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
Other Research Topics and Application Domains
 B1.1.2. Molecular and cellular biology
 B4.3.4. Solar Energy
 B5.3. Nanotechnology
 B5.5. Materials
 B9.5.2. Mathematics
 B9.5.3. Physics
 B9.5.4. Chemistry
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]
 Frédéric Legoll [ENPC, Senior Researcher, HDR]
 Tony Lelièvre [ENPC, Senior Researcher, HDR]
 Antoine Levitt [INRIA, Researcher, until Nov 2022, HDR]
 Gabriel Stoltz [ENPC, Senior Researcher, HDR]
 Urbain Vaes [INRIA, ISFP]
Faculty Members
 Yves Achdou [Université de Paris, Professor, HDR]
 Arnaud Guyader [Sorbonne Université, Professor, from Sep 2022, HDR]
 Alexei Lozinski [Université de FrancheComté, Professor, until Aug 2022, HDR]
PostDoctoral Fellows
 Jad Dabaghi [ENPC, until Sep 2022]
 Mathias Dus [ENPC, from Sep 2022]
 Louis Garrigue [ENPC, until Oct 2022]
 Olga Gorynina [ENPC, until Jan 2022]
 Etienne Polack [ENPC, from Feb 2022]
 Mohamad Rachid [ENPC, from Mar 2022]
 Masaaki Tokieda [INRIA]
PhD Students
 Rutger Biezemans [ENPC]
 Noe Blassel [ENPC, from Oct 2022]
 Andrea Bordignon [ENPC, from Nov 2022]
 Jean CauvinVila [ENPC]
 Shiva Darshan [ENPC, from Oct 2022]
 Renato Freitas Spacek [INRIA]
 Rémy Goudey [ENPC, until Aug 2022]
 Gaspard Kemlin [ENPC, until Nov 2022]
 Alfred Kirsch [ENPC]
 Alberic Lefort [ENPC, from Nov 2022]
 Eloïse Letournel [ENPC]
 Solal PerrinRoussel [ENPC, from Sep 2022]
 Thomas Pigeon [INRIA]
 Simon Ruget [INRIA, from Oct 2022]
 Régis Santet [ENPC]
 LevArcady Sellem [École des Mines]
 Laurent Vidal [ENPC]
Interns and Apprentices
 Guillaume Vigne [ENSMP, until Feb 2022]
 Changhe Yang [Inria, from Mar 2022 until Jun 2022]
Administrative Assistant
 Julien Guieu [INRIA]
2 Overall objectives
The MATHERIALS projectteam was created jointly by the École des Ponts ParisTech (ENPC) and Inria in 2015. It is the followup and an extension of the former projectteam 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 projectteam have positions at CERMICS and at two other laboratories of École des Ponts: Institut Navier and Laboratoire SaintVenant. The scientific focus of the projectteam 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 projectteam 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 projectteam is to have created a corpus of literature, authoring books and research monographs on the subject 1, 2, 3, 4, 6, 5, 7 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 longterm 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 socalled 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 timedependent form or in its timeindependent 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 selfadjoint 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 timedependent equation is a firstorder linear evolution equation, whereas the timeindependent 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 symmetryconstrained subspace of ${L}^{2}\left({\mathbb{R}}^{d}\right)$, 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 twobody 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 BornOppenheimerHartreeFock and the BornOppenheimerKohnSham strategies, end up with large systems of coupled nonlinear partial differential equations, each of these equations being posed on ${L}^{2}\left({\mathbb{R}}^{3}\right)$. 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 projectteam 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 ${\mathcal{N}}_{A}\sim 6\times {10}^{23}$, 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 ${\mathcal{N}}_{A}$ 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 smallscale 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 projectteam 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 coarsescale 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 wellknown that the case of ordered (that is, in this context, periodic) systems is now wellunderstood, 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 New software and platforms
5.1 New software
5.1.1 DFTK

Keywords:
Molecular simulation, Quantum chemistry, Materials

Functional Description:
DFTK, short for the densityfunctional toolkit, is a Julia library implementing planewave 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 2022 has gained support for GPU acceleration, normconserving pseudopotentials, and many other smaller features. It has been used for several publications both inside and outside the projectteam.
 URL:

Contact:
Antoine Levitt
6 New results
6.1 Electronic structure calculations
Participants: Andrea Bordignon, Eric Cancès, Virginie Ehrlacher, Louis Garrigue, Gaspard Kemlin, Antoine Levitt, Eloïse Letournel, Solal PerrinRoussel, Etienne Polack, Laurent Vidal.
6.1.1 Density functional theory
A track of the projectteam's activity this year has been the investigation of continuum eigenstates, as opposed to the bound states that form much of the projectteam's usual focus. Such states are relevant to the study of processes where electrons propagate away from the system under consideration, such as ionization. They are delocalized, complicating their discretization. In 35 and 36, together with colleagues from the Laboratoire de Chimie Théorique at Sorbonne Université, É. Cancès and A. Levitt have proposed a method to compute the photoionization spectrum for atoms in timedependent density functional theory (TDDFT) in the Sternheimer formalism. This method, inspired by similar schemes in numerical wave propagation, employs an analytic DirichlettoNeumann map to impose correct boundary conditions on the Sternheimer equations, which appears mathematically as a perturbation of an Helmholtz equation with a Coulomb potential. In 56, E. Letournel and A. Levitt, together with physicist colleagues from CEA Grenoble, have proposed a method to compute electronic resonances in crystals with defects. The method involves the computation of analytic continuations of Green functions of periodic operators, which is accomplished by a complex deformation of the Brillouin zone.
E. Cancès, G. Kemlin and A. Levitt have studied the numerical properties of response computations in density functional theory at finite temperature. They have proposed a method based on a Schur complement to increase the stability and efficiency of iterative solvers for the Sternheimer equations 49.
Together with D. Gontier (U. Paris Dauphine and ENS Paris), E. Cancès and L. Garrigue provided a formal derivation of a reduced model for twisted bilayer graphene (TBG) from Density Functional Theory. This derivation is based on a variational approximation of the TBG KohnSham Hamiltonian and asymptotic limit techniques. In contrast with other approaches, it does not require the introduction of an intermediate tightbinding model. The soobtained model is similar to that of the BistritzerMacDonald (BM) model but contains additional terms. Its parameters can be easily computed from KohnSham calculations on singlelayer graphene and untwisted bilayer graphene with different stackings. It allows one in particular to estimate the parameters ${w}_{\mathrm{AA}}$ and ${w}_{\mathrm{AB}}$ of the BM model from firstprinciples. The resulting numerical values, namely ${w}_{\mathrm{AA}}={w}_{\mathrm{AB}}\simeq 126$ meV for the experimental interlayer mean distance are in good agreement with the empirical values ${w}_{\mathrm{AA}}={w}_{\mathrm{AB}}=110$ meV obtained by fitting to experimental data. We also show that if the BM parameters are set to ${w}_{\mathrm{AA}}={w}_{\mathrm{AB}}\simeq 126$ meV, the BM model is an accurate approximation of this new reduced model.
With G. Dusson (CNRS and U. of FrancheComté) E. Cancès, G. Kemlin and L. Vidal proposed in 45 general criteria to construct optimal atomic centered basis sets in quantum chemistry. They focuses in particular on two criteria, one based on the groundstate onebody density matrix of the system and the other based on the groundstate energy. The performance of these two criteria was numerically tested and compared on a parametrized eigenvalue problem, which corresponds to a onedimensional toy version of the groundstate dissociation of a diatomic molecule.
In solid state physics, electronic properties of crystalline materials are often inferred from the spectrum of periodic Schrödinger operators. As a consequence of Bloch's theorem, the numerical computation of electronic quantities of interest involves computing derivatives or integrals over the Brillouin zone of socalled energy bands, which are piecewise smooth, Lipschitz continuous periodic functions obtained by solving a parametrized elliptic eigenvalue problem on a Hilbert space of periodic functions. Classical discretization strategies for resolving these eigenvalue problems produce approximate energy bands that are either nonperiodic or discontinuous, both of which cause difficulty when computing numerical derivatives or employing numerical quadrature. In a paper with M. Hassan (Sorbonne Université) 48, E. Cancès and L. Vidal studied an alternative discretization strategy based on an ad hoc operator modification approach. While specific instances of this approach have been proposed in the physics literature, they introduced a systematic formulation of this operator modification approach. They derived a priori error estimates for the resulting energy bands and showed that these bands are periodic and can be made arbitrarily smooth (away from band crossings) by adjusting suitable parameters in the operator modification approach.
6.1.2 Open quantum systems
In his postdoctoral work cosupervised by Claude Le Bris (MATHERIALS) and Pierre Rouchon (Inria QUANTIC), Masaaki Tokieda addresses various issues related to the numerical simulation and the fundamental understanding of several models of physical systems likely candidates to play a crucial role in quantum computing. More specifically, he studies several pathways to efficiently account for adiabatic elimination in the simulation of composite quantum systems in interactions, modeled by Lindblad type master equations. The specific question currently under study is the expansion up to high orders and the compatibility of such an expansion with the formal requirements of consistency of quantum mechanical evolutions. He is also planning to address various other connected issues, all aiming at better fundamental understanding and a more effective simulation of open quantum systems.
6.1.3 Tensor methods
Tensor methods have proved to be very powerful tools in order to represent highdimensional objects with low complexity. Such methods prove to have a wide range of applications in quantum chemistry, for instance for the approximation of the ground state wavefunction of a molecular system when the number of electrons is large. The DMRG method is one example of such a numerical scheme. Research efforts are led in the team so as to propose new methodological developments in order to improve on the current stateoftheart tensor methods.
In 20, V. Ehrlacher, M. FuenteRuiz and D. Lombardi (Inria COMMEDIA) introduce a method to compute an approximation of a given tensor as a sum of Tensor Trains (TTs), where the order of the variates and the values of the ranks can vary from one term to the other in an adaptive way. The numerical scheme is based on a greedy algorithm and an adaptation of the TTSVD method. The proposed approach can also be used in order to compute an approximation of a tensor in a Canonical Polyadic format (CP), as an alternative to standard algorithms like Alternating Linear Squares (ALS) or Alternating Singular Value Decomposition (ASVD) methods. Some numerical experiments are presented, in which the proposed method is compared to ALS and ASVD methods for the construction of a CP approximation of a given tensor and performs particularly well for highorder tensors. The interest of approximating a tensor as a sum of Tensor Trains is illustrated in several numerical test cases.
6.2 Computational statistical physics
Participants: Noé Blassel, Shiva Darshan, Olga Gorynina, Frédéric Legoll, Tony Lelièvre, Antoine Levitt, Thomas Pigeon, Mohamad Rachid, 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 five main topics: (i) the improvement of techniques to sample the configuration space; (ii) the study of simulation methods to efficiently simulate nonequilibrium systems; (iii) the sampling of dynamical properties and rare events; (iv) the use and development of machine learning tools in molecular dynamics and sampling; (v) 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. In 25, Tony Lelièvre together with Lucie Delemotte (KTH, Sweden), J. Hénin (IBPC, France), Michael Shirts (University of Colorado, USA) and Omar Valsson (MPI Mainz, Germany) provide on overview of enhanced sampling algorithms. These algorithms have emerged as powerful methods to extend the potential of molecular dynamics simulations and allow the sampling of larger portions of the configuration space of complex systems. This "living" review is intended to be updated to continue to reflect the wealth of sampling methods as they emerge in the literature.
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. G. Stoltz reviewed in 66 the motivations and mathematical framework involved in the computation of transport coefficients, with a particular emphasis on the numerical analysis of the estimators at hand.
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. Such a method was proposed by G. Pavliotis (Imperial College London, UnitedKingdom), G. Stoltz and U. Vaes in the context of the estimation of the mobility via Einstein's method in 65, although the method can be adapted to other transport coefficients. The fundamental idea is to approximate the solution to some Poisson equation determining the transport coefficient, and relying on Ito calculus to construct a random variable strongly correlated to the square displacement from the position at origin. The motivation of this work was to estimate the mobility of underdamped Langevin dynamics of two dimensional systems for low values of the friction, in an attempt to (in)validate physical conjectures about the divergence of the mobility as the friction goes to zero.
6.2.3 Sampling dynamical properties and rare events
Sampling trajectories which link metastable states of the target probability measure, and estimating the associated transition rates from one metastable state to another, is a difficult task, which requires dedicated numerical methods. Various works along these lines were preprinted this year.
In 62, Tony Lelièvre, together with Mouad Ramil (Seoul National University, South Korea) and Julien Reygner (CERMICS, France), propose and analyze a simple and complete numerical method to estimate statistics of transitions between metastable states for the Langevin dynamics, based on the socalled Hill relation. More precisely, they prove the Hill relation in the fairly general context of positive Harris recurrent chains, and show that this formula applies to the Langevin dynamics. Moreover, they provide an explicit expression of the invariant measure involved in the Hill relation for the Langevin dynamics, and describe an elementary exact simulation procedure.
In 61, Tony Lelièvre, together with Boris Nectoux (Laboratoire de Mathématiques Blaise Pascal, France) and Dorian Le Peutrec (Institut Denis Poisson, France), conclude a series of papers which aim at providing firm mathematical grounds to jump Markov models which are used to model the evolution of molecular systems, as well as to some numerical methods which use these underlying jump Markov models to efficiently sample metastable trajectories of the overdamped Langevin dynamics. More precisely, using the quasistationary distribution approach to analyze the metastable exit from the basin of attraction of a local minimum of the potential energy function, they prove that the exit event (exit position and exit time) of the overdamped Langevin dynamics in the small temperature regime can be accurately modeled by a jump Markov model parameterized by the Eyring–Kramers rates. From a mathematical viewpoint, the proof relies on tools from the semiclassical analysis of Witten Laplacians on bounded domains. The main difficulty is that, since they consider as metastable states the basins of attraction of the local minima of the energy, the exit regions are neighborhoods of saddle points of the energy, and many standard techniques (such as WKB approximations) cannot handle critical points on the boundary.
In 34, Tony Lelièvre, together with Mouad Ramil (Seoul National University, South Korea) and Julien Reygner (CERMICS, France), give an overview of some of the results obtained during the PhD work of Mouad Ramil. More precisely, the paper provides a selfcontained analysis of the Parallel Replica algorithm applied to the Langevin dynamics. This algorithm was designed to efficiently sample metastable trajectories relying on a parallelization in time technique. The analysis relies on results on the existence of quasistationary distributions of the Langevin dynamics in domains bounded in positions. The article also contains some discussions about the overdamped limit of the quasistationary distribution.
Another approach to sampling reactive trajectories is to allow for longer integration times, thanks to dedicated algorithmic developments. In 58, Frédéric Legoll and Tony Lelièvre, together with Olga Gorynina (WSLSLF, Switzerland) and Danny Perez (Los Alamos National Laboratory, USA) numerically investigate an adaptive version of the parareal algorithm in the context of molecular dynamics. This method allows to more efficiently integrate in time the dynamics under consideration. The parareal algorithm uses a family of machinelearning spectral neighbor analysis potentials (SNAP) as fine, reference, potentials and embeddedatom method potentials (EAM) as coarse potentials. The numerical results (obtained using LAMMPS, a very broadly used software within the materials science community) demonstrate significant computational gains when using the adaptive parareal algorithm in comparison to a sequential integration of the Langevin dynamics.
6.2.4 Machinelearning approaches in molecular dynamics
Together with G. Robin (CNRS & Université d'Evry), I. Sekkat (CERMICS) and G. Victorino Cardoso (CMAP, Ecole polytechnique & IHU LIRYC), T. Lelièvre and G. Stoltz considered in 63 how to generate reactive trajectories linking two metastable states. More precisely, they investigated the capabilities and limitations of supervised and unsupervised methods based on variational autoencoders to generate such paths. Bottleneck autoencoders are however somewhat limited in describing reactive paths, which is why alternative approaches based on an importance sampling function determined by a reinforcement learning strategy were also studied. The potential of the approach was demonstrated on simple low dimensional examples.
A. Levitt and G. Stoltz studied in 17 with F. Bottin (CEA/DAM), A. Castellano (CEA/DAM), J. Bouchet (CEA Cadarache) how to train simple empirical force fields on abinitio data, in order to reproduce thermodynamic properties at finite temperature. The method 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 abinitio data. Thermodynamic consistency is ensured via some nonlinear reweighting procedure.
6.2.5 Interacting particle methods for sampling
In some situations, stochastic numerical methods can be made more efficient by using various replicas of the system. The ensemble Kalman filter is a methodology for incorporating noisy data into complex dynamical models to enhance predictive capability. It 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. In order to address this issue, U. Vaes together with J. A. Carrillo (University of Oxford, United Kingdom), F. Hoffmann (Hausdorff Center for Mathematics, Germany) and A. M. Stuart (Caltech, USA) provided in 51 an analysis of the accuracy of the ensemble Kalman filter beyond the Gaussian setting. The analysis is developed for the mean field ensemble Kalman filter, which can be rewritten in terms of maps on probability measures. These maps are proved to be locally Lipschitz in an appropriate weighted total variation metric, which enables to demonstrate that, if the true filtering distribution is close to Gaussian after appropriate lifting to the joint space of state and data, then it is well approximated by the ensemble Kalman filter.
In 60, Tony Lelièvre and Panos Parpas (Imperial College London, United Kingdom) introduce a new stochastic algorithm to locate the index1 saddle points of a potentiel energy function. Finding index1 saddle points is crucial to build kinetic Monte Carlo models. These models describe the evolution of the molecular system by a jump Markov model with values in the local minima of the energy function, the jumps between these states being parameterized by the Eyring–Kramers laws. This paramaterization thus requires to identify the index1 saddle points which connect local minima. The proposed algorithm can be seen as an equivalent of the stochastic gradient descent which is a natural stochastic process to locate local minima. It relies on two ingredients: (i) the concentration properties on index1 saddle points of the first eigenmodes of the Witten Laplacian on 1forms and (ii) a probabilistic representation of the solution to a partial differential equation involving this differential operator. The resulting algorithm is an interacting particle system, where the particles populate neighborhoods of the index1 saddle points. Numerical examples on simple molecular systems illustrate the efficacy of the proposed approach.
6.3 Homogenization
Participants: Yves Achdou, Rutger Biezemans, Rémi Goudey, Claude Le Bris, Albéric Lefort, Frédéric Legoll, Alexei Lozinski, Simon Ruget.
6.3.1 Deterministic nonperiodic systems
From the theoretical viewpoint, the projectteam has pursued the development of a general theory for homogenization of deterministic materials modeled as periodic structures with defects. This work, performed in collaboration with X. Blanc, P.L. Lions and P. Souganidis, has also been the topic of the PhD thesis of R. Goudey, defended this year. We recall that the periodic setting is the oldest traditional setting for homogenization. Alternative settings include the quasi and almostperiodic settings, and the random stationary setting. From a modeling viewpoint, assuming that multiscale materials are periodic is however an idealistic assumption: natural media (such as the subsoil) have no reason to be periodic, and manufactured materials, even though indeed sometimes designed to be periodic, are often not periodic in practice, e.g. because of imperfect manufacturing processes, of small geometric details that break the periodicity and can be critical in terms of industrial performances, ...Quasi and almostperiodic settings are not appropriate answers to this difficulty. Using a random stationary setting may be tempting from a modelization viewpoint (in the sense that all that is not known about the microstructure can be “hidden” in a probabilistic description), but this often leads to prohibitively expensive computations, since the model is very general. The direction explored by the projectteam consists in considering periodic structures with defects, a setting which is rich enough to fit reality while still leading to affordable computations.
Considering defects in the structure raises many mathematical questions. From an overall perspective, homogenization is based upon the determination of corrector functions, useful to compute the homogenized properties of the materials as well as to provide a finescale description of the oscillatory solution. In general, corrector problems are posed on the whole space. In the periodic and random stationary settings, it turns out that the corrector problems can actually be posed on a bounded domain. Powerful tools (e.g. Rellich compactness theorems) can then be used (to establish wellposedness and qualitative properties of the correctors, ...). The presence of defects breaks this property, making the corrector problem noncompact. Additional tools (such as the concentrationcompactness method or the theory of CalderónZygmund operators) are required to circumvent this difficulty.
Starting from the simplest case (localized defects in a purely diffusive equation, a setting for which we were able to show twoscale expansion results), we have followed two directions: (i) considering more complex equations (advectiondiffusion equations, HamiltonJacobi equations, ...) for which the defects, although localized, may have an impact on a larger and larger neighborhood, and (ii) considering more complex (i.e. less localized) defects:
 In line with the first direction, and in the context of the "délégation" of Y. Achdou (on partial leave from Université ParisCité), C. Le Bris has studied in 38 a homogenization theory for a general first order HamiltonJacobi equation in the presence of defects. The study extends to the fully nonlinear setting previous studies performed by X. Blanc, C. Le Bris and P.L. Lions in the linear (mostly elliptic) setting. It also extends the class of problems previously addressed by Y. Achdou and his collaborators in the periodic setting only. The study complements, in a slightly different but related regime, results obtained a few years ago by P.L. Lions and P. Souganidis.
 The works 23 (where defects become increasingly rare but do not decay at infinity) and 24, both by R. Goudey, fall within the second direction. Defects in the form of interfaces between two perfectly periodic materials also fall within this research direction. In the same vein, R. Goudey and C. Le Bris have studied in 59 the homogenized limit of a Schroedinger equation with an highly oscillatory potential, when the latter potential belongs to a general class of nonperiodic functions that are global perturbations of a periodic function. Such a class of functions is reminiscent of classes constructed two decades ago, in collaboration with X. Blanc and P.L. Lions, in the context of thermodynamic limit problems. The result obtained may be seen as a first step toward similar studies for other types of equations.
A monograph that summarizes the contributions of the projectteam on this topic, along with a general perspective on the field, has been written by C. Le Bris, in collaboration with X. Blanc. The French and English versions of this textbook are respectively in print for the series "Maths & Applications" and "MS&A, Modeling, Simulation and Applications", both at Springer. In addition, C. Le Bris has written a short text that summarizes the major results obtained and that will be published in the "Séminaire Laurent Schwartz 20222023 volume".
6.3.2 Inverse multiscale problems
In the context of the PhD of S. Ruget, which started this year, 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 an homogenization approach. This work is a followup on earlier works over the years in collaboration with K. Li, S. Lemaire and O. Gorynina, in the case of a diffusion equation with highly oscillatory diffusion coefficients. Here, this question is revisited in the setting of Schroedinger equations with rapidly oscillating potentials. The motivation for this work is that Schroedinger equations, besides their own interest, show some specific features (in comparison e.g. to diffusion equations) bringing hope that further progress can be achieved. To address these questions is the objective of the PhD of S. Ruget.
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 when 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 precomputed basis, obtained by solving local problems mimicking the problem at hand at the scale of mesh elements. This basis differs from standard (e.g. polynomial) bases that are generally used in existing legacy codes in industry. As a result, the MsFEM approach is intrusive, which hinders its adoption in industrial (and, more generally, nonacademic) environments.
To overcome this obstacle, R. Biezemans, C. Le Bris and F. Legoll, together with A. Lozinski (in delegation in the team for the first half of the year), have designed modified MsFEM approaches that allow for a nonintrusive implementation, i.e., using any existing legacy code for a Galerkin approximation on a piecewise affine basis. The key principles of the approach are presented in 12. The technique is reminiscent of the classical approach to homogenization: "corrector" functions are computed in each element of the mesh, from which slowly varying effective coefficients are computed. This leads to an effective PDE that can indeed be solved by standard finite element approaches.
A more comprehensive study of the nonintrusive MsFEM technique has subsequently been finalized in 41, where R. Biezemans, C. Le Bris, F. Legoll and A. Lozinski show that the nonintrusive approach can be extended to a wide variety of problems and more advanced, state of the art MsFEM variants. Indeed, this work provides nonintrusive MsFEMs for general linear second order PDEs, and with various choices for the local problems, in particular the use of an oversampling technique (where the local problems are solved on domains that are larger than a single mesh element) and a CrouzeixRaviart MsFEM. This work is also the first to propose and test a new MsFEM variant, namely by developing an oversampling technique for the CrouzeixRaviart MsFEM. Further, it is proved there that the nonintrusive approach is equivalent to the original MsFEM for a PetrovGalerkin variant of the MsFEM with affine test functions, and for the Galerkin MsFEM, numerical experiments show that the nonintrusive approach preserves the same accuracy as the original MsFEM.
A second research direction pursued in the PhD of R. Biezemans is the question of how to design accurate MsFEM approaches for various types of equations, beyond the purely diffusive case, and in particular for the case of multiscale advectiondiffusion problems, in the advectiondominated 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 by R. Biezemans, C. Le Bris, F. Legoll and A. Lozinski. The main results are being prepared for publication, showing in particular the stabilization of an MsFEM with weak continuity conditions of CrouzeixRaviart type by adding specific bubble functions, satisfying the same type of weak boundary conditions, to the approximation space.
Finally, R. Biezemans, C. Le Bris, F. Legoll and A. Lozinski have continued their study of the convergence analysis of MsFEMs. Indeed, despite the fact that MsFEM approaches have been proposed more than two decades ago, it turns out that not all specific settings are covered by the numerical analyses existing in the literature. The research team have previously extended the analysis of MsFEM to the case of rectangular meshes and that of periodic diffusion coefficients that are not necessarily Hölder continuous. An ongoing research effort is devoted to further generalizing the analysis to nonperiodic settings and to provide a fully rigorous convergence proof of various MsFEMs with the oversampling technique.
In the context of the PhD of A. Lefort, which started this year, C. Le Bris and F. Legoll have undertaken the study of a multiscale, timedependent, reactiondiffusion equation. This problem is different from the equations previously studied by the team by the fact that it is timedependent and that it includes a reaction term (in addition to the diffusive term). From a numerical perspective, two difficulties are present in 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: a standard marching scheme such as the backward Euler scheme would need a small timestep to provide an accurate solution. Several directions of research have been identified, such as establishing the homogenized limit of the problem and designing efficient numerical approaches.
6.4 Various topics
6.4.1 Complex fluids
Participants: Sébastien Boyaval.
In 2022, S. Boyaval has improved the mathematical understanding of the symmetrichyperbolic system of conservation laws introduced in 2020 to model nonNewtonian fluids 43. Precisely, he has established rigorously the structural stability of the model: Newtonian fluids are recovered in one asymptotic limit of the PDE parameters, while the elastodynamics of hyperelastic materials is also recovered in another asymptotic limit of the parameters 44. Researches are pursued for efficient numerical simulations.6.4.2 Modelorder reduction methods
Participants: Jad Dabaghi, Virginie Ehrlacher.
The objective of a reducedorder model 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 reducedorder model method then consists in constructing, from a few complex simulations which were performed for a small number of wellchosen values of the parameters, a socalled reduced model, much cheaper and quicker to solve from a numerical point of view, and which enables to get an accurate approximation of the solution of the model for any other values of the parameters.
In 64, together with Idrissa Niakh, Guillaume Drouet (EDF) and Alexandre Ern (SERENA), a new stable model reduction method for linear variational inequalities with parameterdependent constraints. The method was applied for the reduction of various parametrized contact mechanical problems.
In 39, a new modelorder reduction method based on optimal transport theory was investigated by Virginie Ehrlacher together with B. Battisti, T. Blickhan (Garching, Germany), G. Enchéry (IFPEN), D. Lombardi (INRIA COMMEDIA) and O. Mula (Eindhoven University). This approach, based on the use of Wasserstein barycenters, was successfully applied to the reduction of parametrized porous medium flow problems.
6.4.3 Crossdiffusion systems
Participants: Jean CauvinVila, Jad Dabaghi, Virginie Ehrlacher.
Crossdiffusion systems are nonlinear degenerate parabolic systems which naturally arise in diffusion models of multispecies 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 55, Jad Dabaghi and Virginie Ehrlacher investigated a new structurepreserving model reduction methods for parametrized crossdiffusion systems. The proposed numerical method is analyzed from a mathematical point of view and proved to satisfy the same mathematical properties as the high fidelity modelity (preserving for instance the nonnegativeness of the solutions). The method consists in introducing an appropriate nonlinear transformation of the set of solutions, based on the use of entropy variables associated to the system, before applying reducedbasis techniques to the parametrized problem.
In 52, Jean CauvinVila, Virginie Ehrlacher and Amaury Hayat analyzed from a mathematical point of view the boundary stabilization of onedimensional (linearized) crossdiffusion system in a moving domain. They prove that the system can be stabilized in any arbitrary finite time by using the socalled backstepping stabilization technique.
7 Bilateral contracts and grants with industry
Many research activities of the projectteam are conducted in close collaboration with private or public companies: CEA, EDF, IFPEN, Sanofi, OSMOS Group, SAFRANTech. The projectteam 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 UrbanaChampaign 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 PhysicoChimique (Paris). The LIA has been renewed for 4 years, starting January 1st, 2018.
Eric Cancès is one of the PIs of the Simons Targeted Grant “Moiré materials magic” (September 2021  August 2026). His coPIs 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 (MaxPlanck Institut), Maciej Zworski (UC Berkeley).
8.2 International research visitors
8.2.1 Visits of international scientists
Danny Perez (Los Alamos National Laboratory, USA) visited the team in September and October. This was the opportunity to discuss new research directions for accelerated molecular dynamics algorithms.
8.2.2 Visits to international teams
Research stays abroad
Claude Le Bris

Visited institution:
Freie Universitat, HumboldtUniversitat, Technische Universitat, Weierstrass Institute for Applied Analysis and Stochastics, and Zuse Institute Berlin

Country:
Germany

Dates:
NovemberDecember

Context of the visit:
MATH+ Distinguished Visiting Scholar, Berlin Mathematics Research Center

Mobility program/type of mobility:
Research stay
8.3 European initiatives
8.3.1 H2020 projects
EMC2
Participants: Noé Blassel, Eric Cancès, Shiva Darshan, Gaspard Kemlin, Alfred Kirsch, Eloïse Letournel, Antoine Levitt, Solal PerrinRoussel, Régis Santet, Renato Spacek, Gabriel Stoltz, Laurent Vidal, Urbain Vaes.
EMC2 project on cordis.europa.eu

Title:
Extremescale Mathematicallybased Computational Chemistry

Duration:
From September 1, 2019 to February 28, 2026

Partners:
 Institut National de Recherche en Informatique et Automatique (INRIA), France
 École Nationale des Ponts et Chaussées (ENPC), France
 Centre National de la Recherche Scientifique (CNRS), France
 Sorbonne Université, France

Inria contact:
Laura Grigori (Inria 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 multimillion 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 stronglycorrelated 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.
TIMEX
Participants: Olga Gorynina, Frédéric Legoll, Tony Lelièvre.

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 parallelintime 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. TIMEX will take the next leap in the development and deployment of this promising new approach for massively parallel HPC simulation, enabling efficient parallelintime integration for reallife applications. We will:
(i) provide software for parallelintime integration on current and future Exascale HPC architectures, delivering substantial improvements in parallel scaling;
(ii) develop novel algorithmic concepts for parallelintime integration, deepening our mathematical understanding of their convergence behaviour and including advances in multiscale methodology;
(iii) demonstrate the impact of parallelintime 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 interdisciplinary TIMEX Consortium unites top researchers from numerical analysis and applied mathematics, computer science and the selected application domains. Europe is leading research in parallelintime integration. TIMEX 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 parallelintime 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.
8.4 National initiatives
The projectteam is involved in several ANR projects:
 S. Boyaval is the PI of the ANR JCJC project SEDIFLO (20162022) to investigate new numerical models of solid transport in rivers.
 V. Ehrlacher is the PI of the ANR project COMODO (20202025) which focuses on the development of efficient numerical methods to simulate crossdiffusion systems on moving domains, with application to the simulation of the fabrication process of thin film solar cells. It includes Inria projectteams from Lille and SophiaAntipolis as well as research teams from Germany.
 V. Ehrlacher is the PI of the ANR TremplinERC project HighDim (20222025) which focuses on the development of efficient numerical methods for the resolution of highdimensional partial Differential Equations, using machine learning and neural networks.
 V. Ehrlacher is a member of the ANR project ADAPT (20182023), PI: D. Lombardi, Inria COMMEDIA teamproject. This project is concerned with the parallelization of tensor methods for highdimensional problems.
 T. Lelièvre is responsible of the node "Ecole des Ponts" of the ANR QuAMProcs (20192023), to which G. Stoltz also participates, PI: L. Michel, Université de Bordeaux.
 G. Stoltz is the PI of the ANR project SINEQ (20222025), 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 ParisDauphine and the SIMSART projectteam of Inria Rennes.
Members of the projectteam are participating in the following GdR:
 AMORE (Advanced Model Order REduction),
 DYNQUA (time evolution of quantum systems),
 EGRIN (gravity flows),
 IAMAT (Artificial Intelligence for MATerials),
 MANU (MAthematics for NUclear applications),
 MASCOTNUM (stochastic methods for the analysis of numerical codes),
 MEPHY (multiphase flows),
 NBODY (electronic structure),
 REST (theoretical spectroscopy).
The projectteam 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 projectteam).
9 Dissemination
9.1 Promoting scientific activities
S. Boyaval
 is the director of Laboratoire d’Hydraulique SaintVenant (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 MFO scientific committee (Oberwolfach),
 is a member of the editorial boards of Mathematical Modelling and Numerical Analysis $\text{(2006}\phantom{\rule{4.pt}{0ex}}\text{)}$, SIAM Journal of Scientific Computing (2008), SIAM Multiscale Modeling and Simulation $\text{(2012)}$, and the Journal of Computational Mathematics (2017),
 is a member of the editorial board of the Springer series “Mathematics and Molecular Modeling”,
 is a member of the committees of the GDRs DynQua, NBody, and REST,
 has coorganized the IPAM program on “Advancing Quantum Mechanics with Mathematics and Statistics” (UCLA, MarchJune 2022), as well as two workshops in this program, a minischool on mathematics for theoretical chemistry and physics (Jussieu, May 30June 1), the 2022 DFTK summer school (Jussieu, Aug. 2931), and the 2022 Solid Math workshop (Trieste, Sept. 69).
V. Ehrlacher
 is a member of the “Conseil d'Administration” of Ecole des Ponts,
 is a member of the “Conseil d'Administration” of the COMUE ParisEst,
 is a member of the CordiS selection committee of INRIA,
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), 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), 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),
 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.
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 review panel for research area proposals for several German universities.
T. Lelièvre
 is a member of the editorial boards of Foundations of Computational Mathematics, 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 (until October 2022),
 is the head of the applied mathematics department (CERMICS) at Ecole des Ponts,
 has coorganized the workshop Machine LearningAssisted Sampling for Scientific Computing  Applications in Physics, Collège de France  site Ulm, Paris, October 34th 2022 (with M. Gabrié and V. de Bortoli).
A. Levitt coorganizes 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),
 is a member of the executive board of GDR IAMAT,
 coorganized with M. Bianciotto (Sanofi), F. Gervasio (UCL London), P. Gkeka (Sanofi), C. Hartmann (BTU Cottbus), a CECAM workshop on “Chasing CVs using Machine Learning: from methods development to biophysical applications”, which took place at Inria Paris from June 28th to 30th.
9.2 Teaching  Supervision  Juries
9.2.1 Teaching
The members of the projectteam have taught the following courses.
At École des Ponts 1st year (equivalent to L3):
 Analyse et calcul scientifique, 30h (R. Biezemans, J. CauvinVila, V. Ehrlacher, E. Letournel, R. Santet, G. Stoltz)
 Équations aux dérivées partielles: approches variationnelles, 15h (R. Biezemans, J. CauvinVila, F. Legoll, R. Santet)
 Probabilités, 24h (N. Blassel)
 Hydraulique numérique, 15h (S. Boyaval)
 Outils mathématiques pour l’ingénieur (E. Cancès: 18h, V. Ehrlacher, E. Letournel, F. Legoll, A. Levitt, G. Stoltz, L. Vidal: 9h)
 Mécanique quantique, 15h (E. Cancès, A. Levitt)
 Méthodes numériques pour les problèmes en grande dimension, 17h30 (V. Ehrlacher)
 Pratique du calcul scientifique, 15h (A. Levitt)
 Initiation au travail en projet, 13h (R. Santet, R. Spacek, L. Vidal)
 PAMS project, 16h (R. Spacek)
At École des Ponts 2nd year (equivalent to M1):
 Contrôle de systèmes dynamiques et équations aux dérivées partielles, 18h (E. Cancès)
 Statistiques numériques et analyse de données, 26h (S. Darshan)
 Problèmes d'évolution, 36h (V. Ehrlacher, F. Legoll)
 Projets Modéliser Programmer Simuler (T. Lelièvre)
 Statistics and data sciences, 30h (G. Stoltz)
 Projet de physique statistique et mécanique quantique, 10h (G. Stoltz, L. Vidal)
At the M2 “Mathématiques de la modélisation” of Sorbonne Université:
 Théorie spectrale et méthodes variationnelles, 10h (E. Cancès)
 Problèmes multiéchelles, aspects théoriques et numériques, 24h (F. Legoll)
 Introduction to computational statistical physics, 20h (G. Stoltz)
At other institutions:
 Modal de Mathématiques Appliquées (MAP473D), 15h, Ecole Polytechnique (T. Lelièvre)
 Aléatoire (MAP361), 40h, Ecole Polytechnique (T. Lelièvre)
 Gestion des incertitudes et analyse de risque (MAP568), 20h, Ecole Polytechnique (T. Lelièvre)
 Théorie spectrale et mécanique quantique, 30h, ENS Paris (S. PerrinRoussel)
 Modélisation, 40h, ENS Paris (S. PerrinRoussel)
 Numerical Analysis (in Spring and Fall), $2\times 56$h, NYU Paris (U. Vaes)
9.2.2 Supervision
The following PhD theses supervised by members of the projectteam have been defended:
 Zineb Belkacemi, funding CIFRE SANOFI, Deciphering protein function with artificial intelligence, cosupervised by T. Lelièvre and G. Stoltz, defended in July
 Raed Blel, funding UM6P, Modelorder reduction methods for stochastic problems, cosupervised by V. Ehrlacher and T. Lelièvre, defended in June
 Rémi Goudey, funding CDSN, Homogenization problems in the presence of defects, supervised by C. Le Bris, defended in October
 Gaspard Kemlin, funding ERC Synergy EMC2, Mathematical and numerical analysis for electronic structures, cosupervised by E. Cancès and A. Levitt, defended in December
 Idrissa Niakh, thèse CIFRE EDF, Reduced basis for variational inequalities, cosupervised by V. Ehrlacher and A. Ern (Inria SERENA), defended in December
 Inass Sekkat, funding UM6P, Large scale Bayesian inference, supervised by G. Stoltz, defended in September
The following PhD theses supervised by members of the projectteam are ongoing:
 Hichem Belbal, thèse CIFRE EDF, Understanding suspended matter measures in Loire river, since September 2022, supervised by S. Boyaval
 Elisa Beteille, thèse CIFRE EDF, Propagation of Urban Flood waves, since November 2021, supervised by S. Boyaval
 Rutger Biezemans, funding DIM Math Innov (Inria), Difficult multiscale problems and nonintrusive approaches, Ecole des Ponts, since October 2020, cosupervised by C. Le Bris and A. Lozinski (University of Besançon)
 Noé Blassel, funding ERC Synergy EMC2, Approximation of the quasistationnary distribution, Ecole des Ponts, since October 2022, cosupervised by T. Lelièvre and G. Stoltz
 Andrea Bordignon, Mathematical and numerical analysis for Density Functional Theory, funding ERC Synergy EMC2, cosupervised by E. Cancès and A. Levitt
 Jean CauvinVila, funding Ecole des Ponts, Crossdiffusion systems on moving boundary domains, since October 2020, cosupervised by V. Ehrlacher and A. Hayat
 Yonah ConjugoTaumhas, thèse CIFRE CEA, Reduced basis methods for nonsymmetric eigenvalue problems, since October 2020, cosupervised by T. Lelièvre and V. Ehrlacher together with G. Dusson (CNRS Besançon) and F. Madiot (CEA)
 Shiva Darshan, funding ANR SINEQ, Linear response of constrained stochastic dynamics, since October 2022, cosupervised by G. Stoltz and S. Olla (Université ParisDauphine)
 Maria FuenteRuiz, funding INRIA, Parallel algorithms for tensor methods, since September 2020, cosupervised by V. Ehrlacher and D. Lombardi (Inria COMMEDIA)
 Abbas Kabalan, thèse CIFRE SAFRANTech, Reducedorder models for problems with nonparametric geometrical variations, since November 2022, cosupervised by V. Ehrlacher and F. Casenave (SAFRANTech)
 Albéric Lefort, funding CERMICSENPC, Multiscale numerical methods for reactiondiffusion equations and related problems, Ecole des Ponts, since November 2022, cosupervised by F. Legoll and C. Le Bris
 Eloïse Letournel, funding DIM Math Innov (Inria), Finite size effects in electronic structure, École des Ponts, since September 2021, supervised by A. Levitt
 Alfred Kirsch, funding Simons foundation, Mathematical and numerical analysis of interacting electrons models, École des Ponts, since September 2021, cosupervised by E. Cancès and D. Gontier (ParisDauphine CEREMADE)
 Solal PerrinRoussel, funding École des Ponts, Mathematical anlaysis and numerical simulation of electronic transport in moiré materials, cosupervised by É. Cances and by D. Gontier (CEREMADE, Université ParisDauphine PSL)
 Thomas Pigeon, funding Inria, Combining machine learning and quantum computations to discover new catalytic mechanisms, Université de Lyon, since October 2020, cosupervised by P. Raybaud (IFPEN) and T. Lelièvre, together with G. Stoltz and M. CorralVallero (IFPEN)
 Simon Ruget, funding Inria, Coarse approximation for a Schrödinger problem with highly oscillatory coefficients, Ecole des Ponts, since October 2022, cosupervised by F. Legoll and C. Le Bris
 Régis Santet, funding Ecole des Ponts, Enhancing the sampling efficiency of reversible and nonreversible dynamics, Ecole des Ponts, since October 2021, cosupervised by T. Lelièvre and G. Stoltz
 LevArcady Sellem, funding Advanced ERC QFeedback (PI: P. Rouchon), Mathematical approaches for simulation and control of open quantum systems, Ecole des Mines de Paris, since October 2020, cosupervised 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 SorbonneUniversité, since November 2021, cosupervised by G. Stoltz and P. Monmarché (Sorbonne Université)
 Jana Tarhini, thèse IFPEN, Fast simulation of CO2/H2 storage in geological bassins, since November 2021, supervised by S. Boyaval
 JeanPaul Travert, thèse CIFRE EDF, Data assimilation for flood predictions, since November 2022, supervised by S. Boyaval
 Laurent Vidal, funding ERC Synergy EMC2, Model reduction in physics and quantum chemistry, since February 2021, supervisd by E. Cancès and A. Levitt.
9.2.3 Juries
Projectteam members have participated in the following PhD juries:
 S. Boyaval, PhD of Omar Mokhtari (“Écoulement de solutions de polymères en milieux poreux: impact des effets viscoélastiques à l’échelle du pore sur les propriétés effectives à l’échelle de Darcy”), defended at INP Toulouse in July
 E. Cancès, PhD of Etienne Polack (“ Development of efficient multiscale methods and extrapolation techniques for multiphysics molecular chemistry”), defended at Sorbonne University in January (chair)
 E. Cancès, PhD of Augustin Blanchet (“De la surface au cœur des étoiles: vers une modélisation unifiée de la matière condensée aux plasmas”), defended at the University of Paris Saclay in February (chair)
 E. Cancès, PhD of Michele Nottoli (“Fast and accurate multi layer polarizable embedding strategies for the static and dynamic modeling of complex systems”), defended at the University of Pisa in February (referee)
 E. Cancès, PhD of Martin Mrovec (“Mathematical Methods of Modelling Electronic Structure of Large Systems”), defended at the University of Ostrava in April (referee)
 V. Ehrlacher, PhD of Kiran Kollepara ("Lowrank and sparse approximations for contact mechanics"), defended at Nantes University in July (referee)
 V. Ehrlacher, PhD of Philip Edel ("Reduced basis method for parameterdependent linear equations. Application to timeharmonic problems in electromagnetism and in aeroacoustics."), defended at Sorbonne University in October (referee)
 V. Ehrlacher, PhD of Emilie Bourne ("NonUniform Numerical Schemes for the Modelling of Turbulence in the 5D GYSELA Code"), defended at CEA Cadarache in December (referee)
 V. Ehrlacher, PhD of Katharina Eichinger ("Problèmes variationnels pour l’interpolation dans l’espace de Wasserstein"), defended at University ParisDauphine in December (referee)
 T. Lelièvre, PhD of Aurélien Enfroy ("Contributions à la conception, l'étude et la mise en œuvre de méthodes de Monte Carlo par chaîne de Markov appliquées à l'inférence bayésienne"), defended at Institut Polytechnique de Paris in July (chair)
 T. Lelièvre, PhD of Fiona Desplats ("Development of a hybrid method coupling deterministic and stochastic neutronic calculations: Applications to heterogeneous PWR and SFR calculations"), defended at Université Grenoble Alpes in October (referee)
 T. Lelièvre, PhD of Loris Felardos ("Datafree Generation of Molecular Configurations with Normalizing Flows"), defended at Université Grenoble Alpes in December (referee)
 A. Levitt, PhD of Jean Cazalis (“Systèmes quantiques non linéaires en dissociation : l'exemple du graphène”), defended at Université Paris Dauphine in July
 G. Stoltz, PhD of Paul Rohrbach ("Multilevel Monte Carlo simulation of soft matter using coarsegrained models"), defended at the University of Cambridge in October (referee)
 G. Stoltz, PhD of Benjamin Stottrup ("Spectral, scattering, and regularity properties related to various functional and differential equations"), defended at Aalborg University in April (referee)
 G. Stoltz, PhD of Lorenzo Campana ("Stochastic modeling of nonspherical particles in turbulence"), defended at Inria Sophia in March (referee)
 G. Stoltz, PhD of Clovis Lapointe ("Modélisation multiéchelles des défauts d’irradiation dans les métaux cubiques centrés"), defended at INSTN Saclay in February (referee)
Projectteam members have participated in the following habilitation juries:
 V. Ehrlacher and T. Lelièvre (referee), HdR of Marie BillaudFriess ("Contributions for the approximation and model order reduction of partial differential equations"), defended at Nantes University in October
Projectteam members have participated in the following selection committees:
 V. Ehrlacher, MCF position at Laboratoire de Mathématiques d'Orsay.
 V. Ehrlacher, member of the 2022 ANR project selection committee CE46 ("Modeling and simulation").
 T. Lelièvre, positions in applied mathematics, Ecole Polytechnique.
9.3 Conference participation
Members of the projectteam have delivered lectures in the following seminars, workshops and conferences:
 R. Biezemans, CERMICS Young Researchers Seminar, ChampssurMarne, June
 R. Biezemans, ECCOMAS Congress 2022, Oslo (Norway), June
 R. Biezemans, CANUM 2022, EvianlesBains, June
 R. Biezemans, SciCADE 2022, Reykjavik (Iceland), July
 R. Biezemans, Journées Scientifiques des Jeunes du Cermics (First edition), Provins, October
 S. Boyaval, HYP 2022, Malaga (Spain), June
 S. Boyaval, WAVES 2022, Palaiseau, July
 E. Cancès, IMA workshop on Nonlocal and Singular Problems: Recent Advances and Outlook, Singapore, February
 E. Cancès, IPAM workshop on Model Reduction in Quantum Mechanics, Los Angeles (USA), April
 E. Cancès, IPAM workshop on Moiré Materials, Los Angeles (USA), May
 E. Cancès, Séminaire de Mathématiques Appliquées, Collège de France, Paris, June
 E. Cancès, Séminaire des Mathématiques, Ecole Normale Supérieure, Paris, November
 E. Cancès, Mathematical Challenges in Quantum Mechanics, online, December
 J. CauvinVila, CANUM 2022, EvianlesBains, June
 J. CauvinVila, Workshop on “Nonlinear evolutionary equations and applications”, TU Chemnitz (Germany), September
 S. Darshan, CECAM MixedGen Season 3 – Session 2: Theory and numerical simulation of transport processes in condensed matter, online, December
 V. Ehrlacher, Séminaire du Collège de France, January
 V. Ehrlacher, SIAM Conference on Imaging Sciences, online, March
 V. Ehrlacher, Erwin Schrödinger Institute Program on "Computational Uncertainty Quantification: Mathematical Foundations, Methodology & Data", Vienna (Austria), May
 V. Ehrlacher, IPAM workshop on LargeScale Certified Numerical Methods in Quantum Mechanics, Los Angeles (USA), May
 V. Ehrlacher, MATHICSE seminar, EPFL (Switzerland), June
 V. Ehrlacher, 2022 Curves and Surfaces conference, Arcachon, June
 V. Ehrlacher, Congrès pour honorer la mémoire de Roland Glowinski, Sorbonne Université, July
 V. Ehrlacher, CEMRACS 2022 seminar, CIRM Luminy, August
 V. Ehrlacher, plenary talk at the 2022 MORE Conference, Berlin (Germany), September
 V. Ehrlacher, GdT Optimal Transport, Orsay, October
 V. Ehrlacher, Journée de la Fédération BourgogneFrancheComté, Besançon, November
 V. Ehrlacher, Séminaire MACS, Lyon, November
 V. Ehrlacher, Séminaire MOCO, Strasbourg, November
 V. Ehrlacher, SFB Colloquium, RWTH Aachen University (Germany), December
 V. Ehrlacher, Journée IFPENINRIA, December
 L. Garrigue, GDR quantum Nbody problem, online, January
 L. Garrigue, Séminaire de théorie spectrale, Institut Henri Poincaré, March
 L. Garrigue, Séminaire de physique mathématique, Dijon, March
 L. Garrigue, Séminaire de physique mathématique, Grenoble, April
 L. Garrigue, Workshop at the Norwegian Center for Advanced Study, Oslo (Norway), June
 L. Garrigue, 92th annual meeting of the International association of applied mathematics and mechanics, Aachen (Germany), August
 L. Garrigue, Conference Aspect'22, Oldenbourg (Germany), September
 L. Garrigue, Séminaire EDP, Besançon, December
 L. Garrigue, Séminaire physique mathématique, Stuttgart (Germany), December
 C. Le Bris, Séminaire Laurent Schwartz, Ecole Polyetchnique / IHES, October
 C. Le Bris, CRC1114 Colloquium, Berlin (Germany), December
 F. Legoll, workshop of the ANR QuAMProcs project, Paris, March
 F. Legoll, European Mechanics of Materials Conference (EMMC), Oxford (United Kingdom), April
 F. Legoll, Annual meeting of the TIMEX project, Leuven (Belgium), April
 F. Legoll, ECCOMAS Congress 2022, Oslo (Norway), June
 F. Legoll, seminar within the CEAEDFINRIA summer school on "Certification d'erreurs dans des simulations numériques", Saclay, June
 F. Legoll, ParallelinTime (PinT) conference, Marseille, July
 F. Legoll, World Congress on Computational Mechanics (WCCM), online meeting organized by Yokohama University (Japan), August
 F. Legoll, Congrès Français de Mécanique (CFM), Nantes, August
 F. Legoll, Numerical Analysis seminar of the Department of Mathematics of the University of Hong Kong, online, October
 T. Lelièvre, Workshop LIA CNRSUniversity of Illinois, Hauteluce, January
 T. Lelièvre, CMAP Seminar, Ecole Polytechnique, January
 T. Lelièvre, Workshop on Pólya urns and quasistationary distributions, Bath (United Kingdom), April
 T. Lelièvre, Journée Analyse Appliquée HautsdeFrance, May
 T. Lelièvre, Conference 30 years of Acta Numerica, Bedelewo (Poland), June
 T. Lelièvre, ICMMES conference, La Rochelle, June
 T. Lelièvre, CECAM meeting on the development of coarsegrained models, Lyon, November
 T. Lelièvre, Thematic meeting of the GT MASIM  GDR BIM, Paris, December
 E. Letournel, IPAM QMM2022, Los Angeles (USA), March to May
 E. Letournel, WAVES 2022, Palaiseau, July
 E. Letournel, CERMICS Young Researchers Seminar, ChampssurMarne, October
 A. Levitt, GDR NBODY, Toulouse, January
 A. Levitt, IPAM QMM2022, Los Angeles (USA), April
 A. Levitt, Collège de France, Paris, May
 A. Levitt, CECAM workshop, Error control in firstprinciples modelling, Lausanne (Switzerland), June
 A. Levitt, Lab seminar, Orsay, September
 R. Santet, CECAM MixedGen Season 2  Session 7: Simulating nonequilibrium phenomena and rareevents, online, April
 R. Santet, MCQMC 2022, Linz (Austria), July
 R. Spacek, CECAM Numerical Techniques for Nonequilibrium Steady States, Mainz (Germany), April
 R. Spacek, MCQMC 2022, Linz (Austria), July
 G. Stoltz, CECAM workshop “Numerical Techniques for Nonequilibrium Steady States”, Mainz (Germany), April
 G. Stoltz, NOMATEN International Conference on Materials Informatics, Warsaw (Poland), June
 G. Stoltz, Probability seminar, Université de Rennes, June
 G. Stoltz, CNRSICL workshop, London (UnitedKingdom), July
 G. Stoltz, plenary talk at MCQMC 2022, Linz (Austria), July
 G. Stoltz, workshop “Machinelearning assisted scientific computing”, Paris, October
 G. Stoltz, MASIM ML & sampling workshop, Paris, December
 U. Vaes, Mathematics of Machine Learning Seminar at UMass Amherst, online, December
 U. Vaes, Applied PDE Seminar, Imperial College London (United Kingdom), September
 U. Vaes, SIAM Mathematics of Data Science, online, September
 U. Vaes, MCQMC 2022, Linz (Austria), July
 U. Vaes, SIAM Annual Meeting, online, July
 U. Vaes, LMSBirmingham Workshop on “Stochastic/Partial Differential Equations: Analysis and Computations”, University of Birmingham (United Kingdom), June
 U. Vaes, Hausdorff Center for Mathematics workshop on “Synergies between Data Science and Partial Differential Equations”, Bonn (Germany), June
 U. Vaes, Erwin Schrödinger Institute workshop on “PDEconstrained Bayesian inverse uncertainty quantification”, online, May
 U. Vaes, Isaac Newton Institute workshop on “Frontiers in kinetic equations for plasmas and collective behaviour”, Cambridge (United Kingdom), April
 L. Vidal, CERMICS Young Researchers Seminar, "Direct minimization for wavefunction methods", ChampssurMarne, April
 L. Vidal, GAMM 2022 meeting, "On the approximation of energy bands in the Brillouin zone", Aachen (Germany), August
Members of the projectteam have delivered the following series of lectures:
 E. Cancès, Mathematical aspects of electronic structure theory, 3h lecture, Aussois, June
 E. Cancès, Planewave DFT calculations, 2h lecture, DFTK summer school, Paris, August
 C. Le Bris, Block course, Berlin Mathematical School, "Multiscale Problems and Homogenization", 15h lecture, Berlin (Germany), November and December
 G. Stoltz, High Dimensional Sampling and Applications, 2h lecture + 1h seminar, MACMIGS tutorial, Edinburgh (United Kingdom), November
Members of the projectteam have presented posters in the following seminars, workshops and international conferences:
 N. Blassel, CECAM online workshop, “Theory and numerical simulation of transport processes in condensed matter”, December
 E. Letournel, DFTK summer school, Sorbonne Université, August
 E. Letournel, ISTCP 2022, Aussois, June
 R. Santet, CECAM workshop “Numerical Techniques for Nonequilibrium Steady States”, Mainz (Germany), April
 R. Santet, Journée des doctorants MSTIC 2022, NoisyleGrand, June
 R. Spacek, CECAM MixedGen Season 2  Session 7: Simulating nonequilibrium phenomena and rareevents, online, April
 R. Spacek, CECAM online workshop, “Theory and numerical simulation of transport processes in condensed matter”, December
Members of the projectteam have participated (without giving talks nor presenting posters) in the following seminars, workshops and international conferences:
 R. Biezemans, Semaine d'Etude MathématiquesEntreprises, Rennes, May
 A. Bordignon, ISTCP 2022, Aussois, June
 A. Bordignon, GDR Nbody minischool, Sorbonne Université, June
 A. Bordignon, Journées Scientifiques des Jeunes du Cermics (First edition), Provins, October
 J. CauvinVila, Hausdorff Center for Mathematics School on “Diffusive Systems Part II”, Bonn (Germany), April
 S. Darshan, Statistical Physics of Complex Systems, Bangalore (India), December
 A. Kirsch, IPAM QMM2022, Los Angeles (USA), April
 A. Kirsch, International Summer School on Computational Quantum Materials, Sherbrooke University (USA), June
 A. Lefort, Journées Scientifiques des Jeunes du Cermics (First edition), Provins, October
 E. Letournel, GDR Nbody minischool, Sorbonne Université, June
 E. Letournel, ERC Synergy EMC2 workshop, September
 E. Polack, L. Vidal, ERC Synergy EMC2 workshop, September
 S. PerrinRoussel, ISTCP 2022, Aussois, June
 S. PerrinRoussel, Solid Math 2022, Trieste (Italy), September
 S. PerrinRoussel, ERC Synergy EMC2 workshop, September
 M. Rachid, Rencontre ANR QuAMProcs, Paris, March
 M. Rachid, EDPs et Probabilités, Bordeaux, October
 S. Ruget, Journées Scientifiques des Jeunes du Cermics (First edition), Provins, October
 L. Vidal, ISTCP 2022, Aussois, June
 L. Vidal, ERC Synergy EMC2 workshop, September
9.4 Popularization
 G. Stoltz corealized a video of the webmagazine Ingenius from Ecole des Ponts on "What is numerical statistical physics?"
 Three researchers of the team (Virginie Ehrlacher, Tony Lelièvre, Gabriel Stoltz) realized videos explaining their scientific activities. Those are available on the webpage of CERMICS.
10 Scientific production
10.1 Major publications
 1 miscComputational Quantum Chemistry: A Primer.2003
 2 bookMathematical Methods in Quantum Chemistry. An Introduction. (Méthodes mathématiques en chimie quantique. Une introduction.).Mathématiques et Applications (Berlin) 53. Berlin: Springer. xvi, 409~p. 2006
 3 bookThe Mathematical Theory of Thermodynamic Limits: ThomasFermi Type Models.Oxford Mathematical Monographs. Oxford: Clarendon Press. xiii, 277~p.1998
 4 bookMathematical Methods for the Magnetohydrodynamics of Liquid Metals.Numerical Mathematics and Scientific Computation. Oxford: Oxford University Press., 324~p.2006
 5 bookParabolic Equations with Irregular Data and Related Issues: Applications to Stochastic Differential Equations.4De Gruyter Series in Applied and Numerical Mathematics2019
 6 bookMultiscale Analysis. Modeling and Simulation. (Systèmes multiéchelles. Modélisation et simulation.).Mathématiques et Applications (Berlin) 47. Berlin: Springer. xi, 212~p.2005
 7 bookFree Energy Computations: A Mathematical Perspective.Imperial College Press, 458~p.2010
10.2 Publications of the year
International journals
 8 articleOn the Hill relation and the mean reaction time for metastable processes.Stochastic Processes and their Applications155January 2023, 393436
 9 articleChasing Collective Variables using Autoencoders and biased trajectories.Journal of Chemical Theory and Computation181January 2022, 5978
 10 articleMulticenter decomposition of molecular densities: a mathematical perspective.Journal of Chemical Physics156April 2022, 164107
 11 articleHypocoercivity with Schur complements.Annales Henri Lebesgue5May 2022, 523557
 12 articleNonintrusive implementation of Multiscale Finite Element Methods: an illustrative example.Journal of Computational Physics2023
 13 articleFiniteVolume approximation of the invariant measure of a viscous stochastic scalar conservation law.IMA Journal of Numerical Analysis423July 2022, 27102770

14
articleWeighted
${L}^{2}$ contractivity of Langevin dynamics with singular potentials.Nonlinearity352January 2022, 9981035  15 articlePractical error bounds for properties in planewave electronic structure calculations.SIAM Journal on Scientific Computing445October 2022
 16 articleConsensus Based Sampling.Studies in Applied Mathematics1483January 2022, 10691140
 17 articleAb initio Canonical Sampling based on Variational Inference.Physical Review B10616October 2022, L161110
 18 articleAn introductory review on a posteriori error estimation in Finite Element computations.SIAM Review2022
 19 articleFinitesize effects in response functions of molecular systems.SMAI Journal of Computational MathematicsDecember 2022
 20 articleSoTT: greedy approximation of a tensor as a sum of Tensor Trains.SIAM Journal on Scientific Computing442March 2022
 21 articleAdaptive force biasing algorithms: new convergence results and tensor approximations of the bias.The Annals of Applied Probability325October 2022
 22 articleMathematical analysis of a coupling method for the practical computation of homogenized coefficients.ESAIM: Control, Optimisation and Calculus of Variations282022, 44
 23 articleA periodic homogenization problem with defects rare at infinity.Networks & Heterogeneous Media1742022, 547592
 24 articleElliptic homogenization with almost translationinvariant coefficients.Asymptotic AnalysisPrepressJune 2022, 142
 25 articleEnhanced sampling methods for molecular dynamics simulations.Living Journal of Computational Molecular Science412022
 26 articleA robust and efficient line search for selfconsistent field iterations.Journal of Computational Physics459June 2022
 27 articleAn adaptive parareal algorithm: application to the simulation of molecular dynamics trajectories.SIAM Journal on Scientific Computing4412022, B146B176
 28 articleAn MsFEM approach enriched using Legendre polynomials.Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal2022022, 798834
 29 articleThe exit from a metastable state: concentration of the exit point distribution on the low energy saddle points, part 2.Stochastics and Partial Differential Equations: Analysis and Computations1012022, 317–357
 30 articleThe Adaptive Biasing Force algorithm with nonconservative forces and related topics.ESAIM: Mathematical Modelling and Numerical Analysis562March 2022, 529564
 31 articleA probabilistic study of the kinetic FokkerPlanck equation in cylindrical domains.Journal of Evolution Equations22382022
 32 articleQuasistationary distribution for the Langevin process in cylindrical domains, part I: existence, uniqueness and longtime convergence.Stochastic Processes and their Applications144February 2022, 173201
 33 articleDerivativefree Bayesian Inversion Using Multiscale Dynamics.SIAM Journal on Applied Dynamical Systems211January 2022, 284326
 34 articleMathematical foundations for the Parallel Replica algorithm applied to the underdamped Langevin dynamics.MRS CommunicationsJuly 2022
 35 articlePhotoionization and core resonances from rangeseparated densityfunctional theory: General formalism and example of the beryllium atom.Journal of Chemical Physics156222022, 224106
 36 articlePhotoionization and core resonances from rangeseparated timedependent densityfunctional theory for openshell states: Example of the lithium atom.Journal of Chemical Physics157December 2022, 244104
International peerreviewed conferences
 37 inproceedingsRéduction de modèles pour les inéquations variationnelles.15e colloque national en calcul des structuresHyèreslesPalmiers, FranceMay 2022
Reports & preprints
 38 miscHomogenization of some periodic HamiltonJacobi equations with defects.November 2022
 39 miscWasserstein model reduction approach for parametrized flow problems in porous media.May 2022
 40 miscDeep learningbased schemes for singularly perturbed convectiondiffusion problems.May 2022
 41 miscNonintrusive implementation of a wide variety of Multiscale Finite Element Methods.December 2022
 42 reportSome solutions for nonhomogeneous smoothing.IRMAR, University of Rennes 1May 2022
 43 miscA viscoelastic flow model of maxwelltype with a symmetrichyperbolic formulation.December 2022
 44 miscAbout the structural stability of Maxwell fluids: convergence toward elastodynamics.December 2022
 45 miscOn basis set optimisation in quantum chemistry.December 2022
 46 miscFinite Volumes for the StefanMaxwell CrossDiffusion System.2022
 47 miscA simple derivation of moiréscale continuous models for twisted bilayer graphene.November 2022
 48 miscModifiedOperator Method for the Calculation of Band Diagrams of Crystalline Materials.October 2022
 49 miscNumerical stability and efficiency of response property calculations in density functional theory.October 2022
 50 miscA priori error analysis of linear and nonlinear periodic Schrödinger equations with analytic potentials.June 2022
 51 miscThe Ensemble Kalman Filter in the NearGaussian Setting.December 2022
 52 miscBoundary stabilization of onedimensional crossdiffusion systems in a moving domain: linearized system.January 2023
 53 miscComputation of the selfdiffusion coefficient with lowrank tensor methods: application to the simulation of a crossdiffusion system.April 2022
 54 miscTensor approximation of the selfdiffusion matrix of tagged particle processes.October 2022
 55 miscStructurepreserving reduced order model for parametric crossdiffusion systems.June 2022
 56 miscEfficient extraction of resonant states in systems with defects.September 2022
 57 miscMixed state representability of entropydensity pairs.March 2022
 58 miscCombining machinelearned and empirical force fields with the parareal algorithm: application to the diffusion of atomistic defects.December 2022
 59 miscLinear elliptic homogenization for a class of highly oscillating nonperiodic potentials.June 2022
 60 miscUsing Witten Laplacians to locate index1 saddle points.December 2022
 61 miscEyringKramers exit rates for the overdamped Langevin dynamics: the case with saddle points on the boundary.July 2022
 62 miscEstimation of statistics of transitions and Hill relation for Langevin dynamics.June 2022
 63 miscGenerative methods for sampling transition paths in molecular dynamics.May 2022
 64 miscStable model reduction for linear variational inequalities with parameterdependent constraints.September 2022
 65 miscMobility estimation for Langevin dynamics using control variates.June 2022
 66 miscError estimates and variance reduction for nonequilibrium stochastic dynamics.November 2022
 67 miscComplete Positivity Violation in Higherorder Quantum Adiabatic Elimination.November 2022