2025​​​‌Activity reportProject-TeamMATHERIALS‌

6.1.1 DFTK

• Keywords:
  Molecular‌ simulation, Quantum chemistry, Materials‌​‌
• Functional Description:
  DFTK, short​​ for the density-functional toolkit,​​​‌ is a Julia library‌ implementing plane-wave density functional‌​‌ theory for the simulation​​ of the electronic structure​​​‌ of molecules and materials.‌ It aims at providing‌​‌ a simple platform for​​ experimentation and algorithm development​​​‌ for scientists of different‌ backgrounds.
• Release Contributions:
  In‌​‌ 2025 DFTK continued to​​ be actively developed, and​​​‌ it received several contributions‌ from members of MATHERIALS.‌​‌ The library has been​​ used for several publications​​​‌ both inside and outside‌ the project-team.
• URL:
  http://­dftk.­org‌​‌
• Contact:
  Antoine Levitt

7.1.1 Density‌​‌ functional theory

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

In a​​ joint work with A.​​​‌ Levitt and J. Thomas‌ (University Paris Saclay), E.‌​‌ Cancès studies the decay​​ of the interatomic force​​​‌ constants (equivalently, the smoothness‌ properties of the dynamical‌​‌ matrix) in perfect crystals​​ both at finite electronic​​​‌ temperature, and for insulators‌ at zero temperature, within‌​‌ the reduced Hartree–Fock approximation​​ (also called Random Phase​​​‌ Approximation). This model is‌ obtained by setting to‌​‌ zero the exchange-correlation energy​​ functional is Kohn–Sham Density​​​‌ Functional Theory. At finite‌ temperature the electrons are‌​‌ mobile, leading to exponential​​ decay of the force​​​‌ constants. In insulators, there‌ is incomplete screening, leading‌​‌ to an algebraic decay​​ of dipole-dipole interaction type.​​​‌ This is the first‌ fully rigorous mathematical proof‌​‌ of a well-known phenomenon​​ in solid-state physics.

7.1.2​​​‌ Electronic excited states

Computing‌ excited states of many-body‌​‌ quantum Hamiltonians is a​​​‌ fundamental challenge in computational​ physics and chemistry, with​‌ state-of-the-art methods broadly classified​​ into variational (critical point​​​‌ search) and linear response​ approaches. The Kähler manifold​‌ formalism provides a uniform​​ framework which naturally accommodates​​​‌ both strategies for a​ wide range of variational​‌ models, including Hartree-Fock, CASSCF,​​ Full CI, and adiabatic​​​‌ TDDFT. In particular, this​ formalism leads to a​‌ systematic and straightforward way​​ to obtain the final​​​‌ equations of linear response​ theory for nonlinear models,​‌ which provides, in the​​ case of mean-field models​​​‌ (Hartree-Fock and DFT), a​ simple alternative to Casida's​‌ derivation. In a joint​​ work with Y. Hu​​​‌ (Ecole des Ponts IPP),​ E. Cancès and L.​‌ Grazioli detail the mathematical​​ structure of Hamiltonian dynamics​​​‌ on Kähler manifolds, establish​ connections to standard quantum​‌ chemistry equations, and provide​​ theoretical and numerical comparisons​​​‌ of excitation energy computation​ schemes at the Hartree-Fock​‌ level of theory.

7.1.3​​ Strongly-correlated systems

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

Quantum embedding​​​‌ methods enable the study​ of large, strongly correlated​‌ quantum systems by (usually​​ self-consistent) decomposition into computationally​​​‌ manageable subproblems, in the​ spirit of divide-and-conquer methods.​‌

The first contribution on​​ the mathematical and numerical​​​‌ analysis of this family​ of methods is concerned​‌ with the Dynamical Mean-Field​​ Theory (DMFT), the most​​​‌ famous quantum embedding methods​ method introduced by A.​‌ Georges and G. Kotliar​​ in 1992. In a​​​‌ previous contribution, E. Cancès,​ A. Kirsch and S.​‌ Perrin-Roussel had proven the​​ existence of a solution​​​‌ to the DMFT equations​ under the Iterated Perturbation​‌ Theory (IPT-DMFT) approximation. In​​ view of numerical simulations,​​​‌ these equations need to​ be discretized. In a​‌ follow-up contribution, they studied​​ a discretization of the​​​‌ IPT-DMFT functional equations, based​ on the restriction of​‌ the hybridization function and​​ local self-energy to a​​​‌ finite number of Matsubara​ frequencies. They first prove​‌ the existence of solutions​​ to the discretized equations​​​‌ in some parameter range​ depending on the Matsubara​‌ frequency cut-off. They then​​ prove uniqueness for a​​​‌ smaller range of parameters.​ They also study more​‌ in depth the case​​ of bipartite systems exhibiting​​​‌ particle-hole symmetry. In this​ case, the discretized IPT-DMFT​‌ equations have purely imaginary​​ solutions, which can be​​​‌ obtained by solving a​ real algebraic system. They​‌ provide a complete characterization​​ of the solutions in​​​‌ the simple case of​ the Hubbard dimer for​‌ discretizations on the lowest​​ two values of the​​​‌ Matsubara frequency cut-off and​ numerical simulations for larger​‌ values. First, they describe​​ how a conductor-to-insulator transition​​​‌ occurs in the Matsubara​ frequency discretized IPT-DMFT model​‌ in the small temperature​​ regime. Finally, they show​​​‌ that for some parameters​ $(U,\mathrm{T‌})$ of the considered​​ Hubbard model, this discretization​​​‌ method may provide values​ of the Green's function​‌ at the lowest Matsubara​​ frequencies which cannot be​​​‌ interpolated by a Pick​ function.

The second contribution​‌ is concerned with Density​​ Matrix Embedding Theory (DMET),​​​‌ a popular method in​ quantum chemistry introduced by​‌ Ksanyi and Chang in​​ 2012. DMET is an​​ efficient approach that enforces​​​‌ self-consistency at the level‌ of one-particle reduced density‌​‌ matrices (1-RDMs), facilitating applications​​ across diverse quantum systems.​​​‌ However, conventional DMET is‌ constrained by the requirement‌​‌ that the global 1-RDM​​ (low-level descriptor) be an​​​‌ orthogonal projector, limiting flexibility‌ in bath construction and‌​‌ potentially impeding accuracy in​​ strongly correlated regimes. In​​​‌ a joint work 58‌ with T. Ayral (Eviden,‌​‌ now Ecole Polytechnique), F.​​ Faulstich (RPI, USA), R.​​​‌ Kim and L. Lin‌ (UC Berkeley, USA), E.‌​‌ Cancès and A. Nègre​​ introduced a generalized DMET​​​‌ framework in which the‌ low-level descriptor can be‌​‌ an arbitrary 1-RDM and​​ the bath construction is​​​‌ based on optimizing a‌ quantitative criterion related to‌​‌ the maximal disentanglement between​​ different fragments. This yields​​​‌ an alternative yet controllable‌ bath space construction for‌​‌ generic 1-RDMs, lifting a​​ key limitation of conventional​​​‌ DMET. They demonstrate its‌ consistency with conventional DMET‌​‌ in appropriate limits and​​ exploring its implications for​​​‌ bath construction, downfolding (impurity‌ Hamiltonian construction), low-level solvers,‌​‌ and adaptive fragmentation. We​​ expect that this more​​​‌ flexible framework, which leads‌ to several new variants‌​‌ of DMET, can improve​​ the robustness and accuracy​​​‌ of DMET.

7.1.4 Electronic‌ structure of moiré materials‌​‌

Computing the electronic structure​​ of incommensurate materials is​​​‌ a central challenge in‌ condensed matter physics, requiring‌​‌ efficient ways to approximate​​ spectral quantities such as​​​‌ the density of states‌ (DoS). In a joint‌​‌ work with D. Massatt​​ (NJIT, USA), L. Meng​​​‌ ( Zhejiang University, China),‌ Etienne Polack (former member‌​‌ of Matherials, now at​​ CEA Grenoble) X. Zhang​​​‌ (Beijing Normal University, China),‌ E. Cancès numerically investigate‌​‌ two distinct approaches for​​ approximating the DoS of​​​‌ incommensurate Hamiltonians for small‌ values of the incommensurability‌​‌ parameters $ϵ$ (e.g., small​​ twist angle, or small​​​‌ lattice mismatch): the first‌ employs a momentum-space decomposition,‌​‌ and the second exploits​​ a semiclassical expansion with​​​‌ respect to $ϵ$.‌ In particular, these two‌​‌ methods are compared using​​ a 1D toy model.​​​‌ We check their consistency‌ by comparing the asymptotic‌​‌ expansion terms of the​​ DoS, and it is​​​‌ shown that, for full‌ DoS, the two methods‌​‌ exhibit good agreement in​​ the small $ϵ$ limit,​​​‌ while discrepancies arise for‌ less small $ϵ$,‌​‌ which indicates the importance​​ of higher-order corrections in​​​‌ the semiclassical method for‌ such regimes. These discrepancies‌​‌ are found to be​​ caused by oscillations in​​​‌ the DoS at the‌ semiclassical analogues of Van‌​‌ Hove singularities, which can​​ be explained qualitatively, and​​​‌ quantitatively for $ϵ$ small‌ enough, by a semiclassical‌​‌ approach.

7.1.5 Optimal transport​​ and quantum chemistry

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

7.2.1​ Sampling of the configuration​‌ space

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

To quantify​‌ the performance of sampling​​ methods based on ergodic​​​‌ stochastic differential equations, bounds​ on the resolvent of​‌ the generator of the​​ dynamics under consideration are​​​‌ useful, as one can​ derive upper bounds on​‌ the asymptotic variance for​​ time averages. Such bounds​​​‌ can in turn be​ deduced from decay estimates​‌ on the semigroup. Together​​ with G. Brigati (ISTA,​​​‌ Austria), A. Wang (University​ of Warwick United-Kingdom) and​‌ L. Wang (NUS, Singapore),​​ G. Stoltz studied in​​​‌ 43 how to obtain​ constructive decay estimates for​‌ the semigroup associated with​​ hypoelliptic generators associated with​​​‌ Langevin dynamics, using hypocoercive​ techniques based on space​‌ time averages and Lions'​​ lemma, in the setting​​​‌ where a Poincaré inequality​ does not hold, but​‌ weak or weighted Poincaré​​ inequalities are satisfied by​​​‌ the marginal distributions in​ positions and momenta.

Besides,​‌ together with X. Lin​​ and P. Monmarché (Université​​​‌ Gustave Eiffel), T. Lelièvre​ explores in 57 a​‌ new approach to study​​ the convergence of free-energy-based​​​‌ adaptive biasing methods, such​ as Metadynamics and their​‌ variants. These are enhanced​​ sampling algorithms which are​​​‌ widely used in molecular​ simulations. Although their efficiency​‌ has been empirically acknowledged​​ for decades, providing theoretical​​​‌ insights via a quantitative​ convergence analysis is a​‌ difficult problem, in particular​​ for the kinetic Langevin​​​‌ diffusion, which is non-reversible​ and hypocoercive. They obtain​‌ the first exponential convergence​​ result for such a​​​‌ process, in an idealized​ setting where the dynamics​‌ can be associated with​​ a mean-field non-linear flow​​​‌ on the space of​ probability measures. A key​‌ of the analysis is​​ the interpretation of the​​​‌ (idealized) algorithm as the​ gradient descent of a​‌ suitable functional over the​​ space of probability distributions.​​​‌

7.2.2 Mathematical understanding and​ efficient simulation of nonequilibrium​‌ systems

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

In‌ 53, R. Gastaldello,‌​‌ G. Stoltz and U.​​ Vaes proposed a variance​​​‌ reduction method for calculating‌ transport coefficients using an‌​‌ importance sampling method via​​ Girsanov's theorem applied to​​​‌ Green–Kubo's formula. They optimized‌ the magnitude of the‌​‌ perturbation applied to the​​ reference dynamics by means​​​‌ of a scalar parameter‌ $\alpha$ and proposed an‌​‌ asymptotic analysis to fully​​ characterize the long-time behavior​​​‌ in order to evaluate‌ the possible variance reduction.‌​‌ Theoretical results were corroborated​​ by numerical results and​​​‌ show that this method‌ allows for some reduction‌​‌ in variance, although rather​​ modest in most situations.​​​‌

In 17, S.‌ Darshan and G. Stoltz‌​‌ studied with A. Iacobucci​​ and S. Olla (CEREMADE,​​​‌ France) the behavior of‌ $\beta$-FPUT chains under‌​‌ time-periodic forcings. Recent works​​ proved a hydrodynamic limit​​​‌ for periodically forced atom‌ chains with harmonic interaction‌​‌ and pinning, together with​​ momentum flip. When energy​​​‌ is the only conserved‌ quantity, one would expect‌​‌ similar results in the​​ anharmonic case, as conjectured​​​‌ for the temperature profile‌ and energy flux. However,‌​‌ outside the harmonic case,​​ explicit computations are generally​​​‌ no longer possible, thus‌ making a rigorous proof‌​‌ of this hydrodynamic limit​​ difficult. This motivated the​​​‌ numerical investigation of the‌ plausibility of this limit‌​‌ for the particular case​​ of a chain with​​​‌ $\beta$-FPUT interactions and‌ harmonic pinning. The results‌​‌ suggest that the conjectured​​ PDE for the limiting​​​‌ temperature profile and Green–Kubo‌ type formula for the‌​‌ limiting energy current conjectured​​ are correct. This Green–Kubo​​​‌ type formula is then‌ used to investigate the‌​‌ relationship between the energy​​ current and period of​​​‌ the forcing. This relationship‌ is investigated in the‌​‌ case of significant rate​​ of momentum flip, small​​​‌ rate of momentum flip‌ and no momentum flip.‌​‌

7.2.3 Modeling and sampling​​ dynamical properties and rare​​​‌ events

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

In 39 N. Blassel,‌​‌ T. Lelièvre and G.​​ Stoltz studied the optimization​​​‌ of the shape of‌ metastable states. The work‌​‌ is motivated by–and is​​ relevant to–the problem of​​​‌ finding optimal hyperparameters for‌ accelerated molecular dynamics algorithms.‌​‌ The definition of metastable​​ states is indeed an​​​‌ ubiquitous task in the‌ design and analysis of‌​‌ molecular simulation, and is​​ a crucial input in​​​‌ a variety of acceleration‌ methods for the sampling‌​‌ of long configurational trajectories.​​ Although standard definitions based​​​‌ on local energy minimization‌ procedures can sometimes be‌​‌ used, these definitions are​​ typically suboptimal, or entirely​​​‌ inadequate when entropic effects‌ are significant, or when‌​‌ the lowest energy barriers​​ are quickly overcome by​​​‌ thermal fluctuations. This work‌ proposed an approach to‌​‌ the definition of metastable​​ states, based on the​​​‌ shape-optimization of a local‌ separation of timescale metric‌​‌ directly linked to the​​​‌ efficiency of a class​ of accelerated molecular dynamics​‌ algorithms. To realize this​​ approach, analytic expressions were​​​‌ derived for shape-variations of​ Dirichlet eigenvalues for a​‌ class of operators associated​​ with reversible elliptic diffusions,​​​‌ and used to construct​ a local ascent algorithm,​‌ explicitly treating the case​​ of multiple eigenvalues. Two​​​‌ methods were proposed to​ make the method tractable​‌ in high-dimensional systems: one​​ based on dynamical coarse-graining,​​​‌ the other on recently​ obtained low-temperature shape-sensitive spectral​‌ asymptotics. The method was​​ validated on a benchmark​​​‌ biomolecular system, showcasing a​ significant improvement over conventional​‌ definitions of metastable states.​​

In 38 N. Blassel,​​​‌ T. Lelièvre and G.​ Stoltz derived novel sharp​‌ low-temperature asymptotics for the​​ spectrum of the infinitesimal​​​‌ generator of the overdamped​ Langevin dynamics. The novelty​‌ was that the operator​​ is endowed with homogeneous​​​‌ Dirichlet conditions at the​ boundary of a domain​‌ which depends on the​​ temperature, having in mind​​​‌ the shape optimization problem​ discussed in the previous​‌ paragraph. From the point​​ of view of stochastic​​​‌ processes, this gives information​ on the long-time behavior​‌ of the diffusion conditioned​​ on non-absorption at the​​​‌ boundary, in the so-called​ quasistationary regime. The results​‌ provide sharp estimates of​​ the spectral gap and​​​‌ principal eigenvalue, extending the​ Eyring–Kramers formula. The phenomenology​‌ is richer than in​​ the case of a​​​‌ fixed boundary and gives​ new insight into the​‌ sensitivity of the spectrum​​ with respect to the​​​‌ shape of the domain​ near critical points of​‌ the energy function.

7.2.4​​ Machine learning approaches in​​​‌ molecular dynamics

The team​ is investigating how machine​‌ learning techniques can be​​ used in order to​​​‌ design new sampling techniques.​ Indeed, neural networks can​‌ efficiently approximate high dimensional​​ functions which can then​​​‌ be used as importance​ functions in various sampling​‌ algorithms. Moreover, generative models​​ can complement standard sampling​​​‌ techniques in order to​ propose innovative moves.

In​‌ 60 T. Lelièvre and​​ G. Stoltz developed with​​​‌ C. Schönle (Ecole polytechnique),​ D. Carbone and M.​‌ Gabrié (ENS Paris) Monte-Carlo​​ samplers for metastable systems​​​‌ using non-local collective variable​ updates. This addresses the​‌ problem of sampling a​​ high dimensional multimodal target​​​‌ probability measure, assuming that​ a good proposal kernel​‌ to move only a​​ subset of the degrees​​​‌ of freedoms (also known​ as collective variables) is​‌ known a priori. This​​ proposal kernel can for​​​‌ example be built using​ normalizing flows. The work​‌ shows how to generalize​​ the extension of the​​​‌ move from the collective​ variable space to the​‌ full space and how​​ to implement an accept-reject​​​‌ step in order to​ get a reversible chain​‌ with respect to a​​ target probability measure –​​​‌ namely for situations when​ non-linear CVs and underdamped​‌ Langevin dynamics are considered.​​ Performance is demonstrated on​​​‌ several numerical examples, observing​ a substantial improvement compared​‌ to methods based on​​ overdamped Langevin dynamics as​​​‌ considered previously.

In 35​ G. Stoltz proposed together​‌ with F.-Z. Akhyar, W.​​ Zhang (ZIB, Germany) and​​​‌ C. Schütte (FU Berlin,​ Germany) a generative method​‌ for conditional probability distributions​​ on the level-sets of​​ collective variables. More precisely,​​​‌ given a probability distribution‌ $\mu$ in ${ℝ}^{\mathrm{d‌‌}}$ represented by data, they​​ studied the generative modeling​​​‌ of its conditional probability‌ distributions on the level-sets‌​‌ of a collective variable​​ $\xi$. They developed​​​‌ a general and efficient‌ learning approach able to‌​‌ learn generative models on​​ different level-sets of $\mathrm{\xi ‌}$ simultaneously. To improve the‌ learning quality on level-sets‌​‌ in low-probability regions, they​​ also proposed a strategy​​​‌ for data enrichment by‌ utilizing data from enhanced‌​‌ sampling techniques. They demonstrated​​ the effectiveness of the​​​‌ proposed learning approach through‌ concrete numerical examples, including‌​‌ molecular systems in biophysics.​​

7.2.5 Interacting particle methods​​​‌ for sampling

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

A well-known numerical‌​‌ method based on an​​ interacting particle system is​​​‌ the ensemble Kalman filter,‌ a methodology for incorporating‌​‌ noisy data into complex​​ dynamical models to enhance​​​‌ predictive capability. This filter‌ is widely adopted in‌​‌ the geophysical sciences, and​​ is starting to be​​​‌ used throughout the sciences‌ and engineering, but there‌​‌ is no theory which​​ quantifies its accuracy as​​​‌ an approximation of the‌ true filtering distribution, except‌​‌ in the Gaussian setting.​​ Study of the statistical​​​‌ accuracy of the ensemble‌ Kalman filter is inherently‌​‌ technical, as it involves​​ the evolution of probability​​​‌ measures according to a‌ nonlinear and nonautonomous dynamical‌​‌ system. In 44,​​ U. Vaes together with​​​‌ E. Calvello (Caltech, USA),‌ José A. Carrillo (University‌​‌ of Oxford, United Kingdom),​​ Franca Hoffmann (Caltech, USA),​​​‌ P. Monmarché (Université Gustave‌ Eiffel) and Andrew M.‌​‌ Stuart (Caltech, USA) provide​​ an accessible overview of​​​‌ previous work in which‌ they took first steps‌​‌ to analyze the accuracy​​ of the filter beyond​​​‌ the linear Gaussian setting‌ 63.

In a‌​‌ different work 54,​​ U. Vaes together with​​​‌ N. J. Gerber (Hausdorff‌ Center for Mathematics, Germany),‌​‌ D. Kim (Caltech, USA)​​ and F. Hoffmann (Caltech,​​​‌ USA) prove a quantitative,‌ uniform-in-time propagation of chaos‌​‌ result for a consensus-based​​ optimization method, generalizing previous​​​‌ results in 64.‌ The analysis relies on‌​‌ the classical synchronous coupling​​ method by Sznitman and​​​‌ McKean, together with several‌ novel auxiliary results, including‌​‌ a stability estimate for​​ the weighted mean and​​​‌ novel concentration estimates for‌ the interacting particle system.‌​‌

In a third work​​ on interacting particle systems​​​‌ 55, U. Vaes‌ together with N. J.‌​‌ Gerber (Hausdorff Center for​​ Mathematics, Germany) establish uniform-in-time​​​‌ propagation of chaos for‌ the Cucker–Smale model, a‌​‌ well-known model for flocking​​ behavior. This extends an​​​‌ existing finite-time propagation of‌ chaos result from 62‌​‌ to infinite time horizons,​​ by leveraging long-time stability​​​‌ properties of the Cucker–Smale‌ interacting particle system. The‌​‌ result can be interpreted​​ as a long-time error​​​‌ estimate for a stochastic‌ particle method for the‌​‌ Vlasov–Fokker–Planck equation associated with​​​‌ the Cucker–Smale model.

7.3.1 Homogenization theory​​

In collaboration with Yves​​​‌ Achdou (Université Paris Cité),​ C. Le Bris has​‌ pursued the study of​​ some homogenization problems for​​​‌ a class of stationary​ Hamilton-Jacobi equations in which​‌ the Hamiltonian is obtained​​ by perturbing an otherwise​​​‌ periodic Hamiltonian. Homogenization then​ leads to an effective​‌ Hamilton-Jacobi equation supplemented with​​ effective Dirichlet boundary conditions.​​​‌ After the case of​ a perturbation at the​‌ origin, the case of​​ a perturbation near a​​​‌ half-line of the state​ space has been considered.​‌ In all these cases,​​ the limiting problem belongs​​​‌ to the class of​ stratified problems introduced by​‌ Alberto Bressan and Yunho​​ Hong and later studied​​​‌ by Guy Barles and​ Emmanuel Chasseigne. The work​‌ has been published in​​ Nonlinear Differential Equations and​​​‌ Applications, Volume 32, 119​ (2025).

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

A third research effort,​​ also connected to the​​​‌ area of homogenization theory,​ has been conducted by​‌ C. Le Bris in​​ collaboration with Anthony T.​​​‌ Patera (MIT), Kento Kaneko​ (MIT) and Theron Guo​‌ (MIT). The general context​​ of the study is​​​‌ heat transfer, and more​ specifically the dunking problem:​‌ a solid body, at​​ uniform temperature and possibly​​​‌ with heterogeneous material composition,​ is placed in an​‌ environment characterized by far​​ field temperature and spatially​​​‌ uniform time independent heat​ transfer coefficient. The problem​‌ is described by a​​ heat equation with Robin​​​‌ boundary conditions. The crucial​ parameter is the Biot​‌ number – a nondimensional​​ heat transfer (Robin) coefficient.​​​‌ Various approximations were introduced,​ justified theoretically and tested​‌ numerically. Error estimates for​​ these approximations were also​​​‌ provided and companion numerical​ solutions of the heat​‌ equation confirm the effectiveness​​ of the error estimates​​ for small Biot number.​​​‌ The work is described‌ in the following three‌​‌ documents: a chapter submitted​​ to Springer MS&A series,​​​‌ a second preprint and‌ a manuscript submitted to‌​‌ ESAIM: M2AN.

V. Ehrlacher​​ and F. Legoll, together​​​‌ with A. Lesage and‌ A. Lebée (ENPC), have‌​‌ considered in 52 elasticity​​ equations posed in thin​​​‌ domains (such as plates)‌ and including highly oscillatory‌​‌ periodic coefficients. The aim​​ of the work is​​​‌ to establish strong convergence‌ results on the difference‌​‌ between the solution to​​ the reference oscillatory problem​​​‌ and its two-scale expansion.‌ Under some classical assumptions‌​‌ on the symmetries of​​ the elasticity tensor, the​​​‌ problem can be split‌ into two independent problems,‌​‌ the membrane problem and​​ the bending problem. The​​​‌ membrane case can actually‌ be addressed using a‌​‌ careful adaptation of classical​​ arguments. The bending case​​​‌ turned out to be‌ much more challenging. A‌​‌ different strategy of proof​​ is introduced to handle​​​‌ it.

7.3.2 Inverse multiscale‌ problems

In the context‌​‌ of the PhD of​​ S. Ruget, C. Le​​​‌ Bris and F. Legoll‌ have completed their work‌​‌ on the question of​​ how to determine the​​​‌ homogenized coefficient of a‌ multiscale problem without explicitly‌​‌ performing a homogenization approach.​​ This work is a​​​‌ follow-up on earlier works‌ over the years in‌​‌ collaboration with Kun Li,​​ Simon Lemaire and Olga​​​‌ Gorynina. The robustness of‌ the proposed approach with‌​‌ respect to the available​​ data, and with respect​​​‌ to noise in measurements,‌ has been thoroughly investigated.‌​‌ In particular, an attractive​​ situation is to assume​​​‌ to only have access‌ to coarse information on‌​‌ the solution, such as​​ the global energy stored​​​‌ in the physical system‌ for any given load‌​‌ (such inverse problems with​​ partial information available are​​​‌ indeed relevant to engineering‌ situations). While the reconstruction‌​‌ of the microstructure is​​ known to be an​​​‌ ill-posed problem, the reconstruction‌ of effective parameters based‌​‌ only on this aggregated​​ information is possible. A​​​‌ manuscript collecting various theoretical‌ and numerical results is‌​‌ currently in preparation (see​​ also 34).

7.3.3​​​‌ Multiscale Finite Element approaches‌

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

In the context of​​​‌ the PhD of A.‌ Lefort, C. Le Bris‌​‌ and F. Legoll have​​ undertaken the study of​​​‌ a multiscale, reaction-diffusion equation.‌ This problem is different‌​‌ from the equations previously​​ studied by the team​​​‌ by the fact that‌ it includes a large‌​‌ reaction term which competes​​ with the diffusive term.​​​‌ To start with, the‌ associated eigenvalue problem has‌​‌ been considered. A promising​​​‌ MsFEM-type approach, which uses​ an oversampling procedure based​‌ on filtering ideas investigated​​ by the team several​​​‌ years ago, has been​ introduced. This MsFEM approach​‌ has next been successfully​​ extended to vector-valued reaction-diffusion​​​‌ eigenvalue problems, a very​ relevant case from the​‌ application viewpoint which presents​​ the mathematical difficulty of​​​‌ being non self-adjoint. The​ time-dependent version of the​‌ problem has next been​​ investigated, using two possible​​​‌ approaches: a spectral approach,​ where the solution is​‌ written as a linear​​ combination of the eigenmodes​​​‌ of the stationary problem,​ or a time-marching scheme.​‌ A manuscript collecting the​​ theoretical and numerical results​​​‌ is currently being prepared.​

In the context of​‌ the M2 internship and​​ then PhD of Y.​​​‌ Bouchereau, C. Le Bris​ and F. Legoll have​‌ studied the wave equation​​ in heterogeneous periodic media.​​​‌ Classical homogenization results indicate​ that the solution to​‌ the wave equation converges​​ to the solution to​​​‌ a simple homogenized problem​ when the final time​‌ is fixed and independent​​ of the characteristic length​​​‌ $\epsilon$ of the medium.​ It is also well-known​‌ that, over larger time​​ horizons (say times of​​​‌ the order of ${\mathrm{\epsilon }}^{-2}$), the​‌ homogenized equation should be​​ complemented by corrective terms​​​‌ to remain an accurate​ approximation of the reference​‌ model. Y. Bouchereau, C.​​ Le Bris and F.​​​‌ Legoll have investigated how​ to design efficient MsFEM-type​‌ approaches in that context.​​ Adapted basis functions are​​​‌ introduced on the basis​ of the sole diffusion​‌ operator. A spatial Galerkin​​ approximation of the wave​​​‌ equation is then performed,​ leading to a second-order​‌ ODE which can be​​ integrated using different techniques.​​​‌ Different regimes of time​ horizons have been explored,​‌ as well as different​​ settings such as periodic,​​​‌ quasi-periodic and general heterogeneous​ media. Encouraging preliminary results​‌ have been obtained that​​ now need to be​​​‌ consolidated.

In parallel to​ the exploration of reaction-diffusion​‌ equations and wave equations,​​ another direction of research​​​‌ is devoted to hyperbolic​ multiscale conservation laws. From​‌ the viewpoint of asymptotic​​ analysis, it is known​​​‌ that introducing a vanishing​ viscosity in the conservation​‌ law leads to a​​ problem, the homogenized limit​​​‌ of which again reads​ as a conservation law​‌ with effective parameters defined​​ in terms of a​​​‌ corrector function, solution to​ an elliptic cell problem​‌ encoding the fine-scale structure​​ of the flux. Based​​​‌ on that observation, C.​ Le Bris, F. Legoll​‌ and G. Merlini have​​ considered a strategy aiming​​​‌ at efficiently computing an​ approximation of the numerical​‌ solution given by a​​ fine-scale finite volume scheme​​​‌ applied to the original​ hyperbolic multiscale conservation law.​‌ The idea is to​​ homogenize the numerical scheme​​​‌ discretizing the PDE, rather​ than the PDE itself,​‌ by exploiting the numerical​​ viscosity naturally present in​​​‌ the numerical scheme. This​ strategy has been first​‌ put in practice for​​ the classical Lax–Friedrichs scheme.​​​‌ Numerical results demonstrate that​ the resulting homogenized model​‌ provides an accurate and​​ computationally efficient approximation of​​​‌ the reference solution in​ one and two space​‌ dimensions for a broad​​ class of multiscale conservation​​ laws. Current and future​​​‌ works include extending this‌ strategy to higher-order numerical‌​‌ schemes and investigating specific​​ regimes or examples (such​​​‌ as shear flows), for‌ which the asymptotic homogenized‌​‌ model is not of​​ the form of a​​​‌ (standard) conservation law.

7.4.1 Complex‌​‌ fluids

Participants: Sébastien Boyaval​​.

In 2025, S.​​​‌ Boyaval has pursued the‌ modelling of viscoelastic flows‌​‌ using symmetric-hyperbolic balance laws​​ consistent with polyconvex elastodynamics.​​​‌ In 65, he‌ proved with Na Wang‌​‌ and Yuxi Hu that​​ for one-dimensional flows, the​​​‌ symmetric-hyperbolic conservation laws are‌ asymptotically consistent with the‌​‌ standard viscous compressible Navier-Stokes​​ equaions, when the viscoelastic​​​‌ stress relaxes infinitely fast.‌ In 36, he‌​‌ investigated, with Emmanuel Audusse,​​ Virgile Dubos and Minh​​​‌ Hoang Le, the construction‌ of kinetic schemes for‌​‌ the numerical simulation of​​ (linear) elastodynamics, formulated as​​​‌ a symmetric-hyperbolic system of‌ PDEs.

In 61,‌​‌ S. Boyaval and his​​ co-authors Jean-Paul Travert, Cédric​​​‌ Goeury, Vito Bacchi and‌ Fabrice Zaoui study numerically,‌​‌ for various metrics, the​​ sensitivity to post-processing parameters​​​‌ of the SAR (images)‌ datas used for flood‌​‌ mapping.

7.4.2 Model-order reduction​​ methods

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

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

In 12​​, R. Blel, T.​​​‌ Lelièvre and V. Ehrlacher‌ perform a numerical analysis‌​‌ of a variance reduction​​ technique for the computation​​​‌ of parameter-dependent expectations using‌ a reduced basis paradigm.‌​‌ The idea is to​​ build online a control​​​‌ variate using expectations computed‌ precisely in an offline‌​‌ stage, with a greedy​​ procedure to select the​​​‌ parameters where thoses expectations‌ are evaluated. Using concentration‌​‌ inequalities, they provide sufficient​​ conditions on the number​​​‌ of samples used for‌ the computation of empirical‌​‌ variances at each iteration​​ of the greedy procedure​​​‌ to guarantee that the‌ resulting method algorithm is‌​‌ a weak greedy algorithm​​ with high probability.

7.4.3​​​‌ Numerical approaches for SPDEs‌

Participants: Claude Le Bris‌​‌.

In collaboration with​​​‌ Ana Djurdjevac (FU Berlin)​ and Endre Suli (Oxford​‌ University), C. Le Bris​​ has considered some dedicated​​​‌ numerical approaches for a​ large class of parabolic​‌ SPDEs with multiplicative noise.​​ The specificity of the​​​‌ class considered is that​ it preserves positivity at​‌ the continuous level. Inspired​​ by well-established techniques for​​​‌ the deterministic case, a​ FEM discretization was introduced​‌ that both is accurate​​ and, unconditionally in the​​​‌ discretization parameters, also preserves​ positivity. Besides the numerical​‌ analysis of the semi-discrete​​ algorithm, some numerical experiments​​​‌ were provided, which illustrate​ the added value with​‌ respect to other approaches​​ in the literature. The​​​‌ work has been submitted​ to Stochastics and Partial​‌ Differential Equations: Analysis and​​ Computations. A follow-up, devoted​​​‌ to the analysis of​ the fully discrete scheme​‌ and presenting other numerical​​ experiments, has been concluded​​​‌ in collaboration with Ana​ Djurdjevac and Owen Hearder​‌ FU Berlin). It is​​ submitted to Communications in​​​‌ Applied Mathematics and Computational​ Science.

7.4.4 Quantum optimal​‌ transport and applications

Participants:​​ Thomas Borsoni, Virginie​​​‌ Ehrlacher.

In 41​, T. Borsoni has​‌ contributed to the current​​ development of quantum optimal​​​‌ transport formulations, by introducing​ the framework of folded​‌ optimal transport, which relies​​ on Choquet theory of​​​‌ boundary representations. This framework​ encompasses classical and entanglement-free​‌ quantum optimal transport Kantorovich​​ formulations. Also recovered is​​​‌ the semi-classical optimal transport​ cost, used in 40​‌ by T. Borsoni and​​ V. Ehrlacher to obtain​​​‌ an observability-type result for​ the von Neumann equation​‌ in a crystal setting.​​ In this work, various​​​‌ concepts are also developed​ to fit the periodic​‌ case, such as the​​ Husimi and Töplitz transforms.​​​‌

Other international​ visits to the team​‌

9.1.1 Visits to international​​ teams

9.2.1 H2020 projects

10.2.1​‌ Teaching

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

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

• Équations aux dérivées partielles:​​​‌ approches variationnelles, 15h (S.​ Darshan, T. Duez, F.​‌ Legoll)
• Mécanique des milieux​​ continus fluides, 25h (S.​​​‌ Boyaval)
• Mécanique quantique, 15h​ (E. Cancès, A. Negre)​‌
• Mise à niveau en​​ mathématiques, 16h (E. Cancès)​​​‌
• Modèles et équations aux​ dérivées partielles, 18h (L.​‌ Carillo, T. Duez, T.​​ Lelièvre, G. Merlini)
• Pratique​​​‌ du calcul scientifique, 18h​ (L. Carillo, U. Vaes)​‌
• Programmation en Python, 24h​​ (L. Carillo, G. Sambataro)​​​‌
• Projets de première année,​ 15h (L. Carillo, T.​‌ Duez, R. Gastaldello, A.​​ Negre, S. Ruget)

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

• Contrôle de systèmes dynamiques​​ et équations aux dérivées​​​‌ partielles, 18h (E. Cancès)​
• Problèmes d'évolution, 10h (F.​‌ Legoll)

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

• Introduction​ to computational statistical physics,​‌ 20h (G. Stoltz and​​ T. Lelièvre)
• Méthodes de​​​‌ tenseurs par la résolution​ d'équations aux dérivées partielles​‌ en grande dimension, 20h​​ (V. Ehrlacher)
• Principes de​​​‌ modélisation, 30h (F. Legoll)​
• Théorie spectrale et méthodes​‌ variationnelles, 10h (E. Cancès)​​

At other institutions:

• Aléatoire​​​‌ (MAP361), 40h, Ecole Polytechnique​ (V. Ehrlacher)
• Calcul Différentiel​‌ et Optimisation, 36h, Université​​ Paris-Dauphine, PSL (D. Gontier)​​​‌
• Gestion des incertitudes et​ analyse de risque (MAP568),​‌ 20h, Ecole Polytechnique (T.​​ Lelièvre)
• Introduction to Machine​​​‌ Learning, 64h, Institut polytechnique​ Paris, M1 Applied mathematics​‌ and statistics (G. Stoltz)​​
• Modal de Mathématiques Appliquées​​​‌ (MAP473D), 15h, Ecole Polytechnique​ (T. Lelièvre)
• Modélisation de​‌ phénomènes aléatoires (MAP432), 40h,​​ Ecole Polytechnique (V. Ehrlacher,​​​‌ T. Lelièvre)
• Optimisation et​ Transport Optimal, 38h, ENS​‌ Paris (S. Perrin-Roussel)
• Numerical​​ Analysis, 56h, NYU Paris​​​‌ (U. Vaes)

10.2.2 Supervision​

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

• Noé Blassel, funding​ ERC Synergy EMC2, Analysis​‌ and sampling of metastable​​ and nonequilibrium stochastic dynamics,​​​‌ Ecole des Ponts, co-supervised​ by T. Lelièvre and​‌ G. Stoltz, defended in​​ December.
• Shiva Darshan, funding​​​‌ ANR SINEQ, Linear response​ of constrained stochastic dynamics,​‌ co-supervised by G. Stoltz​​ and S. Olla (Université​​​‌ Paris-Dauphine), defended in December.​
• Arthur Guillot – Le​‌ Goff, funding ENPC, Hydrodynamic​​ and microbiological modelling of​​​‌ urban water bodies for​ the prevention of health​‌ risks in open water​​ bathing, Ecole des Ponts,​​​‌ co-supervised by B. Vinçon​ Leite, R. Carmigniani and​‌ S. Boyaval, defended in​​ December.
• Abbas Kabalan, thèse​​​‌ CIFRE SAFRANTech, Reduced-order models​ for problems with non-parametric​‌ geometrical variations, co-supervised by​​ V. Ehrlacher and F.​​​‌ Casenave (SAFRANTech), defended in​ December.
• Albéric Lefort, funding​‌ CERMICS-ENPC, Multiscale approaches for​​ reaction-diffusion equations and applications,​​ Ecole des Ponts, co-supervised​​​‌ by F. Legoll and‌ C. Le Bris, defended‌​‌ in December.
• Simon Ruget,​​ funding Inria, Effective approximations​​​‌ for multiscale PDEs based‌ on limited information, Ecole‌​‌ des Ponts, co-supervised by​​ F. Legoll and C.​​​‌ Le Bris, defended in‌ December.
• Jean-Paul Travert, funding‌​‌ CIFRE EDF, Data assimilation​​ for flood predictions, since​​​‌ November 2022, supervised by‌ S. Boyaval (and C.‌​‌ Goeury, F. Zaoui and​​ V. Bacchi from EDF),​​​‌ defended in October.

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

• Yann​​​‌ Bouchereau, funding ENPC, Multiscale‌ approaches in neutronics, and‌​‌ related problems, Institut Polytechnique​​ de Paris, since September​​​‌ 2025, co-supervised by F.‌ Legoll and C. Le‌​‌ Bris
• Thomas Brunel, funded​​ by ANR Neptune, Paddle​​​‌ sports physics: Velocity–stroke rate‌ and active drags, supervised‌​‌ by S. Boyaval (and​​ R. Carmigniani from ENPC)​​​‌
• Louis Carillo, funded by‌ a CDSN fellowship with‌​‌ additional funding from ENPC,​​ Mathematical analysis and numerical​​​‌ methods for metastable systems‌ in statistical physics, since‌​‌ September 2024, co-supervised by​​ T. Lelièvre and U.​​​‌ Vaes
• Charlotte Chapellier, funding‌ CIFRE Sanofi, Generative methods‌​‌ for drug design, since​​ October 2023, co-supervised by​​​‌ T. Lelièvre and G.‌ Stoltz
• Antonin Coatantiec, funding‌​‌ AMX, Mathematical and numerical​​ analysis for sampling metastable​​​‌ dynamics, since October 2025,‌ co-supervised by T. Lelièvre‌​‌ and G. Stoltz
• Théo​​ Duez, funding CNRS, Contributions​​​‌ to the development of‌ new approximations and numerical‌​‌ methods for Time-Dependent Density-Functional​​ Theory (TDDFT) for molecules​​​‌ and materials, since October‌ 2024, co-supervised by E.‌​‌ Cancès and M. Lewin​​ (CEREMADE, CNRS and Université​​​‌ Paris-Dauphine PSL)
• François Escolan,‌ funding ERC HighLEAP, Stochastic‌​‌ particle methods for optimal​​ transport, since November 2024,​​​‌ co-supervised by V. Ehrlacher,‌ J. Reygner (CERMICS) and‌​‌ A. Alfonsi (MATHRISK).
• Sofiane​​ Ezzehi, funding Région Île-de-France​​​‌ and IFPEN, Nonlinear reduced-order‌ modeling techniques for underground‌​‌ CO2 storage applications, since​​ November 2024, co-supervised with​​​‌ G. Enchéry (IFPEN).
• Raphaël‌ Gastaldello, funding CNRS, Variance‌​‌ reduction methods for the​​ computation of transport coefficients,​​​‌ since December 2023, co-supervised‌ by G. Stoltz and‌​‌ U. Vaes
• Baptiste Guilbery,​​ funding Inria, Applications of​​​‌ model reduction techniques to‌ thermo-hydro-mechanical simulations in porous‌​‌ media, since November 2025,​​ co-supervised by S. Boyaval​​​‌ and G. Enchéry (IFPEN)‌
• Clément Guillot, funding ENPC,‌​‌ Space-time variational principles for​​ the Schrödinger equation in​​​‌ large dimension, since November‌ 2023, co-supervised by V.‌​‌ Ehrlacher and M. Dupuy​​ (Sorbonne Université)
• Jean-Baptiste Himbert,​​​‌ funding Ministry of Ecology,‌ Machine learning for molecular‌​‌ dynamics in materials science,​​ since September 2025, co-supervised​​​‌ by T. Lelièvre and‌ G. Stoltz
• Pierre Marmey,‌​‌ funding IFPEN, Evaluation of​​ reaction constants using approaches​​​‌ coupling machine learning and‌ quantum chemistry, since October‌​‌ 2023, co-supervised by T.​​ Lelièvre and P. Raybaud​​​‌ (IFPEN), together with G.‌ Stoltz and M. Corral-Valero‌​‌ (IFPEN)
• Alicia Negre, funding​​ Inria, Quantum computing for​​​‌ quantum embedding methods, since‌ October 2023, co-supervised by‌​‌ E. Cancès and T.​​ Ayral (Ecole Polytechnique)
• Solal​​​‌ Perrin-Roussel, funding ENPC and‌ ERC Synergy EMC2, Mathematical‌​‌ analysis and numerical simulation​​ of electronic transport in​​​‌ moiré materials, since September‌ 2022, co-supervised by É.‌​‌ Cances and by D.​​​‌ Gontier
• Henri Pinsolle, funding​ Onera, Calcul quantique appliqué​‌ à la résolution d’équations​​ aux dérivées partielles linéaires​​​‌ et non linéaires, since​ November 2024, co-supervised by​‌ E. Cancès and F.​​ Renac (Onera)
• Thaddeus Roussigne,​​​‌ funding Université Paris-Dauphine, Etude​ mathématique des distorsions dans​‌ le graphene, since Septembre​​ 2023, co-supervised by D.​​​‌ Gontier and E. Séré​ (Université Paris-Dauphine).
• Jean Ruel,​‌ funding ENS-Saclay, Certified and​​ robust reduced models for​​​‌ the simulation of elongated​ structures, Ecole des Ponts,​‌ since October 2023, co-supervised​​ by F. Legoll, L.​​​‌ Chamoin (ENS Paris-Saclay) and​ A. Lebée (Ecole des​‌ Ponts)
• Jonte Weixler, funding​​ Studienstiftung scholarship, New spectral​​​‌ methods for eigenvalue optimization,​ since Septembre 2025, co-supervised​‌ by D. Gontier and​​ J. Dolbeault (Université Paris-Dauphine).​​​‌

10.2.3 Juries

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

• T.​​ Lelièvre, PhD of Nicolaï​​​‌ Gouraud ("Stochastic algorithms for​ high-dimensional sampling in molecular​‌ dynamics: mathematical analysis and​​ scalable implementation"), defended at​​​‌ Sorbonne Université in June​ (examiner)
• T. Lelièvre, PhD​‌ of Loïs Delande ("Hypocoercivité​​ semiclassique et loi d'Eyring-Kramers​​​‌ pour des opérateurs de​ Fokker-Planck dégénérés") defended at​‌ Université de Bordeaux in​​ June (examiner)

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

• E.​‌ Cancès, HdR of Ivan​​ Duchemin ("Algorithms and Methodologies​​​‌ for Large Scale Simulations​ in Many Body Perturbation​‌ Theory"), defended at Université​​ Grenoble Alpes in June​​​‌ (referee)

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

• T. Lelièvre,​​ professor position at Ecole​​​‌ Polytechnique

International journals

• 10‌​‌ articleE.Elisa Beteille​​, F.Frederique Larrarte​​​‌, S.Sebastien Boyaval‌, E.Eric Demay‌​‌ and M.-H.Minh-Hoang Le​​. Dam-break flow over​​​‌ various obstacles configurations.‌Journal of Ecohydraulics63‌​‌March 2025, 156-170​​HALDOI
• 11 article​​​‌R. A.Rutger A.‌ Biezemans, C.Claude‌​‌ Le Bris, F.​​Frédéric Legoll and A.​​​‌Alexei Lozinski. MsFEM‌ for advection-dominated problems in‌​‌ heterogeneous media: Stabilization via​​ nonconforming variants.Computer​​​‌ Methods in Applied Mechanics‌ and Engineering4332025‌​‌, 117496HALDOI​​
• 12 articleM.-R.Mohamed-Raed​​​‌ Blel, V.Virginie‌ Ehrlacher and T.Tony‌​‌ Lelièvre. Influence of​​ sampling on the convergence​​​‌ rates of greedy algorithms‌ for parameter-dependent random variables‌​‌.Mathematics of Computation​​2025HALDOIback​​​‌ to text
• 13 article‌A.Andrea Bordignon,‌​‌ É.Éric Cancès,​​ G.Geneviève Dusson,​​​‌ G.Gaspard Kemlin,‌ R. A.Rafael Antonio‌​‌ Lainez Reyes and B.​​Benjamin Stamm. Fully​​​‌ guaranteed and computable error‌ bounds on the energy‌​‌ for periodic Kohn-Sham equations​​​‌ with convex density functionals​.SIAM Journal on​‌ Scientific Computing475​​October 2025, A2881-A2905​​​‌HALDOI
• 14 article​G.Giovanni Brigati and​‌ G.Gabriel Stoltz.​​ How to construct decay​​​‌ rates for kinetic Fokker--Planck​ equations?SIAM Journal on​‌ Mathematical Analysis574​​July 2025, 3587-3622​​​‌HALDOI
• 15 article​C.Clément Cancès,​‌ J.Jean Cauvin-Vila,​​ C.Claire Chainais-Hillairet and​​​‌ V.Virginie Ehrlacher.​ Cross-diffusion systems coupled via​‌ a moving interface.​​Interfaces and Free Boundaries​​​‌ : Mathematical Analysis, Computation​ and Applications2025.​‌ In press. HALDOI​​
• 16 articleA.Aloïs​​​‌ Castellano, R.Romuald​ Béjaud, P.Pauline​‌ Richard, O.Olivier​​ Nadeau, C.Clément​​​‌ Duval, G.Grégory​ Geneste, G.Gabriel​‌ Antonius, J.Johann​​ Bouchet, A.Antoine​​​‌ Levitt, G.Gabriel​ Stoltz and F.François​‌ Bottin. Machine learning​​ assisted canonical sampling (MLACS)​​​‌.Computer Physics Communications​316July 2025,​‌ 109730HALDOI
• 17​​ articleS.Shiva Darshan​​​‌, A.Alessandra Iacobucci​, S.Stefano Olla​‌ and G.Gabriel Stoltz​​. beta-FPUT chains under​​​‌ time-periodic forcing.Journal​ of Statistical Physics2025​‌. In press. HAL​​back to text
• 18​​​‌ articleA.Alain Durmus​, A.Aurélien Enfroy​‌, É.Éric Moulines​​ and G.Gabriel Stoltz​​​‌. Uniform minorization condition​ and convergence bounds for​‌ discretizations of kinetic Langevin​​ dynamics.Annales de​​​‌ l'Institut Henri Poincaré (B)​ Probabilités et Statistiques61​‌1February 2025HAL​​DOI
• 19 articleM.​​​‌Mathias Dus, G.​Geneviève Dusson, V.​‌Virginie Ehrlacher, C.​​Clément Guillot and J.​​​‌ P.Joel Pascal Soffo​ Wambo. Comparison between​‌ tensor methods and neural​​ networks in electronic structure​​​‌ calculations.ESAIM: Proceedings​ and Surveys81October​‌ 2025, 55-76HAL​​DOI
• 20 articleV.​​​‌Virginie Ehrlacher, F.​Frederic Legoll, B.​‌Benjamin Stamm and S.​​Shuyang Xiang. Embedded​​​‌ corrector problems for homogenization​ in linear elasticity.​‌Multiscale Modeling and Simulation:​​ A SIAM Interdisciplinary Journal​​​‌2332025,​ 1236--1273HAL
• 21 article​‌L.Lucas Journel,​​ T.Tony Lelièvre and​​​‌ J.Julien Reygner.​ Condensation in Fleming--Viot particle​‌ systems with fast selection​​ mechanism.ALEA :​​​‌ Latin American Journal of​ Probability and Mathematical Statistics​‌2222025,​​ 825HALDOI
• 22​​​‌ articleA.Abbas Kabalan​, F.Fabien Casenave​‌, F.Felipe Bordeu​​, V.Virginie Ehrlacher​​​‌ and A.Alexandre Ern​. Elasticity-based mesh morphing​‌ technique with application to​​ reduced-order modeling.Applied​​​‌ Mathematical Modelling1412025​, 115929, 25 pp.​‌HALDOI
• 23 article​​D.Dante Kalise,​​​‌ L.Lucas Moschen,​ G.Grigorios Pavliotis and​‌ U.Urbain Vaes.​​ A Spectral Approach to​​​‌ Optimal Control of the​ Fokker–Planck Equation.IEEE​‌ Control Systems Letters9​​2025, 504-509HAL​​​‌DOI
• 24 articleT.​Tony Lelièvre, D.​‌Dorian Le Peutrec and​​ B.Boris Nectoux.​​​‌ Eyring-Kramers exit rates for​ the overdamped Langevin dynamics:​‌ the case with saddle​​ points on the boundary​​.Journal de l'École​​​‌ polytechnique — Mathématiques12‌June 2025, 881-982‌​‌HALDOI
• 25 article​​T.Tony Lelièvre,​​​‌ G. A.Grigorios A.‌ Pavliotis, G.Geneviève‌​‌ Robin, R.Régis​​ Santet and G.Gabriel​​​‌ Stoltz. Optimizing the‌ diffusion coefficient of overdamped‌​‌ Langevin dynamics.Mathematics​​ of ComputationMay 2025​​​‌HALDOI
• 26 article‌T.Tony Lelièvre,‌​‌ R.Régis Santet and​​ G.Gabriel Stoltz.​​​‌ Improving sampling by modifying‌ the effective diffusion.‌​‌Journal of Computational Physics​​541November 2025,​​​‌ 114313HALDOI
• 27‌ articleR.Rémi Robin‌​‌, P.Pierre Rouchon​​ and L.-A.Lev-Arcady Sellem​​​‌. Convergence of bipartite‌ open quantum systems stabilized‌​‌ by reservoir engineering.​​Annales Henri Poincaré26​​​‌May 2025, 1769–1819‌HALDOI
• 28 article‌​‌C.Christoph Schönle,​​ M.Marylou Gabrié,​​​‌ T.Tony Lelièvre and‌ G.Gabriel Stoltz.‌​‌ Sampling metastable systems using​​ collective variables and Jarzynski-Crooks​​​‌ paths.Journal of‌ Computational Physics527April‌​‌ 2025, 113806HAL​​DOI
• 29 articleL.-A.​​​‌Lev-Arcady Sellem, A.‌Alain Sarlette, Z.‌​‌Zaki Leghtas, M.​​Mazyar Mirrahimi, P.​​​‌Pierre Rouchon and P.‌Philippe Campagne-Ibarcq. Dissipative‌​‌ Protection of a GKP​​ Qubit in a High-Impedance​​​‌ Superconducting Circuit Driven by‌ a Microwave Frequency Comb‌​‌.Physical Review X​​1512025,​​​‌ 011011HALDOI
• 30‌ articleR.Renato Spacek‌​‌, P.Pierre Monmarché​​ and G.Gabriel Stoltz​​​‌. Transient subtraction: A‌ control variate method for‌​‌ computing transport coefficients.​​Journal of Statistical Physics​​​‌1924April 2025‌, 53HALDOI‌​‌

International peer-reviewed conferences

• 31​​ inproceedingsJ.Jonas Aparicio​​​‌, V.Virginie Ehrlacher‌, G.Gwendal Cumunel‌​‌, T.Tien Hoang​​ and G.Gilles Foret​​​‌. Enhancing Vibration-Based Tension‌ Measurement in External Prestressing‌​‌ Tendons Using a Monte​​ Carlo Markov Chain Algorithm​​​‌.Lecture Notes in‌ Civil EngineeringEvaces 2025‌​‌ - 11th International Conference​​ on Experimental Vibration Analysis​​​‌ for Civil Engineering Structures‌675Lecture Notes in‌​‌ Civil EngineeringPorto, Portugal​​Springer Nature SwitzerlandOctober​​​‌ 2025, 268-277HAL‌DOI
• 32 inproceedingsE.‌​‌Elisa Beteille, S.​​Sébastien Boyaval, F.​​​‌Frédérique Larrarte and E.‌Eric Demay. Experimental‌​‌ analysis on the influence​​ of urban forms on​​​‌ unsteady urban flooding.‌Proceedings of the 12th‌​‌ International Conference on Fluvial​​ Hydraulics, Liverpool, UK, 2nd-​​​‌ 6th September, 2024River‌ Flow 2024 - 12th‌​‌ International Conference on Fluvial​​ HydraulicsLiverpool, United Kingdom​​​‌April 2025HAL
• 33‌ inproceedingsJ.Jean Ruel‌​‌, F.Frédéric Legoll​​, A.Arthur Lebée​​​‌ and L.Ludovic Chamoin‌. Vers une nouvelle‌​‌ stratégie PGD pour la​​ simulation numérique de structures​​​‌ élancées.CFM 2025‌ - 26e Congrès Français‌​‌ de MécaniqueMetz, France​​August 2025HAL

Doctoral​​​‌ dissertations and habilitation theses‌

• 34 thesisS.Simon‌​‌ Ruget. Effective approximations​​ for multiscale PDEs based​​​‌ on limited information.‌École des Ponts ParisTech‌​‌December 2025HALback​​ to text

Reports &​​​‌ preprints

• 35 miscF.-Z.‌Fatima-Zahrae Akhyar, W.‌​‌Wei Zhang, G.​​​‌Gabriel Stoltz and C.​Christof Schütte. Generative​‌ modeling of conditional probability​​ distributions on the level-sets​​​‌ of collective variables.​December 2025HALback​‌ to text
• 36 misc​​E.Emmanuel Audusse,​​​‌ S.Sébastien Boyaval,​ V.Virgile Dubos and​‌ M.-H.Minh-Hoang Le.​​ Construction and performance of​​​‌ kinetic schemes for linear​ systems of conservation laws​‌.December 2025HAL​​back to text
• 37​​​‌ miscM.Marzia Bisi​, T.Thomas Borsoni​‌ and M.Maria Groppi​​. On modeling polyatomic​​​‌ molecules in kinetic theory​.December 2025HAL​‌
• 38 miscN.Noé​​ Blassel, T.Tony​​​‌ Lelièvre and G.Gabriel​ Stoltz. Quantitative low-temperature​‌ spectral asymptotics for reversible​​ diffusions in temperature-dependent domains​​​‌.January 2025HAL​back to text
• 39​‌ miscN.Noé Blassel​​, T.Tony Lelièvre​​​‌ and G.Gabriel Stoltz​. Shape optimization of​‌ metastable states.July​​ 2025HALback to​​​‌ text
• 40 miscT.​Thomas Borsoni and V.​‌Virginie Ehrlacher. Observability​​ inequality for the von​​​‌ Neumann equation in crystals​.December 2025HAL​‌back to text
• 41​​ miscT.Thomas Borsoni​​​‌. Folded optimal transport​ and its application to​‌ separable quantum optimal transport​​.December 2025HAL​​​‌back to text
• 42​ miscS.Sébastien Boyaval​‌, Y.Yuxi Hu​​ and N.Na Wang​​​‌. Global solutions and​ uniform convergence stability for​‌ compressible Navier-Stokes equations with​​ oldroyd-type constitutive law.​​​‌September 2025HAL
• 43​ miscG.Giovanni Brigati​‌, G.Gabriel Stoltz​​, A. Q.Andi​​​‌ Q. Wang and L.​Lihang Wang. Explicit​‌ convergence rates of underdamped​​ Langevin dynamics under weighted​​​‌ and weak Poincaré-Lions inequalities​.June 2025HAL​‌back to text
• 44​​ miscE.Edoardo Calvello​​​‌, J. A.José​ Antonio Carrillo, F.​‌Franca Hoffmann, P.​​Pierre Monmarché, A.​​​‌ M.Andrew M. Stuart​ and U.Urbain Vaes​‌. Statistical accuracy of​​ the ensemble Kalman filter​​​‌ in the near-linear setting​.March 2025HAL​‌back to text
• 45​​ miscÉ.Éric Cancès​​​‌, T.Théo Duez​, J.Jari van​‌ Gog, A. B.​​Asbjørn Bækgaard Lauritsen,​​​‌ M.Mathieu Lewin and​ J.Julien Toulouse.​‌ Geometric Time-Dependent Density Functional​​ Theory.January 2026​​​‌HAL
• 46 miscE.​Eric Cancès, T.​‌Théo Duez, J.​​Jari van Gog,​​​‌ A. B.Asbjørn Bækgaard​ Lauritsen, M.Mathieu​‌ Lewin and J.Julien​​ Toulouse. Geometric theory​​​‌ of constrained Schrödinger dynamics​ with application to time-dependent​‌ density-functional theory on a​​ finite lattice.January​​​‌ 2026HAL
• 47 misc​F.Frédérique Charles,​‌ A.Annamaria Massimini and​​ F.Francesco Salvarani.​​​‌ Diffusion asymptotics of a​ kinetic model for gas-particle​‌ mixtures with energy exchanges​​.March 2025HAL​​​‌
• 48 miscM.Maxime​ Dalery, G.Geneviève​‌ Dusson, V.Virginie​​ Ehrlacher and A.Alexei​​​‌ Lozinski. Nonlinear reduced​ basis using mixture Wasserstein​‌ Barycenters:application to an eigenvalue​​ problem inspired from quantumchemistry​​​‌.December 2025HAL​
• 49 miscM.Maxime​‌ Dalery, G.Geneviève​​ Dusson and V.Virginie​​ Ehrlacher. Marginal-constrained modified​​​‌ Wasserstein barycenters for Gaussian‌ distributions and Gaussian mixtures‌​‌.June 2025HAL​​
• 50 miscM.-S.Mi-Song​​​‌ Dupuy, V.Virginie‌ Ehrlacher and C.Clément‌​‌ Guillot. A space-time​​ variational formulation for the​​​‌ many-body electronic Schrödinger evolution‌ equation.September 2025‌​‌HAL
• 51 miscG.​​Geneviève Dusson, V.​​​‌Virginie Ehrlacher and N.‌Nathalie Nouaime. A‌​‌ Wasserstein-type metric for generic​​ mixture models, including location-scatter​​​‌ and group invariant measures‌.November 2025HAL‌​‌
• 52 miscV.Virginie​​ Ehrlacher, A.Arthur​​​‌ Lebée, F.Frédéric‌ Legoll and A.Adrien‌​‌ Lesage. Convergence of​​ two-scale expansions for elastic​​​‌ heterogeneous plates.July‌ 2025HALback to‌​‌ text
• 53 miscR.​​Raphaël Gastaldello, G.​​​‌Gabriel Stoltz and U.‌Urbain Vaes. Dynamical‌​‌ reweighting for estimation of​​ fluctuation formulas.April​​​‌ 2025HALback to‌ text
• 54 miscN.‌​‌Nicolai Gerber, F.​​Franca Hoffmann, D.​​​‌Dohyeon Kim and U.‌Urbain Vaes. Uniform-in-time‌​‌ propagation of chaos for​​ Consensus-Based Optimization.May​​​‌ 2025HALDOIback‌ to text
• 55 misc‌​‌N. J.Nicolai Jurek​​ Gerber and U.Urbain​​​‌ Vaes. Uniform-in-time propagation‌ of chaos for the‌​‌ Cucker-Smale model.January​​ 2025HALback to​​​‌ text
• 56 miscA.‌Abbas Kabalan, F.‌​‌Fabien Casenave, F.​​Felipe Bordeu, V.​​​‌Virginie Ehrlacher and A.‌Alexandre Ern. Model-order‌​‌ reduction with optimal morphings​​ for poorly reducible problems​​​‌ with geometric variability.‌December 2025HAL
• 57‌​‌ miscT.Tony Lelièvre​​, X.Xuyang Lin​​​‌ and P.Pierre Monmarché‌. Convergence rates for‌​‌ an Adaptive Biasing Potential​​ scheme from a Wasserstein​​​‌ optimization perspective.January‌ 2025HALback to‌​‌ text
• 58 miscA.​​Alicia Negre, F.​​​‌Fabian Faulstich, R.‌Raehyun Kim, T.‌​‌Thomas Ayral, L.​​Lin Lin and É.​​​‌Éric Cancès. New‌ perspectives on Density-Matrix Embedding‌​‌ Theory.March 2025​​HALback to text​​​‌
• 59 miscR.Rémi‌ Robin, P.Pierre‌​‌ Rouchon and L.-A.Lev-Arcady​​ Sellem. Unconditionally stable​​​‌ time discretization of Lindblad‌ master equations in infinite‌​‌ dimension using quantum channels​​.March 2025HAL​​​‌
• 60 miscC.Christoph‌ Schönle, D.Davide‌​‌ Carbone, M.Marylou​​ Gabrié, T.Tony​​​‌ Lelièvre and G.Gabriel‌ Stoltz. Efficient Monte-Carlo‌​‌ sampling of metastable systems​​ using non-local collective variable​​​‌ updates.December 2025‌HALback to text‌​‌
• 61 miscJ.-P.Jean-Paul​​ Travert, C.Cédric​​​‌ Goeury, S.Sébastien‌ Boyaval, V.Vito‌​‌ Bacchi and F.Fabrice​​ Zaoui. Evaluating the​​​‌ effects of preprocessing, method‌ selection, and hyperparameter tuning‌​‌ on SAR-based flood mapping​​ and water depth estimation​​​‌.2025HALback‌ to text