2023Activity 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 2023 DFTK has gained support for phonon computations, nonlinear core correction and improved Wannier integration. It also received many other smaller improvements, and 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.

E. Cancès co-edited with G. Friesecke (TU Munich, Germany) a book on Density Functional Theory reviewing modeling aspects, mathematical and numerical analysis results, computational methods, and state-of-the-art applications (Springer 2023, 40). In Chapter 7 of this book 41, E. Cancès, A. Levitt, Y. Maday (Sorbonne University), and C. Yang (LBNL, Berkeley, USA) discuss the main algorithms used to solve the Kohn–Sham models, as well as the recent numerical analysis of these models and algorithms.

Together with G. Kemlin (former PhD student in the project-team and now at University of Amiens) and A. Levitt (University Paris-Saclay), E. Cancès has studied the numerical analysis of linear and nonlinear Schrödinger equations with periodic analytic potentials 21. They prove that, for linear equations, when the potential is analytic in a strip of width $A$ of the complex plane, the solution is analytic in the same strip, ensuring an exponential convergence of the planewave discretization of the equation with rate $A$. On the other hand, for nonlinear equations such as the periodic Kohn-Sham equations with Goedecker-Teter-Hutter (GTH) pseudopotentials, they find that the solution may be analytic only in a strip of width smaller than $A$. This behavior is illustrated by two examples using a combination of numerical and analytical arguments.

Together with G. Dusson (CNRS and University of Besançon), B. Stamm (University of Stuttgart), and F. Lipparini, P. Mazzeo, and F. Pes (quantum chemists from the university of Pisa), E. Polack proposed a scheme based on Grassmann extrapolation of density matrices for an accurate calculation of initial guesses in Born-Oppenheimer Molecular Dynamics simulations 32. The method shows excellent results on large quantum mechanics/molecular mechanics systems simulated with Kohn-Sham density functional theory .

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

7.1.2 Quantum embedding methods

The treatment of strongly correlated quantum systems is a long-standing challenge in computational chemistry and physics. The application of high-accuracy first-principle methods that are able to capture the electronic correlation effects at chemical accuracy is commonly stymied by a steep computational scaling with respect to system size. A potential remedy is provided by quantum embedding theories, which can be somehow interpreted as domain decomposition methods for the quantum many-body problem in the Fock space. Such approaches include the dynamical mean-field theory (DMFT) and the density matrix embedding theory (DMET). Together with F. Faultsich (UC Berkeley, USA) and A. Levitt, E. Cancès, A. Kirsch and E. Letournel provided the first mathematical analysis of DMET 47. They prove that, under certain assumptions, (i) the exact ground-state density matrix is a fixed-point of the DMET map for non-interacting systems, (ii) there exists a unique physical solution in the weakly-interacting regime, and (iii) DMET is exact at first order in the coupling parameter. They provide numerical simulations to support their results and comment on the physical meaning of the assumptions under which they hold true. They show that the violation of these assumptions may yield multiple solutions of the DMET equations.

7.1.3 Moiré materials

Twisted bilayer graphene (TBG) is obtained by stacking two identical graphene sheets on top of one another and rotating them in opposite directions around the transverse direction by a relative, typically small, angle $\theta$. Seen from above, this gives rise to a moiré pattern whose diameter scales as ${\theta }^{-1}$. TBG is a controllable quantum system, in the sense that its properties can be finely tuned by playing with the twist angle (with, currently, an experimental accuracy of $\sim 0.1^{\circ }$), the transverse external electric field generated by gating (which allows one to control the doping, that is the density of charge carriers), the in-plane external electric field, the external magnetic field, the temperature, the interlayer distance (which can be changed by applying a pressure field), etc. The theoretical and experimental study of TBG has been one of the hottest topics in condensed matter physics since notably, the experimental discovery of supposedly unconventional superconducting regions in the phase diagram of TBG at “magic” twist angle $\theta \simeq 1.{08}^{\circ }$. Together with L. Meng (Ecole des Ponts), E. Cancès has investigated the mathematical properties of independent-electron models for TBG by examining the density-of-states of corresponding single-particle Hamiltonians using tools from semiclassical analysis 48. This study focuses on a specific atomic-scale Hamiltonian constructed from Density-Functional Theory, and a family of moiré-scale Hamiltonians containing the Bistritzer-MacDonald model. It is shown that the density-of-states of these Hamiltonians admit asymptotic expansions in the twist angle parameter $ϵ:=sin(\theta /2)$. The proof relies on a twisted version of the Weyl calculus and a trace formula for an exotic class of pseudodifferential operators suitable for the study of twisted 2D materials.

7.1.4 Open quantum systems and quantum computing

In his post-doctoral work co-supervised by Claude Le Bris (MATHERIALS) and Pierre Rouchon (Inria QUANTIC), Masaaki Tokieda has addressed 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.

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.

On the one hand, appropriate modified Wasserstein barycenters have been used in order to design new reduced-order model reduction methods to accelerate electronic structure calculations of molecules. First, together with Geneviève Dusson and Nathalie Nouaime, V. Ehrlacher developed new types of mixture Wasserstein distances and barycenters adapted to electronic densities that can be written as squares of linear combination of Slater determinants of Gaussian functions in 55. Second, in 53, V. Ehrlacher (together with M. Dalery, G. Dusson and A. Lozinski) worked on the design of new greedy algorithms using the latter mixture Wasserstein barycenters and on the proof of estimates on the decay of Kolmogorov $n$-widths related to these Wasserstein-type distances on some one-dimensional electronic structure calculations toy models.

On the other hand, V. Ehrlacher and L. Nenna proved in 57 that moment-constrained approximations of the Lieb functional (which may be seen as a particular instance of quantum optimal transport problems) enjoyed similar sparsity properties as moment-constrained approximation of classical optimal transport problems.

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. In 61, T. Lelièvre, R. Santet and G. Stoltz develop and study a new variant of the Hamiltonian Monte Carlo algorithm which can be used for nonseparable Hamiltonian. Such Hamiltonian functions naturally appear in many contexts, for numerical or physical reasons, for example when a position-dependent mass is considered. To get an unbiased sampling method, a reversibility check has to be enforced, because of the implicitness of the Störmer–Verlet integrator.

In order to improve the efficiency of sampling, variance reduction techniques should be used. In 50, T. Lelièvre, G. Stoltz and U. Vaes, together with M. Chak (Sorbonne Université, France) analyze an importance sampling approach for Markov chain Monte Carlo methods that relies on the overdamped Langevin dynamics. More precisely, they study an estimator based on an ergodic average along a realization of an overdamped Langevin process for a modified potential. An explicit expression for the biasing potential that minimizes the asymptotic variance of the estimator is obtained in dimension 1, and a general numerical approach for approximating the optimal potential in the multi-dimensional setting is proposed. The capabilities of the proposed method are demonstrated by means of numerical experiments.

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 (Université Paris Dauphine, France), G. Stoltz studied in 46 how to obtain constructive decay estimates for the semigroup corresponding to hypoelliptic generators associated with Langevin dynamics, using hypocoercive techniques based on space time averages and Lions' lemma.

Finally, let us mention that the stochastic dynamics used to sample probability measures in statistical physics can also be used to minimize functions when the temperature is sent to 0. This idea is used in the work 27 by G. Stoltz together with K. Karoni and B. Leimkuhler (University of Edinburgh, United-Kingdom), where variations of the adaptive Langevin dynamics (an underdamped Langevin dynamics where the friction is adjusted through some feedback term à la Nosé–Hoover) are considered to minimize high dimensional objectives.

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. R. Gastaldello is starting his PhD work on this topic. Let us next describe the efforts of the team done in the previous year.

In 35, R. Spacek and G. Stoltz proposed to add an additional perturbation to the system (so-called synthetic forcing), which perserves the invariant measure and hence does not change the linear response, but which allows to limit the nonlinearity of the response. This makes it possible to resort to larger forcings in practice, and hence more easily determine the response of the nonequilibrium system. Several classes of admissible synthetic forcings are systematically studied, and their performance is assessed on toy systems.

Another way to possibly reduce the variance of estimators of transport coefficients based on nonequilibrium molecular dynamics is to resort to a dual strategy. Whereas standard non-equilibrium approaches fix the forcing and measure the average flux induced in the system driven out of equilibrium, a dual philosophy consists in fixing the value of the flux, and measuring the average magnitude of the forcing needed to induce it. A deterministic version of this approach, named Norton dynamics, was studied in the 1980s by Evans and Morriss. In 45, N. Blassel and G. Stoltz introduce a stochastic version of this method, first developing a general formal theory for a broad class of diffusion processes, and then specializing it to underdamped Langevin dynamics, which are commonly used for molecular dynamics simulations. Numerical evidence demonstrates that the stochastic Norton method provides an equivalent measure of the linear response, and in fact that this equivalence extends well beyond the linear response regime. This work raises many intriguing questions, both from the theoretical and the numerical perspectives.

On the applicative side, G. Stoltz studied in 26 with T. Hoang Ngoc Minh and B. Rotenberg (Sorbonne Université, France) the effet of confinement, adsorption on surfaces, and ion-ion interactions on the response of confined electrolyte solutions to oscillating electric fields in the direction perpendicular to the confining walls. Nonequilibrium simulations allows to characterize the transitions between linear and nonlinear regimes when varying the magnitude and frequency of the applied field, but the linear response, characterized by the frequency-dependent conductivity, is more efficiently predicted from the equilibrium current fluctuations. To that end, a Green–Kubo relation appropriate for overdamped dynamics is rederived for time periodic forcings. The expression highlights the contributions of the underlying Brownian fluctuations and of the interactions of the particles between them and with external potentials. The frequency-dependent conductivity always decays from a bulk-like behavior at high frequency to a vanishing conductivity at low frequency due to the confinement of the charge carriers by the walls.

7.2.3 Sampling dynamical properties and rare events

Sampling transitions from one metastable state to another is a difficult task. In 33 T. Lelièvre, T. Pigeon and G. Stoltz, together with A. Anciaux-Sekadrian, M. Corral-Valero, M. Moreaud (collaborators at IFPEN, France) apply for the first time the Adapive Multilevel Splitting (AMS) method to study catalytic reactions. Computing accurate rate constants for catalytic events occurring at the surface of a given material represents a challenging task with multiple potential applications in chemistry. The AMS method requires a one dimensional reaction coordinate to index the progress of the transition. To build such a reaction coordinate, various approaches are tested, including Support Vector Machine and path collective variables. The calculated rate constants and transition mechanisms are discussed and compared to those obtained by a conventional static approach based on the Eyring-Polanyi equation with harmonic approximation. The AMS method is able to better take into account entropic effects as well as complex transition mechanisms, e.g. those involving multiple pathways.

On the methodological side, A. Guyader and T. Lelièvre are currently exploring how AMS and Importance Sampling (IS) are affected by an importance function which is not exactly the committor ${\xi }^{*}$ but a perturbation of it. AMS is expected to be less sensitive than IS, but IS to be better when the perturbation is small (zero variance principle). Specifically, one can think of an infinitesimal calculation and look at how the variance of AMS/IS degrades when considering an importance function of the form ${\xi }_{\eta }\left(x\right)={\xi }^{*}\left(x\right)+\eta \delta \xi \left(x\right)$, in the limit $\eta \to 0$. In this respect, is it possible to combine AMS and IS in order to use "more IS" when the perturbation is small and "more AMS" when the perturbation is large?

More generally, finding collective variables to describe some important coarse-grained information on physical systems, in particular metastable states, remains a key issue in molecular dynamics. Collective variables allow to compute free energy differences and/or bias the dynamics to observe transitions. In 59, T. Lelièvre, T. Pigeon and G. Stoltz, together with W. Zhang (FU Berlin, Germany) analyze the performances of autoencoders to construct collective variables. They study some relevant mathematical properties of the loss function considered for training autoencoders, and provide physical interpretations based on conditional variances and minimum energy paths. They also consider various extensions in order to better describe physical systems, by incorporating more information on transition states at saddle points, and/or allowing for multiple decoders in order to describe several transition paths. On the application side, T. Lelièvre and G. Stoltz, together with Z. Belkacemi, M. Bianciotto, P. Gkeka, H. Minoux (collaborators at Sanofi, France) characterize in 12 the dynamics of the N-terminal domain of the heat shock protein 90 (Hsp90) using an autoencoder-learned collective variable in conjunction with adaptive biasing force Langevin dynamics. Using this machine-learnt collective variable is a crucial ingredient to generate transitions between native states.

7.2.4 Interacting particle methods for sampling

In some situations, stochastic numerical methods can be made more efficient by using various replicas of the system. For algorithms based on interacting particle systems that admit a mean-field description, convergence analysis is often more accessible at the mean-field level. In order to transpose convergence results obtained at the mean-field level to the finite ensemble size setting, it is desirable to show that the particle dynamics converge in an appropriate sense to the corresponding mean-field dynamics. In 58, U. Vaes together with N. J. Gerber (Hausdorff Center for Mathematics, Germany) and F. Hoffmann (Caltech, USA) proved quantitative mean-field limit results for two related interacting particle systems: Consensus-Based Optimization and Consensus-Based Sampling. The approach employed to this end is based a generalization of Sznitman’s classical argument: in order to circumvent issues related to the lack of global Lipschitz continuity of the coefficients, an event of small probability is discarded, the contribution of which is controlled using moment estimates for the particle systems. In addition, their work presents new results on the well-posedness of the particle systems and their mean-field limit, and provides novel stability estimates for the weighted mean and the weighted covariance.

7.3.1 Deterministic non-periodic systems

From the theoretical viewpoint, the project-team has pursued the development of a general theory for homogenization of deterministic materials modeled as periodic structures with defects. This series of works is performed in collaboration with X. Blanc (Université Paris-Cité), P.-L. Lions (Collège de France) and P. Souganidis (Chicago, USA). The periodic setting is the oldest traditional setting for homogenization. Alternative settings include the quasi- and almost-periodic 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 almost-periodic settings are not appropriate answers to this difficulty. Using a random stationary setting may be tempting from a modeling 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 project-team consists in considering periodic structures with defects, a setting which is rich enough to fit reality while still leading to affordable computations.

In that direction, Y. Achdou (partially on leave at INRIA from Université Paris-Cité in 2022-2023) and C. Le Bris have studied in 10 the homogenization of a class of stationary Hamilton-Jacobi equations in which the Hamiltonian is obtained by perturbing near the origin an otherwise periodic Hamiltonian. Homogenization leads to an effective Hamilton-Jacobi equation supplemented with an effective Dirichlet boundary condition at the origin: this boundary value problem has to be understood in the sense of the stratified problems introduced by Bressan et al and later studied by Barles and Chasseigne. The effective Dirichlet data is obtained as the limit of a sequence of ergodic constants associated to truncated cell problems posed in balls with diameter tending to infinity. Several research directions stem from this work. In particular, an ongoing research deals with situations in which homogenization leads to stratified problems with more complex geometries, for example a stratification of the plane composed of three manifolds: an open half-line, the end-point of the latter and finally the complement of the closed half-line. Other open questions concern the rate of convergence to the solution of the effective problem and the behavior of the above-mentioned ergodic constants as the diameter of the truncated cell tends to infinity. These questions seem difficult and a sensible way to tackle them would be to carry out numerical simulations first.

Also in that direction of research, C. Le Bris has co-authored with X. Blanc two textbooks on Homogenization Theory, one in French 1 and one in English 2. The two books mostly present the same material. The French version however dwells a bit more on the theoretical aspects while the English version is slightly more focused on computational issues. This action testifies to the wish of the team to reach the largest possible audience and introduce them to the challenging field of multiscale science. In addition, a short text that summarizes the major results obtained has been written by C. Le Bris and has been published in the "Séminaire Laurent Schwartz 2022-2023 volume" 28.

7.3.2 Inverse multiscale problems

In the context of the PhD of S. Ruget, C. Le Bris and F. Legoll have pursued their work on the question of how to determine the homogenized coefficient of a multiscale problem without explicitly performing an homogenization approach. This work is a follow-up on earlier works over the years in collaboration with K. Li, S. Lemaire and O. Gorynina. They have first extended their approach to the case of Schroedinger equations with rapidly oscillating potentials. Current efforts are focused on investigating the robustness of the approach with respect to the available data, which, in practice, may be noisy, blurred, incomplete, or only available in the form of averages over domains of large size (compared to the size of the microstructure).

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 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 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 R. Biezemans, C. Le Bris and F. Legoll have addressed 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 advection-diffusion problems, in the advection-dominated regime. Thin boundary layers are present in the exact solution, and numerical approaches should be carefully adapted to this situation, e.g. using stabilization. How stabilization and the multiscale nature of the problem interplay with one another is a challenging question, and several MsFEM variants have been compared by R. Biezemans, C. Le Bris, F. Legoll and A. Lozinski. The main results have been collected in Part II of the PhD manuscript of R. Biezemans (who defended his PhD thesis in September 2023) and a manuscript is about to be submitted for publication. It is shown there how MsFEM with weak continuity conditions of Crouzeix-Raviart type can be stabilized by adding specific bubble functions, satisfying the same type of weak boundary conditions.

In the context of the PhD of A. Lefort, C. Le Bris and F. Legoll have undertaken the study of a multiscale, reaction-diffusion equation. This problem is different from the equations previously studied by the team by the fact that it includes a large reaction term which competes with the diffusive term. From a numerical perspective, two difficulties are present in the time-dependent version of the problem. First, the coefficients of the equation (and therefore the solution) oscillate at a small spatial scale. In addition, the problem in time is stiff. In order to not address all difficulties at the same time, the associated eigenvalue problem has been considered, for which a promising MsFEM-type approach was introduced. The robustness of the method is currently being investigated.

In parallel to the exploration of advection-diffusion equations and reaction-diffusion equations, another direction of research is focused on hyperbolic multiscale conservation laws. The homogenized limit of a large class of such conservation laws has recently been established in the literature. As a preliminary work on this topic, A. Boucart (currently post-doc in the project-team), C. Le Bris and F. Legoll have put in action the classical homogenization approach, and numerically demonstrated that the two-scale approximation provided by homogenization theory is indeed an accurate approximation of the reference solution, however for a ratio between the macro and the micro scales which needs to be larger than for elliptic problems. A manuscript collecting their conclusions (on this topic as well as on related questions) is being prepared for publication.

7.3.4 Stochastic homogenization

Using standard homogenization theory, one knows that the homogenized tensor, which is a deterministic matrix, depends on the solution of a stochastic equation, the so-called corrector problem, which is posed on the whole space ${ℝ}^d$. This equation is therefore delicate and expensive to solve. A standard approach consists in truncating the space ${ℝ}^d$ to some bounded domain, on which the corrector problem is numerically solved.

In collaboration with B. Stamm and S. Xiang (both at Aachen University, Germany), V. Ehrlacher and F. Legoll have studied in 56 new alternatives for the approximation of the homogenized matrix in the case of the (vector-valued) linear elasticity equation. This work extends previous works dedicated to scalar-valued equations. These alternative definitions rely on the use of an embedded corrector problem, where a finite-size domain made of the highly oscillatory material is embedded in a homogeneous infinite medium whose diffusion coefficients have to be appropriately determined.

In collaboration with M. Bertin and S. Brisard (ENPC), F. Legoll has introduced in 16 a variance reduction approach for the computation of the homogenized coefficients. The method is based on concurrently using (in a control variate fashion) the reference computations (solving the corrector problem on a large but finite domain) together with some inexpensive mean-field approximations often used in the computational mechanics community, such as the ones stemming from the Hashin-Shtrikman principle. The numerical efficiency of the approach has been demonstrated on several examples, including cases with large contrasts.

7.4.1 Complex fluids

Participants: Sébastien Boyaval.

In 2023, S. Boyaval has generalized the symmetric-hyperbolic system of conservation laws introduced in 2020 to a whole class of models for non-Newtonian fluids 15. Precisely, a family of quasilinear PDEs is proved symmetric-hyperbolic and endowed with an additional dissipation inequality (a formulation of the second- principle) whatever the choice of a "stored energy functional"–"relaxation functional" couple within that class. This defines a framework for numerical methods.

7.4.2 Model-order reduction methods

Participants: Virginie Ehrlacher, Tony Lelièvre, Giulia Sambattaro.

The objective of a reduced-order 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 reduced-order model method then consists in constructing, from a few complex simulations which 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 62, V. Ehrlacher together with I. Niakh, G. Drouet (EDF) and A. Ern (SERENA) introduced a new model reduction method for parametrized linear variational inequalities of the first and second kind for mechanical contact and friction problems. The method relies on the use of a so-called Nitsche formulation of the problem. This avoids the use of dual variables, which are well-known not to be easily reducible for these types of problems.

In 52 and 51, V. Ehrlacher and T. Lelièvre, together with G.Dusson (Université de Besançon) and Y. Conjungo Taumhas, and F. Madiot (CEA) develop a reduced basis method for parametrized non-symmetric eigenvalue problems arising in the loading pattern optimization of a nuclear core in neutronics. This requires to derive a posteriori error estimates for the eigenvalue and left and right eigenvectors. A first implementation of the method in APOLLO3, the CEA/DES deterministic multi-purpose code for reactor physics analysis, is presented.

7.4.3 Cross-diffusion systems

Participants: Jean Cauvin-Vila, Virginie Ehrlacher.

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

In 49, J. Cauvin-Vila, V. Ehrlacher, G. Marino (Augsburg) and J.-F. Pietschmann (Augsburg) studied mathematically a model of a multicomponent mixture where cross-diffusion effects occur between the different species but only one species separates from the other. The evolution of the system is modeled by a gradient flow, in an appropriate metric, of a degenerate Ginzburg–Landau energy, which yields a system of coupled partial differential equations of Cahn–Hilliard type. Local minimizers of the energy functional are shown to exist and to qualify as classical stationary solutions of the system. Finally, the authors prove exponential convergence to a constant stationary solution in a particular parameter regime, and they introduce a novel structure-preserving finite volume scheme to approximate the evolution numerically.

In 37, C. Cancès, J. Cauvin-Vila, C. Chainais-Hillairet and V. Ehrlacher developed a new structure-preserving numerical scheme for a model of a physical vapor deposition process used for the fabrication of thin film layers. The model involves two different types of cross-diffusion systems coupled by an evolving interface. The moving interface is addressed with a cut-cell approach, where the mesh is locally deformed around the interface. The scheme is shown to preserve the structure of the continuous system, namely: mass conservation, nonnegativity, volume-filling constraints and decay of the free energy.

9.2.1 Visits to international teams

9.3.1 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):

• Analyse et équations aux dérivées partielles, 30h (M. Dus, V. Ehrlacher, A. Lefort, E. Letournel, S. Ruget, G. Sambataro)
• Équations aux dérivées partielles: approches variationnelles, 15h (S. Darshan, F. Legoll, S. Ruget, R. Santet)
• Initiation au travail en projet, 15h (R. Santet, R. Spacek, U. Vaes, L. Vidal)
• Introduction à l'optimisation (M. Dus)
• Mathématiques en action (R. Biezemans, N. Blassel, A. Bordignon, E. Cancès, S. Darshan, É. Polack, R. Santet, G. Stoltz, L. Vidal)
• Mécanique des milieux continus fluides, 25h (S. Boyaval)
• Mécanique quantique, 15h (E. Cancès, A. Kirsch)
• PAMS project, 16h (N. Blassel, R. Spacek)
• Pratique du calcul scientifique, 18h (N. Blassel, S. Darshan, R. Santet, R. Spacek, U. Vaes)
• Probabilités, 24h (N. Blassel, S. Darshan)
• Tutorats d’analyse et calcul scientifique, 18h (M. Rachid)

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

• Contrôle de systèmes dynamiques et équations aux dérivées partielles, 18h (E. Cancès)
• Problèmes d'évolution, 36h (V. Ehrlacher, F. Legoll)
• Projets Modéliser Programmer Simuler (T. Lelièvre)
• Statistics and data sciences, 30h (G. Stoltz)
• Techniques de developpement logiciel, 32h30 (É. Polack)

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

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

At other institutions:

• Aléatoire (MAP361), 40h, Ecole Polytechnique (T. Lelièvre)
• 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)
• Modélisation, 40h, ENS Paris (S. Perrin-Roussel)
• Numerical Analysis, 56h, NYU Paris (U. Vaes)
• Réduction d’Endomorphismes (18h), Analyse numérique (18h), ESILV Paris (M. Rachid)
• Technologies for quantum computing: a mathematical perspective, 5 lectures, block course at Berlin Mathematical School, Winter 2022-2023 (C. Le Bris)
• Théorie spectrale et mécanique quantique, 30h, ENS Paris (S. Perrin-Roussel)

10.2.2 Supervision

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

• Rutger Biezemans, funding DIM Math Innov (Inria), Méthodes multi-échelles : approches non-intrusives, problèmes advection-dominés et questions reliées, co-supervised by C. Le Bris and A. Lozinski (University of Besançon), defended in September.
• Jean Cauvin-Vila, funding Ecole des Ponts, Cross-diffusion systems in moving-boundary domains, co-supervised by V. Ehrlacher and A. Hayat, defended in December.
• Yonah Conjugo-Taumhas, thèse CIFRE CEA, Criticality calculations in neutronics: model order reduction and a posteriori error estimators, co-supervised by T. Lelièvre and V. Ehrlacher together with G. Dusson (CNRS Besançon) and F. Madiot (CEA), defended in December.
• Maria Fuente-Ruiz, funding INRIA (COMMEDIA team), Adaptive tensor methods for high dimensional problems, co-supervised by D. Lombardi and V. Ehrlacher, defended in March.
• Thomas Pigeon, funding INRIA, Rare event sampling methods and machine learning to study catalytic reaction mechanisms, co-supervised by P. Raybaud (IFPEN) and T. Lelièvre, together with G. Stoltz and M. Corral-Vallero (IFPEN), defended in October.

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

• Hichem Belbal, thèse CIFRE EDF, Understanding suspended matter measures in Loire river, since September 2022, supervised by S. Boyaval (and O. Cerdan from BRGM)
• Elisa Beteille, thèse CIFRE EDF, Propagation of Urban Flood waves, since November 2021, supervised by S. Boyaval (and F. Larrarte from UGE)
• Noé Blassel, funding ERC Synergy EMC2, Approximation of the quasi-stationnary distribution, Ecole des Ponts, since October 2022, co-supervised by T. Lelièvre and G. Stoltz
• Andrea Bordignon, Mathematical and numerical analysis for Density Functional Theory, funding ERC Synergy EMC2, co-supervised by E. Cancès and A. Levitt
• Thomas Brunel, funded by ANR Neptune, Paddle sports physics: Velocity–stroke rate and active drags, supervised by S. Boyaval (and R. Carmigniani from ENPC)
• Charlotte Chapellier, Generative methods for drug design, funding CIFRE Sanofi, since October 2023, co-supervised by T. Lelièvre and G. Stoltz
• Shiva Darshan, funding ANR SINEQ, Linear response of constrained stochastic dynamics, since October 2022, co-supervised by G. Stoltz and S. Olla (Université Paris-Dauphine)
• Raphaël Gastaldello, funding CNRS, Variance reduction methods for the computation of transport coefficients, since December 2023, co-supervised by G. Stoltz and U. Vaes
• Clément Guillot, funding ENPC, Space-time variational principles for the Schrödinger equation in large dimension, since November 2023, supervised by V. Ehrlacher and M. Dupuy (Sorbonne Université)
• Abbas Kabalan, thèse CIFRE SAFRANTech, Reduced-order models for problems with non-parametric geometrical variations, since November 2022, co-supervised by V. Ehrlacher and F. Casenave (SAFRANTech)
• Alfred Kirsch, funding Simons foundation, Mathematical and numerical analysis of interacting electrons models, École des Ponts, since September 2021, co-supervised by E. Cancès and D. Gontier (Paris-Dauphine CEREMADE)
• Albéric Lefort, funding CERMICS-ENPC, Multiscale numerical methods for reaction-diffusion equations and related problems, Ecole des Ponts, since November 2022, co-supervised by F. Legoll and C. Le Bris
• Eloïse Letournel, funding DIM Math Innov (Inria), Finite size effects in electronic structure, École des Ponts, since September 2021, supervised by A. Levitt
• 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 Octobre 2023, co-supervised by E. Cancès and T. Ayral (Eviden)
• Solal Perrin-Roussel, funding École des Ponts, Mathematical anlaysis and numerical simulation of electronic transport in moiré materials, co-supervised by É. Cances and by D. Gontier (CEREMADE, Université Paris-Dauphine PSL)
• Jean Ruel, funding ENS-Saclay, Certified and robust reduced models for the simulation of elongated structures, Ecole des Ponts, since October 2023, co-supervised by F. Legoll, L. Chamoin (ENS-Saclay) and A. Lebée (Ecole des Ponts)
• Simon Ruget, funding Inria, Coarse approximation for a Schrödinger problem with highly oscillatory coefficients, Ecole des Ponts, since October 2022, co-supervised by F. Legoll and C. Le Bris
• Régis Santet, funding Ecole des Ponts, Enhancing the sampling efficiency of reversible and non-reversible dynamics, Ecole des Ponts, since October 2021, co-supervised by T. Lelièvre and G. Stoltz
• Lev-Arcady Sellem, funding Advanced ERC Q-Feedback (PI: P. Rouchon), Mathematical approaches for simulation and control of open quantum systems, Ecole des Mines de Paris, since October 2020, co-supervised by C. Le Bris and P. Rouchon (Inria QUANTIC)
• Renato Spacek, funding FSMP CoFund, Efficient computation of linear response of nonequilibrium stochastic dynamics, ED 386 Sorbonne-Université, since November 2021, co-supervised by G. Stoltz and P. Monmarché (Sorbonne Université)
• Jana Tarhini, funded by Inria-IFPEN, Fast simulation of CO2/H2 storage in geological bassins, since November 2021, supervised by S. Boyaval (and G. Enchéry, H. Tran from IFPEN)
• Jean-Paul Travert, thèse CIFRE EDF, Data assimilation for flood predictions, since November 2022, supervised by S. Boyaval (and C. Goeury from EDF)
• Laurent Vidal, funding ERC Synergy EMC2, Model reduction in physics and quantum chemistry, since February 2021, supervised by E. Cancès and A. Levitt.

10.2.3 Juries

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

• E. Cancès, PhD of Thiago Carvalho Corso ("Mathematical contributions to static and time-dependent density functional theory") defended at TU Munich in July (referee)
• E. Cancès, PhD of Adechola Kouande ("Étude mathématique de l'instabilité de Peierls dans le modèle discret du polyacétylène"), defended at University Paris-Dauphine PSL in October (president)
• V. Ehrlacher, PhD of Yipeng Wang ("Estimation d’erreur a posteriori pour des calculs de structure électronique par des méthodes ab initio et son application pour diminuer le cout de calcul"), defended at Sorbonne University in December (president)
• V. Ehrlacher, PhD of Anatole Gallouët ("Problèmes inverses en optique anidolique et équations de jacobien généré"), defended at Grenoble University in October (president)
• V. Ehrlacher, PhD of Gong Chen ("Force Field Parameterization in Molecular Simulation by Machine Learning Methods"), defended at Sorbonne Université in September
• V. Ehrlacher, PhD of Rodrigue Lelotte ("Sur les gaz de Coulomb et le transport optimal multimarge"), defended at Dauphine University in October (referee)
• V. Ehrlacher, PhD of Hannes Vandecasteele ("Micro-Macro accelerated Markov Chain Monte Carlo Methods with Applications in Molecular Dynamics"), defended at Leuven University in June
• V. Ehrlacher, PhD of Matthieu Dolbeault ("Echantillonnage optimal et réduction de modèle"), defended at Sorbonne University in May (referee)
• V. Ehrlacher, PhD of Mathias Beaupère ("Algorithmes parallèles pour le calcul des décompositions de rang faible des matrices et tenseurs"), defended at INRIA in March
• C. Le Bris, PhD of Alexandre Girodroux-Lavigne (“Derivation and study of effective models for heterogeneous media and fluid suspensions”), defended at Université Paris-Cité in September
• C. Le Bris, PhD of Pierre Amenoagbadji (“Propagation des ondes dans des milieux quasi-périodiques”), defended at ENSTA Paris in December
• F. Legoll, PhD of Amandine Boucart (“Diffraction électromagnétique par une couche mince de nanoparticules réparties aléatoirement: développement asymptotique, conditions effectives et simulations”), defended at ENSTA Paris in April (referee)
• T. Lelièvre, PhD of Maksim Kaledin ("Development and Theoretical Analysis of the Algorithms for Optimal Control and Reinforcement Learning"), defended at National Research University Higher School of Economics in January (referee)
• T. Lelièvre, PhD of Kevin Fröhlicher ("Improving Monte Carlo reactor physics simulations using adaptive sampling of neutron histories"), defended at Université Paris Saclay in February (president)
• T. Lelièvre, PhD of Athina Monemvassitis ("Analysis and development of non-reversible samplers: Applications in Statistical and High-energy Physics"), defended at Université Clermont Auvergne in March (referee)
• T. Lelièvre, PhD of Claudia Fonte Sanchez ("Constructive and asymptotic estimates for the solutions of some linear and nonlinear PDEs"), defended at Université PSL in April (examiner)
• T. Lelièvre, PhD of Matthieu Dolbeault ("Optimal Sampling and Model Order Reduction"), defended at Sorbonne Université in June (examiner)
• T. Lelièvre, PhD of Iosif Lytras ("Solving sampling and optimization problems via Tamed Langevin MCMC algorithms in the presence of super-linearities"), defended at the University of Edinburgh, in September (referee)
• T. Lelièvre, PhD of Ashot Aleksian ("Exit-problem for Self-interacting and Self-stabilizing Diffusion Processes"), defended at Université Jean Monnet in November (referee)
• G. Stoltz, PhD of Thimotée Devergne ("Machine learning methods for computational studies in origins of life"), defended at Sorbonne Université in September
• G. Stoltz, PhD of Gong Chen ("Force Field Parameterization in Molecular Simulation by Machine Learning Methods"), defended at Sorbonne Université in September (referee)
• G. Stoltz, PhD of Théo Jaffrelot–Inizan ("Machine learning for new generation molecular dynamics"), defended at Sorbonne Université in October

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

• E. Cancès, HdR of Bérengère Delourme ("Quelques contributions à l'étude théorique et numérique des phénomènes de propagation des ondes dans des milieux périodiques"), defended at Université Sorbonne Paris Nord in October
• T. Lelièvre, HdR of Davide Mancusi ("Reactors, reactions, and random numbers"), defended at Université Paris Saclay in September

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

• E. Cancès, Poste de Professeur en mathématiques appliquées, Université de Rouen
• V. Ehrlacher, Chaire professeur junior, Ecole des Ponts
• V. Ehrlacher, Member of the INRIA-CordiS selection committee
• V. Ehrlacher, Poste de Maître de Conférences en mathématiques appliquées, Centrale Lyon
• V. Ehrlacher, Poste de Professeur en mathématiques appliquées, Université de Rennes
• V. Ehrlacher, Vice-president of the ANR project selection committee CE46
• T. Lelièvre, Poste de Professeur en mathématiques appliquées, Ecole Polytechnique
• T. Lelièvre, Postes CRCN et ISFP, Inria Paris
• L. Nenna, Poste MCF, Université Paris-Saclay
• G. Stoltz, Chaire professeur junior, Sorbonne Université
• G. Stoltz, Chaire professeur junior, Ecole des Ponts

International journals

• 10 articleY.Yves Achdou and C.Claude Le Bris. Homogenization of some periodic Hamilton-Jacobi equations with defects.Communications in Partial Differential Equations486August 2023HALDOIback to text
• 11 articleM.Manon Baudel, A.Arnaud Guyader and T.Tony Lelièvre. On the Hill relation and the mean reaction time for metastable processes.Stochastic Processes and their Applications155January 2023, 393-436HALDOI
• 12 articleZ.Zineb Belkacemi, M.Marc Bianciotto, H.Hervé Minoux, T.Tony Lelièvre, G.Gabriel Stoltz and P.Paraskevi Gkeka. Autoencoders for dimensionality reduction in molecular dynamics: collective variable dimension, biasing and transition states.Journal of Chemical Physics159July 2023, 024122HALDOIback to text
• 13 articleR.Rutger Biezemans, C.Claude Le Bris, F.Frédéric Legoll and A.Alexei Lozinski. Non-intrusive implementation of Multiscale Finite Element Methods: an illustrative example.Journal of Computational Physics4772023, 111914HALDOI
• 14 articleR. A.Rutger A. Biezemans, C.Claude Le Bris, F.Frédéric Legoll and A.Alexei Lozinski. Non-intrusive implementation of a wide variety of Multiscale Finite Element Methods.Comptes Rendus. Mécanique35112023HALDOI
• 15 articleS.Sébastien Boyaval. A class of symmetric-hyperbolic PDEs modelling fluid and solid continua.ESAIM: Proceedings and Surveys2023HALback to text
• 16 articleS.Sebastien Brisard, M.Michael Bertin and F.Frederic Legoll. A variance reduction strategy for numerical random homogenization based on the equivalent inclusion method.Computer Methods in Applied Mechanics and Engineering4172023, 116389HALDOIback to text
• 17 articleE.Eric Cancès, G.Geneviève Dusson, G.Gaspard Kemlin and L.Laurent Vidal. On basis set optimisation in quantum chemistry.ESAIM: Proceedings and Surveys73August 2023, 107-129HALDOI
• 18 articleC.Clément Cancès, V.Virginie Ehrlacher and L.Laurent Monasse. Finite Volumes for the Stefan-Maxwell Cross-Diffusion System.IMA Journal of Numerical AnalysisJune 2023HALDOI
• 19 articleÉ.Éric Cancès, L.Louis Garrigue and D.David Gontier. Simple derivation of moiré-scale continuous models for twisted bilayer graphene.Physical Review B10715April 2023, 155403HALDOI
• 20 articleE.Eric Cancès, M. F.Michael F. Herbst, G.Gaspard Kemlin, A.Antoine Levitt and B.Benjamin Stamm. Numerical stability and efficiency of response property calculations in density functional theory.Letters in Mathematical Physics1131February 2023HALDOI
• 21 articleE.Eric Cancès, G.Gaspard Kemlin and A.Antoine Levitt. A priori error analysis of linear and nonlinear periodic Schrödinger equations with analytic potentials.Journal of Scientific Computing981December 2023, 25HALDOIback to text
• 22 articleF.-B.Francois-Baptiste Cartiaux, A.Alain Ehrlacher, F.Frédéric Legoll, A.Alex Libal and J.Julien Reygner. Probabilistic formulation of Miner's rule and application to structural fatigue.Probabilistic Engineering Mechanics742023, 103500HALDOI
• 23 articleJ.Jean Cauvin-Vila, V.Virginie Ehrlacher and A.Amaury Hayat. Boundary stabilization of one-dimensional cross-diffusion systems in a moving domain: linearized system.Journal of Differential Equations350March 2023, 251-307HALDOI
• 24 articleL.Ludovic Chamoin and F.Frederic Legoll. An introductory review on a posteriori error estimation in Finite Element computations.SIAM Review6542023, 963-1028HALDOI
• 25 articleO.Olga Gorynina, F.Frédéric Legoll, T.Tony Lelièvre and D.Danny Perez. Combining machine-learned and empirical force fields with the parareal algorithm: application to the diffusion of atomistic defects.Comptes Rendus. Mécanique35112023HALDOI
• 26 articleT.Thê Hoang Ngoc Minh, G.Gabriel Stoltz and B.Benjamin Rotenberg. Frequency and field-dependent response of confined electrolytes from Brownian dynamics simulations.Journal of Chemical Physics15810March 2023, 104103HALDOIback to text
• 27 articleA.Aikaterini Karoni, B.Benedict Leimkuhler and G.Gabriel Stoltz. Friction-adaptive descent: a family of dynamics-based optimization methods.Journal of Computational Dynamics104October 2023, 450-484HALDOIback to text
• 28 articleC.Claude Le Bris. Defects in homogenization theory.Séminaire Laurent Schwartz - EDP et applicationsJuly 2023, 1-17HALDOIback to text
• 29 articleT.Tony Lelièvre, G.Geneviève Robin, I.Inass Sekkat, G.Gabriel Stoltz and G.Gabriel Victorino Cardoso. Generative methods for sampling transition paths in molecular dynamics.ESAIM: Proceedings73August 2023, 238-256HALDOI
• 30 articleT.Tony Lelièvre, G.Gabriel Stoltz and W.Wei Zhang. Multiple projection MCMC algorithms on submanifolds.IMA Journal of Numerical Analysis432March 2023, 737-788HALDOI
• 31 articleI.Idrissa Niakh, G.Guillaume Drouet, V.Virginie Ehrlacher and A.Alexandre Ern. Stable model reduction for linear variational inequalities with parameter-dependent constraints.ESAIM: Mathematical Modelling and Numerical Analysis5712023, 167--189HAL
• 32 articleF.Federica Pes, É.Étienne Polack, P.Patrizia Mazzeo, G.Geneviève Dusson, B.Benjamin Stamm and F.Filippo Lipparini. A Quasi Time-Reversible scheme based on density matrix extrapolation on the Grassmann manifold for Born-Oppenheimer Molecular Dynamics.Journal of Physical Chemistry Letters1443July 2023, 9720-9726HALDOIback to text
• 33 articleT.Thomas Pigeon, G.Gabriel Stoltz, M.Manuel Corral-Valero, A.Ani Anciaux-Sedrakian, M.Maxime Moreaud, T.Tony Lelièvre and P.Pascal Raybaud. Computing Surface Reaction Rates by Adaptive Multilevel Splitting Combined with Machine Learning and Ab Initio Molecular Dynamics.Journal of Chemical Theory and Computation1912June 2023, 3538-3550HALDOIback to text
• 34 articleI.Inass Sekkat and G.Gabriel Stoltz. Removing the mini-batching error in Bayesian inference using Adaptive Langevin dynamics.Journal of Machine Learning Research24329December 2023, 1-58HAL
• 35 articleR.Renato Spacek and G.Gabriel Stoltz. Extending the regime of linear response with synthetic forcings.Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal214November 2023, 1602-1643HALDOIback to text

Invited conferences

• 36 inproceedingsD.Damiano Lombardi, V.Virginie Ehrlacher and M.Maria Fuente-Ruiz. SoTT: a greedy construction of a sum of Tensor Trains.ICOSAHOM 2023 - International Conference on Spectral and High Order MethodsSeoul, South KoreaAugust 2023HAL

International peer-reviewed conferences

• 37 inproceedingsC.Clément Cancès, J.Jean Cauvin-Vila, C.Claire Chainais-Hillairet and V.Virginie Ehrlacher. Structure Preserving Finite Volume Approximation of Cross-Diffusion Systems Coupled by a Free Interface.Finite Volume for Complex Applications X -- Volume 1, Elliptic and parabolic problemsFinite Volumes for Complex Applications X432Springer Proceedings in Mathematics & StatisticsStrasbourg, FranceSpringer Nature SwitzerlandOctober 2023, 205-213HALDOIback to text
• 38 inproceedingsL.-A.Lev-Arcady Sellem, R.Rémi Robin, P.Philippe Campagne-Ibarcq and P.Pierre Rouchon. Stability and decoherence rates of a GKP qubit protected by dissipation.22nd World Congress of the International Federation of Automatic ControlIFAC 2023 - 22nd World Congress of the International Federation of Automatic ControlYokohama, JapanApril 2023HALDOI

Scientific books

• 39 bookX.Xavier Blanc and C.Claude Le Bris. Homogenization Theory for Multiscale Problems: An introduction.21MS&ASpringer Nature Switzerland2023HALDOI
• 40 bookE.Eric Cancès and G.Gero Friesecke (eds.). Density Functional Theory.Springer International Publishing2023HALDOIback to text

Scientific book chapters

• 41 inbookE.Eric Cancès, A.Antoine Levitt, Y.Yvon Maday and C.Chao Yang. Numerical Methods for Kohn–Sham Models: Discretization, Algorithms, and Error Analysis.Density Functional TheorySpringer International PublishingDecember 2023, 333-400HALDOIback to text
• 42 inbookC.Christophe Chipot, P.Paraskevi Gkeka, T.Tony Lelièvre and G.Gabriel Stoltz. Equilibrium and Nonequilibrium Methods for Free-Energy Calculations With Molecular Dynamics.3Comprehensive Computational ChemistryElsevier; Elsevier2024, 384-400HALDOIback to text
• 43 inbookD.Danny Perez and T.Tony Lelièvre. Recent advances in Accelerated Molecular Dynamics Methods: Theory and Applications.3Comprehensive Computational ChemistryElsevier2024, 360-383HALDOIback to text

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

• 44 proceedingsB.Beatrice BattistiT.Tobias BlickhanG.Guillaume EncheryV.Virginie EhrlacherD.Damiano LombardiO.Olga MulaWasserstein model reduction approach for parametrized flow problems in porous media.CEMRACS 2021 - Data Assimilation and Reduced Modeling for High Dimensional Problems73EDP SciencesAugust 2023, 28-47HALDOI

Reports & preprints

• 45 miscN.Noé Blassel and G.Gabriel Stoltz. Fixing the flux: A dual approach to computing transport coefficients.May 2023HALback to text
• 46 misc G.Giovanni Brigati and G.Gabriel Stoltz. How to construct decay rates for kinetic Fokker--Planck equations? February 2023 HAL back to text
• 47 miscE.Eric Cancès, F. M.Fabian M. Faulstich, A.Alfred Kirsch, E.Eloïse Letournel and A.Antoine Levitt. Some mathematical insights on Density Matrix Embedding Theory.October 2023HALback to text
• 48 miscE.Eric Cancès and L.Long Meng. Semiclassical analysis of two-scale electronic Hamiltonians for twisted bilayer graphene.November 2023HALback to text
• 49 miscJ.Jean Cauvin-Vila, V.Virginie Ehrlacher, G.Greta Marino and J.-F.Jan-Frederik Pietschmann. Stationary solutions and large time asymptotics to a cross-diffusion-Cahn-Hilliard system.July 2023HALback to text
• 50 miscM.Martin Chak, T.Tony Lelièvre, G.Gabriel Stoltz and U.Urbain Vaes. Optimal importance sampling for overdamped Langevin dynamics.July 2023HALback to text
• 51 miscY.Yonah Conjungo Taumhas, G.Geneviève Dusson, V.Virginie Ehrlacher, T.Tony Lelièvre and F.François Madiot. An Application of Reduced Basis Methods to Core Computation in APOLLO3 ®.November 2023HALback to text
• 52 miscY.Yonah Conjungo Taumhas, G.Geneviève Dusson, V.Virginie Ehrlacher, T.Tony Lelièvre and F.François Madiot. Reduced basis method for non-symmetric eigenvalue problems: application to the multigroup neutron diffusion equations.July 2023HALback to text
• 53 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 quantum chemistry.July 2023HALback to text
• 54 miscM.Mathias Dus and V.Virginie Ehrlacher. Numerical solution of Poisson partial differential equation in high dimension using two-layer neural networks.May 2023HAL
• 55 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.January 2023HALback to text
• 56 miscV.Virginie Ehrlacher, F.Frederic Legoll, B.Benjamin Stamm and S.Shuyang Xiang. Embedded corrector problems for homogenization in linear elasticity.July 2023HALback to text
• 57 miscV.Virginie Ehrlacher and L.Luca Nenna. A sparse approximation of the Lieb functional with moment constraints.June 2023HALback to text
• 58 miscN. J.Nicolai Jurek Gerber, F.Franca Hoffmann and U.Urbain Vaes. Mean-field limits for Consensus-Based Optimization and Sampling.December 2023HALback to text
• 59 miscT.Tony Lelièvre, T.Thomas Pigeon, G.Gabriel Stoltz and W.Wei Zhang. Analyzing multimodal probability measures with autoencoders.October 2023HALback to text
• 60 miscT.Tony Lelièvre, M.Mouad Ramil and J.Julien Reygner. Estimation of statistics of transitions and Hill relation for Langevin dynamics.May 2023HAL
• 61 miscT.Tony Lelièvre, R.Régis Santet and G.Gabriel Stoltz. Unbiasing Hamiltonian Monte Carlo algorithms for a general Hamiltonian function.March 2023HALback to text
• 62 miscI.Idrissa Niakh, G.Guillaume Drouet, V.Virginie Ehrlacher and A.Alexandre Ern. A reduced basis method for frictional contact problems formulated with Nitsche's method.July 2023HALback to text
• 63 miscL.-A.Lev-Arcady Sellem, A.Alain Sarlette, Z.Zaki Leghtas, M.Mazyar Mirrahimi, P.Pierre Rouchon and P.Philippe Campagne-Ibarcq. A GKP qubit protected by dissipation in a high-impedance superconducting circuit driven by a microwave frequency comb.April 2023HALDOI