, online, September
• G. Kemlin, SIAM MS, online, May
• G. Kemlin, SIAM CSE, online, March
• C. Le Bris, invited keynote lecture at the 2nd Conference of the Collaborative Research Center 1114 "Scaling Cascades in Complex Systems" (Freie Universitat - Zuse Institute Berlin), online, March
• C. Le Bris, Mittag-Leffler Institute workshop on "New trends in numerical multiscale methods and beyond", online, July
• T. Lelièvre, IMSI workshop on Mathematical and Computational Materials Science, online, February
• T. Lelièvre, SIAM MS conference, online, May
• T. Lelièvre, IPAM conference, online, June
• T. Lelièvre, Franco-Dutch Eurandom meeting, IHP, July
• T. Lelièvre, CEMRACS conference, CIRM, July
• T. Lelièvre, Lorentz Center, August
• T. Lelièvre, Analysis and probability seminar, Université Mohammed VI Polytechnique, online, September
• T. Lelièvre, CECAM Workshop on Generalized Langevin Equations, Berlin, online, September
• T. Lelièvre, Swiss numerics day, EPFL, September
• T. Lelièvre, Neutron Transport Days of the Bath-Beijing-Paris Branching Structures Meeting, online, September
• T. Lelièvre, Probability and Stochastic Processes seminar, University of Delaware, September
• T. Lelièvre, Workshop On Future Synergies for Stochastic and Learning Algorithms, CIRM, September
• T. Lelièvre, Statistical modeling for stochastic processes and related fields, ISM Japan, online, September
• T. Lelièvre, CECAM discussion meeting Generalized Langevin Equations in classical and quantum simulations, Paris, October
• T. Lelièvre, Séminaire LAMA, Université Gustave Eiffel, November
• T. Lelièvre, Séminaire PHIMECA l'aléatoire par les chemins de traverse, November
• T. Lelièvre, Séminaire LAGA, Université Sorbonne Paris Nord, November
• T. Lelièvre, Matherials Research Society fall meeting, online, December
• T. Lelièvre, Workshop Interaction Mechanisms in Monte Carlo Beyond Resampling, Bristol, online, December
• A. Levitt, GDR DYNQUA general meeting, online, March
• A. Levitt, UM6P seminar, online, April
• A. Levitt, GDR REST general meeting, online, May
• A. Levitt, SIAM Linear Algebra, online, May
• A. Levitt, Solid Math 2021, ENPC, August
• A. Levitt, Numerical analysis seminar, University of Besançon, October
• A. Levitt, Numerical analysis seminar, IMO, University Paris-Saclay, October
• A. Levitt, Chemistry seminar, ICP, University Paris-Saclay, October
• F. Legoll, WCCM congress, online, January
• F. Legoll, SIAM conference on Mathematical Aspects of Materials Science, online, May
• F. Legoll, 10th International Conference On Adaptive Modeling and Simulation (ADMOS), online, June
• F. Legoll, Mittag-Leffler Institute workshop on "New trends in numerical multiscale methods and beyond", online, July
• F. Legoll, 16th US National Congress on Computational Mechanics, online, July
• F. Legoll, Complas conference, Barcelona, September
• F. Legoll, 2nd French-German Workshop on Multiscale Problems, Besançon, September
• F. Legoll, workshop on "Generalized Langevin Equations in classical and quantum simulations", Paris, October
• E.Letournel, MOANSI meeting 2021, online, September
• G. Stoltz, MPI Polymer Research public seminar (online), January
• G. Stoltz, ERC Synergy workshop (online), February
• G. Stoltz, Probability seminar, University of Delaware (online), April
• G. Stoltz, SIAM Materials (online), May
• G. Stoltz, Probability seminar, University of Bristol (online), May
• G. Stoltz, MAC-MIGS afternoon on Mathematical Research Topics in Machine Learning (online), June
• G. Stoltz, Summer school “From kinetic equations to statistical mechanics”, St Jean de Monts, France, June
• G. Stoltz, One World Stochastic Numerics and Inverse Problems seminar (online), July
• G. Stoltz, Workshop “On Future Synergies for Stochastic and Learning Algorithms”, CIRM, France, October
• G. Stoltz, Workshop ANR EFI, Université Paris Dauphine, October
• G. Stoltz, Data Science and Computational Statistics Seminar, University of Birmingham, October
• U. Vaes, Séminaire de Dynamique Quantique et Classique, CPT (online), April
• U. Vaes, CECAM workshop on generalized Langevin equations, Sorbonne Université, October
• U. Vaes, Working group on consensus-based optimization and sampling, Simons Institute (online), October
• U. Vaes, Rencontres Inria–LJLL, Inria Paris, October

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

• T. Lelièvre, T. Pigeon and G. Stoltz, Constructing collective variables using Machine Learning and free energy biased simulations, 3h lecture + 3h hands-on session, AlgoSB Winter school 2021, CIRM, France, November
• T. Lelièvre, Brummer & Partners MathDataLab, "Sampling problems in computational statistical physics" (3h), KTH (online), Sweden, January 2021.
• T. Lelièvre, Lecture Series in Probability Theory, "Jump Markov models and transition state theory: the quasi-stationary distribution approach", Seoul National University, South Korea, August 2021.
• T. Lelièvre, T. Pigeon and G. Stoltz, Constructing collective variables using Machine Learning and free energy biased simulations, 3h lecture + 3h hands-on session, AlgoSB Winter school 2021, CIRM, France, November 2021.

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

• J. Cauvin-Vila, Congrès SMAI 2021, la Grande-Motte, June.

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

• R. Biezemans, Le Monde des Mathématiques Industrielles, online, March
• R. Biezemans, ECCOMAS Young Investigators Conference, online, July
• R. Biezemans, CEMRACS Summer School, July
• R. Biezemans, Congrès des Jeunes Chercheuses et Chercheurs en Mathématiques Appliquées, Palaiseau, October
• G. Kemlin, Online workshop on wave-function methods in quantum chemistry and nuclear physics, online, Frebruary
• G. Kemlin, CEMRACS 2021, Marseille, July-August
• G. Kemlin, Congrès des Jeunes Chercheuses et Chercheurs en Mathématiques Appliquées, Palaiseau, October
• E. Letournel, Solid Math 2021, Marne la Vallée, August
• R. Santet, QuAMProcs working group #3, seminar on metastable processes, Institut Denis Poisson, France, October
• L. Vidal, CEMRACS 2021, Marseille, July-August

9.4 Popularization

• A. Levitt is a member of the editorial board of Interstices, Inria’s popularization website.

10.1 Major publications

• 1 miscE.Eric Cancès, M.Mireille Defranceschi, W.Werner Kutzelnigg, C.Claude Le Bris and Y.Yvon Maday. Computational Quantum Chemistry: A Primer.2003
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• 2 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. 2006
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• 3 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.1998
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• 4 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.2006
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• 5 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 Mathematics2019
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• 6 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.2005
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• 7 bookT.Tony Lelièvre, M.Mathias Rousset and G.Gabriel Stoltz. Free Energy Computations: A Mathematical Perspective.Imperial College Press, 458~p.2010
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10.2 Publications of the year