5.1.1 DFTK

• Keywords:
  Molecular simulation, Quantum chemistry, Materials
• Functional Description:
  DFTK, short for the density-functional toolkit, is a Julia library implementing plane-wave density functional theory for the simulation of the electronic structure of molecules and materials. It aims at providing a simple platform for experimentation and algorithm development for scientists of different backgrounds.
• Release Contributions:
  In 2022 has gained support for GPU acceleration, norm-conserving pseudopotentials, and many other smaller features. It has been used for several publications both inside and outside the project-team.
• URL:
  http://­dftk.­org
• Contact:
  Antoine Levitt

6.1.1 Density functional theory

A track of the project-team's activity this year has been the investigation of continuum eigenstates, as opposed to the bound states that form much of the project-team's usual focus. Such states are relevant to the study of processes where electrons propagate away from the system under consideration, such as ionization. They are delocalized, complicating their discretization. In 35 and 36, together with colleagues from the Laboratoire de Chimie Théorique at Sorbonne Université, É. Cancès and A. Levitt have proposed a method to compute the photoionization spectrum for atoms in time-dependent density functional theory (TDDFT) in the Sternheimer formalism. This method, inspired by similar schemes in numerical wave propagation, employs an analytic Dirichlet-to-Neumann map to impose correct boundary conditions on the Sternheimer equations, which appears mathematically as a perturbation of an Helmholtz equation with a Coulomb potential. In 56, E. Letournel and A. Levitt, together with physicist colleagues from CEA Grenoble, have proposed a method to compute electronic resonances in crystals with defects. The method involves the computation of analytic continuations of Green functions of periodic operators, which is accomplished by a complex deformation of the Brillouin zone.

E. Cancès, G. Kemlin and A. Levitt have studied the numerical properties of response computations in density functional theory at finite temperature. They have proposed a method based on a Schur complement to increase the stability and efficiency of iterative solvers for the Sternheimer equations 49.

Together with D. Gontier (U. Paris Dauphine and ENS Paris), E. Cancès and L. Garrigue provided a formal derivation of a reduced model for twisted bilayer graphene (TBG) from Density Functional Theory. This derivation is based on a variational approximation of the TBG Kohn-Sham Hamiltonian and asymptotic limit techniques. In contrast with other approaches, it does not require the introduction of an intermediate tight-binding model. The so-obtained model is similar to that of the Bistritzer-MacDonald (BM) model but contains additional terms. Its parameters can be easily computed from Kohn-Sham calculations on single-layer graphene and untwisted bilayer graphene with different stackings. It allows one in particular to estimate the parameters $w_{\mathrm{AA}}$ and $w_{\mathrm{AB}}$ of the BM model from first-principles. The resulting numerical values, namely $w_{\mathrm{AA}}=w_{\mathrm{AB}}\simeq 126$ meV for the experimental interlayer mean distance are in good agreement with the empirical values $w_{\mathrm{AA}}=w_{\mathrm{AB}}=110$ meV obtained by fitting to experimental data. We also show that if the BM parameters are set to $w_{\mathrm{AA}}=w_{\mathrm{AB}}\simeq 126$ meV, the BM model is an accurate approximation of this new reduced model.

With G. Dusson (CNRS and U. of Franche-Comté) E. Cancès, G. Kemlin and L. Vidal proposed in 45 general criteria to construct optimal atomic centered basis sets in quantum chemistry. They focuses in particular on two criteria, one based on the ground-state one-body density matrix of the system and the other based on the ground-state energy. The performance of these two criteria was numerically tested and compared on a parametrized eigenvalue problem, which corresponds to a one-dimensional toy version of the ground-state dissociation of a diatomic molecule.

In solid state physics, electronic properties of crystalline materials are often inferred from the spectrum of periodic Schrödinger operators. As a consequence of Bloch's theorem, the numerical computation of electronic quantities of interest involves computing derivatives or integrals over the Brillouin zone of so-called energy bands, which are piecewise smooth, Lipschitz continuous periodic functions obtained by solving a parametrized elliptic eigenvalue problem on a Hilbert space of periodic functions. Classical discretization strategies for resolving these eigenvalue problems produce approximate energy bands that are either non-periodic or discontinuous, both of which cause difficulty when computing numerical derivatives or employing numerical quadrature. In a paper with M. Hassan (Sorbonne Université) 48, E. Cancès and L. Vidal studied an alternative discretization strategy based on an ad hoc operator modification approach. While specific instances of this approach have been proposed in the physics literature, they introduced a systematic formulation of this operator modification approach. They derived a priori error estimates for the resulting energy bands and showed that these bands are periodic and can be made arbitrarily smooth (away from band crossings) by adjusting suitable parameters in the operator modification approach.

6.1.2 Open quantum systems

In his post-doctoral work co-supervised by Claude Le Bris (MATHERIALS) and Pierre Rouchon (Inria QUANTIC), Masaaki Tokieda addresses various issues related to the numerical simulation and the fundamental understanding of several models of physical systems likely candidates to play a crucial role in quantum computing. More specifically, he studies several pathways to efficiently account for adiabatic elimination in the simulation of composite quantum systems in interactions, modeled by Lindblad type master equations. The specific question currently under study is the expansion up to high orders and the compatibility of such an expansion with the formal requirements of consistency of quantum mechanical evolutions. He is also planning to address various other connected issues, all aiming at better fundamental understanding and a more effective simulation of open quantum systems.

6.1.3 Tensor methods

Tensor methods have proved to be very powerful tools in order to represent high-dimensional objects with low complexity. Such methods prove to have a wide range of applications in quantum chemistry, for instance for the approximation of the ground state wavefunction of a molecular system when the number of electrons is large. The DMRG method is one example of such a numerical scheme. Research efforts are led in the team so as to propose new methodological developments in order to improve on the current state-of-the-art tensor methods.

In 20, V. Ehrlacher, M. Fuente-Ruiz and D. Lombardi (Inria COMMEDIA) introduce a method to compute an approximation of a given tensor as a sum of Tensor Trains (TTs), where the order of the variates and the values of the ranks can vary from one term to the other in an adaptive way. The numerical scheme is based on a greedy algorithm and an adaptation of the TT-SVD method. The proposed approach can also be used in order to compute an approximation of a tensor in a Canonical Polyadic format (CP), as an alternative to standard algorithms like Alternating Linear Squares (ALS) or Alternating Singular Value Decomposition (ASVD) methods. Some numerical experiments are presented, in which the proposed method is compared to ALS and ASVD methods for the construction of a CP approximation of a given tensor and performs particularly well for high-order tensors. The interest of approximating a tensor as a sum of Tensor Trains is illustrated in several numerical test cases.

6.2.1 Sampling of the configuration space

There is still a need to improve techniques to sample the configuration space. In 25, Tony Lelièvre together with Lucie Delemotte (KTH, Sweden), J. Hénin (IBPC, France), Michael Shirts (University of Colorado, USA) and Omar Valsson (MPI Mainz, Germany) provide on overview of enhanced sampling algorithms. These algorithms have emerged as powerful methods to extend the potential of molecular dynamics simulations and allow the sampling of larger portions of the configuration space of complex systems. This "living" review is intended to be updated to continue to reflect the wealth of sampling methods as they emerge in the literature.

6.2.2 Mathematical understanding and efficient simulation of nonequilibrium systems

Many systems in computational statistical physics are not at equilibrium. This is in particular the case when one wants to compute transport coefficients, which determine the response of the system to some external perturbation. For instance, the thermal conductivity relates an applied temperature difference to an energy current through Fourier's law, while the mobility coefficient relates an applied external constant force to the average velocity of the particles in the system. G. Stoltz reviewed in 66 the motivations and mathematical framework involved in the computation of transport coefficients, with a particular emphasis on the numerical analysis of the estimators at hand.

The main limitations of usual methods to compute transport coefficients is the large variance of the estimators, which motivates searching for dedicated variance reduction strategies. Such a method was proposed by G. Pavliotis (Imperial College London, United-Kingdom), G. Stoltz and U. Vaes in the context of the estimation of the mobility via Einstein's method in 65, although the method can be adapted to other transport coefficients. The fundamental idea is to approximate the solution to some Poisson equation determining the transport coefficient, and relying on Ito calculus to construct a random variable strongly correlated to the square displacement from the position at origin. The motivation of this work was to estimate the mobility of underdamped Langevin dynamics of two dimensional systems for low values of the friction, in an attempt to (in)validate physical conjectures about the divergence of the mobility as the friction goes to zero.

6.2.3 Sampling dynamical properties and rare events

Sampling trajectories which link metastable states of the target probability measure, and estimating the associated transition rates from one metastable state to another, is a difficult task, which requires dedicated numerical methods. Various works along these lines were preprinted this year.

In 62, Tony Lelièvre, together with Mouad Ramil (Seoul National University, South Korea) and Julien Reygner (CERMICS, France), propose and analyze a simple and complete numerical method to estimate statistics of transitions between metastable states for the Langevin dynamics, based on the so-called Hill relation. More precisely, they prove the Hill relation in the fairly general context of positive Harris recurrent chains, and show that this formula applies to the Langevin dynamics. Moreover, they provide an explicit expression of the invariant measure involved in the Hill relation for the Langevin dynamics, and describe an elementary exact simulation procedure.

In 61, Tony Lelièvre, together with Boris Nectoux (Laboratoire de Mathématiques Blaise Pascal, France) and Dorian Le Peutrec (Institut Denis Poisson, France), conclude a series of papers which aim at providing firm mathematical grounds to jump Markov models which are used to model the evolution of molecular systems, as well as to some numerical methods which use these underlying jump Markov models to efficiently sample metastable trajectories of the overdamped Langevin dynamics. More precisely, using the quasi-stationary distribution approach to analyze the metastable exit from the basin of attraction of a local minimum of the potential energy function, they prove that the exit event (exit position and exit time) of the overdamped Langevin dynamics in the small temperature regime can be accurately modeled by a jump Markov model parameterized by the Eyring–Kramers rates. From a mathematical viewpoint, the proof relies on tools from the semiclassical analysis of Witten Laplacians on bounded domains. The main difficulty is that, since they consider as metastable states the basins of attraction of the local minima of the energy, the exit regions are neighborhoods of saddle points of the energy, and many standard techniques (such as WKB approximations) cannot handle critical points on the boundary.

In 34, Tony Lelièvre, together with Mouad Ramil (Seoul National University, South Korea) and Julien Reygner (CERMICS, France), give an overview of some of the results obtained during the PhD work of Mouad Ramil. More precisely, the paper provides a self-contained analysis of the Parallel Replica algorithm applied to the Langevin dynamics. This algorithm was designed to efficiently sample metastable trajectories relying on a parallelization in time technique. The analysis relies on results on the existence of quasi-stationary distributions of the Langevin dynamics in domains bounded in positions. The article also contains some discussions about the overdamped limit of the quasi-stationary distribution.

Another approach to sampling reactive trajectories is to allow for longer integration times, thanks to dedicated algorithmic developments. In 58, Frédéric Legoll and Tony Lelièvre, together with Olga Gorynina (WSL-SLF, Switzerland) and Danny Perez (Los Alamos National Laboratory, USA) numerically investigate an adaptive version of the parareal algorithm in the context of molecular dynamics. This method allows to more efficiently integrate in time the dynamics under consideration. The parareal algorithm uses a family of machine-learning spectral neighbor analysis potentials (SNAP) as fine, reference, potentials and embedded-atom method potentials (EAM) as coarse potentials. The numerical results (obtained using LAMMPS, a very broadly used software within the materials science community) demonstrate significant computational gains when using the adaptive parareal algorithm in comparison to a sequential integration of the Langevin dynamics.

6.2.4 Machine-learning approaches in molecular dynamics

Together with G. Robin (CNRS & Université d'Evry), I. Sekkat (CERMICS) and G. Victorino Cardoso (CMAP, Ecole polytechnique & IHU LIRYC), T. Lelièvre and G. Stoltz considered in 63 how to generate reactive trajectories linking two metastable states. More precisely, they investigated the capabilities and limitations of supervised and unsupervised methods based on variational autoencoders to generate such paths. Bottleneck autoencoders are however somewhat limited in describing reactive paths, which is why alternative approaches based on an importance sampling function determined by a reinforcement learning strategy were also studied. The potential of the approach was demonstrated on simple low dimensional examples.

A. Levitt and G. Stoltz studied in 17 with F. Bottin (CEA/DAM), A. Castellano (CEA/DAM), J. Bouchet (CEA Cadarache) how to train simple empirical force fields on ab-initio data, in order to reproduce thermodynamic properties at finite temperature. The method iterates between exploration phases where new configurations are efficiently sampled and generated, using the current version of the simple empirical potential at hand, and a training phase where the empirical potential is updated with new ab-initio data. Thermodynamic consistency is ensured via some nonlinear reweighting procedure.

6.2.5 Interacting particle methods for sampling

In some situations, stochastic numerical methods can be made more efficient by using various replicas of the system. The ensemble Kalman filter is a methodology for incorporating noisy data into complex dynamical models to enhance predictive capability. It is widely adopted in the geophysical sciences, underpinning weather forecasting for example, and is starting to be used throughout the sciences and engineering. For high dimensional filtering problems, the ensemble Kalman filter has a robustness that is not shared by the particle filter; in particular it does not suffer from weight collapse. However, there is no theory which quantifies its accuracy as an approximation of the true filtering distribution, except in the Gaussian setting. In order to address this issue, U. Vaes together with J. A. Carrillo (University of Oxford, United Kingdom), F. Hoffmann (Hausdorff Center for Mathematics, Germany) and A. M. Stuart (Caltech, USA) provided in 51 an analysis of the accuracy of the ensemble Kalman filter beyond the Gaussian setting. The analysis is developed for the mean field ensemble Kalman filter, which can be rewritten in terms of maps on probability measures. These maps are proved to be locally Lipschitz in an appropriate weighted total variation metric, which enables to demonstrate that, if the true filtering distribution is close to Gaussian after appropriate lifting to the joint space of state and data, then it is well approximated by the ensemble Kalman filter.

In 60, Tony Lelièvre and Panos Parpas (Imperial College London, United Kingdom) introduce a new stochastic algorithm to locate the index-1 saddle points of a potentiel energy function. Finding index-1 saddle points is crucial to build kinetic Monte Carlo models. These models describe the evolution of the molecular system by a jump Markov model with values in the local minima of the energy function, the jumps between these states being parameterized by the Eyring–Kramers laws. This paramaterization thus requires to identify the index-1 saddle points which connect local minima. The proposed algorithm can be seen as an equivalent of the stochastic gradient descent which is a natural stochastic process to locate local minima. It relies on two ingredients: (i) the concentration properties on index-1 saddle points of the first eigenmodes of the Witten Laplacian on 1-forms and (ii) a probabilistic representation of the solution to a partial differential equation involving this differential operator. The resulting algorithm is an interacting particle system, where the particles populate neighborhoods of the index-1 saddle points. Numerical examples on simple molecular systems illustrate the efficacy of the proposed approach.

6.3.1 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 work, performed in collaboration with X. Blanc, P.-L. Lions and P. Souganidis, has also been the topic of the PhD thesis of R. Goudey, defended this year. We recall that the periodic setting is the oldest traditional setting for homogenization. Alternative settings include the quasi- and 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 modelization viewpoint (in the sense that all that is not known about the microstructure can be “hidden” in a probabilistic description), but this often leads to prohibitively expensive computations, since the model is very general. The direction explored by the project-team consists in considering periodic structures with defects, a setting which is rich enough to fit reality while still leading to affordable computations.

Considering defects in the structure raises many mathematical questions. From an overall perspective, homogenization is based upon the determination of corrector functions, useful to compute the homogenized properties of the materials as well as to provide a fine-scale description of the oscillatory solution. In general, corrector problems are posed on the whole space. In the periodic and random stationary settings, it turns out that the corrector problems can actually be posed on a bounded domain. Powerful tools (e.g. Rellich compactness theorems) can then be used (to establish well-posedness and qualitative properties of the correctors, ...). The presence of defects breaks this property, making the corrector problem non-compact. Additional tools (such as the concentration-compactness method or the theory of Calderón-Zygmund operators) are required to circumvent this difficulty.

Starting from the simplest case (localized defects in a purely diffusive equation, a setting for which we were able to show two-scale expansion results), we have followed two directions: (i) considering more complex equations (advection-diffusion equations, Hamilton-Jacobi equations, ...) for which the defects, although localized, may have an impact on a larger and larger neighborhood, and (ii) considering more complex (i.e. less localized) defects:

• In line with the first direction, and in the context of the "délégation" of Y. Achdou (on partial leave from Université Paris-Cité), C. Le Bris has studied in 38 a homogenization theory for a general first order Hamilton-Jacobi equation in the presence of defects. The study extends to the fully nonlinear setting previous studies performed by X. Blanc, C. Le Bris and P.-L. Lions in the linear (mostly elliptic) setting. It also extends the class of problems previously addressed by Y. Achdou and his collaborators in the periodic setting only. The study complements, in a slightly different but related regime, results obtained a few years ago by P.-L. Lions and P. Souganidis.
• The works 23 (where defects become increasingly rare but do not decay at infinity) and 24, both by R. Goudey, fall within the second direction. Defects in the form of interfaces between two perfectly periodic materials also fall within this research direction. In the same vein, R. Goudey and C. Le Bris have studied in 59 the homogenized limit of a Schroedinger equation with an highly oscillatory potential, when the latter potential belongs to a general class of non-periodic functions that are global perturbations of a periodic function. Such a class of functions is reminiscent of classes constructed two decades ago, in collaboration with X. Blanc and P.-L. Lions, in the context of thermodynamic limit problems. The result obtained may be seen as a first step toward similar studies for other types of equations.

A monograph that summarizes the contributions of the project-team on this topic, along with a general perspective on the field, has been written by C. Le Bris, in collaboration with X. Blanc. The French and English versions of this textbook are respectively in print for the series "Maths & Applications" and "MS&A, Modeling, Simulation and Applications", both at Springer. In addition, C. Le Bris has written a short text that summarizes the major results obtained and that will be published in the "Séminaire Laurent Schwartz 2022-2023 volume".

6.3.2 Inverse multiscale problems

In the context of the PhD of S. Ruget, which started this year, C. Le Bris and F. Legoll have pursued their work on the question of how to determine the homogenized coefficient of a multiscale problem without explicitly performing an homogenization approach. This work is a follow-up on earlier works over the years in collaboration with K. Li, S. Lemaire and O. Gorynina, in the case of a diffusion equation with highly oscillatory diffusion coefficients. Here, this question is revisited in the setting of Schroedinger equations with rapidly oscillating potentials. The motivation for this work is that Schroedinger equations, besides their own interest, show some specific features (in comparison e.g. to diffusion equations) bringing hope that further progress can be achieved. To address these questions is the objective of the PhD of S. Ruget.

6.3.3 Multiscale Finite Element approaches

From a numerical perspective, the Multiscale Finite Element Method (MsFEM) is a classical strategy to address the situation when the homogenized problem is not known (e.g. in difficult nonlinear cases), or when the scale of the heterogeneities, although small, is not considered to be zero (and hence the homogenized problem cannot be considered as a sufficiently accurate approximation).

The MsFEM approach uses a Galerkin approximation of the problem on a pre-computed basis, obtained by solving local problems mimicking the problem at hand at the scale of mesh elements. This basis differs from standard (e.g. polynomial) bases that are generally used in existing legacy codes in industry. As a result, the MsFEM approach is intrusive, which hinders its adoption in industrial (and, more generally, non-academic) environments.

To overcome this obstacle, R. Biezemans, C. Le Bris and F. Legoll, together with A. Lozinski (in delegation in the team for the first half of the year), have designed modified MsFEM approaches that allow for a non-intrusive implementation, i.e., using any existing legacy code for a Galerkin approximation on a piecewise affine basis. The key principles of the approach are presented in 12. The technique is reminiscent of the classical approach to homogenization: "corrector" functions are computed in each element of the mesh, from which slowly varying effective coefficients are computed. This leads to an effective PDE that can indeed be solved by standard finite element approaches.

A more comprehensive study of the non-intrusive MsFEM technique has subsequently been finalized in 41, where R. Biezemans, C. Le Bris, F. Legoll and A. Lozinski show that the non-intrusive approach can be extended to a wide variety of problems and more advanced, state of the art MsFEM variants. Indeed, this work provides non-intrusive MsFEMs for general linear second order PDEs, and with various choices for the local problems, in particular the use of an oversampling technique (where the local problems are solved on domains that are larger than a single mesh element) and a Crouzeix-Raviart MsFEM. This work is also the first to propose and test a new MsFEM variant, namely by developing an oversampling technique for the Crouzeix-Raviart MsFEM. Further, it is proved there that the non-intrusive approach is equivalent to the original MsFEM for a Petrov-Galerkin variant of the MsFEM with affine test functions, and for the Galerkin MsFEM, numerical experiments show that the non-intrusive approach preserves the same accuracy as the original MsFEM.

A second research direction pursued in the PhD of R. Biezemans is the question of how to design accurate MsFEM approaches for various types of equations, beyond the purely diffusive case, and in particular for the case of multiscale 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 are being prepared for publication, showing in particular the stabilization of an MsFEM with weak continuity conditions of Crouzeix-Raviart type by adding specific bubble functions, satisfying the same type of weak boundary conditions, to the approximation space.

Finally, R. Biezemans, C. Le Bris, F. Legoll and A. Lozinski have continued their study of the convergence analysis of MsFEMs. Indeed, despite the fact that MsFEM approaches have been proposed more than two decades ago, it turns out that not all specific settings are covered by the numerical analyses existing in the literature. The research team have previously extended the analysis of MsFEM to the case of rectangular meshes and that of periodic diffusion coefficients that are not necessarily Hölder continuous. An ongoing research effort is devoted to further generalizing the analysis to non-periodic settings and to provide a fully rigorous convergence proof of various MsFEMs with the oversampling technique.

In the context of the PhD of A. Lefort, which started this year, C. Le Bris and F. Legoll have undertaken the study of a multiscale, time-dependent, reaction-diffusion equation. This problem is different from the equations previously studied by the team by the fact that it is time-dependent and that it includes a reaction term (in addition to the diffusive term). From a numerical perspective, two difficulties are present in the problem. First, the coefficients of the equation (and therefore the solution) oscillate at a small spatial scale. In addition, the problem in time is stiff: a standard marching scheme such as the backward Euler scheme would need a small time-step to provide an accurate solution. Several directions of research have been identified, such as establishing the homogenized limit of the problem and designing efficient numerical approaches.

6.4.1 Complex fluids

Participants: Sébastien Boyaval.

6.4.2 Model-order reduction methods

Participants: Jad Dabaghi, Virginie Ehrlacher.

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 get an accurate approximation of the solution of the model for any other values of the parameters.

In 64, together with Idrissa Niakh, Guillaume Drouet (EDF) and Alexandre Ern (SERENA), a new stable model reduction method for linear variational inequalities with parameter-dependent constraints. The method was applied for the reduction of various parametrized contact mechanical problems.

In 39, a new model-order reduction method based on optimal transport theory was investigated by Virginie Ehrlacher together with B. Battisti, T. Blickhan (Garching, Germany), G. Enchéry (IFPEN), D. Lombardi (INRIA COMMEDIA) and O. Mula (Eindhoven University). This approach, based on the use of Wasserstein barycenters, was successfully applied to the reduction of parametrized porous medium flow problems.

6.4.3 Cross-diffusion systems

Participants: Jean Cauvin-Vila, Jad Dabaghi, Virginie Ehrlacher.

Cross-diffusion systems are nonlinear degenerate parabolic systems which 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 55, Jad Dabaghi and Virginie Ehrlacher investigated a new structure-preserving model reduction methods for parametrized cross-diffusion systems. The proposed numerical method is analyzed from a mathematical point of view and proved to satisfy the same mathematical properties as the high fidelity modelity (preserving for instance the non-negativeness of the solutions). The method consists in introducing an appropriate nonlinear transformation of the set of solutions, based on the use of entropy variables associated to the system, before applying reduced-basis techniques to the parametrized problem.

In 52, Jean Cauvin-Vila, Virginie Ehrlacher and Amaury Hayat analyzed from a mathematical point of view the boundary stabilization of one-dimensional (linearized) cross-diffusion system in a moving domain. They prove that the system can be stabilized in any arbitrary finite time by using the so-called backstepping stabilization technique.

8.2.1 Visits of international scientists

Danny Perez (Los Alamos National Laboratory, USA) visited the team in September and October. This was the opportunity to discuss new research directions for accelerated molecular dynamics algorithms.

8.2.2 Visits to international teams

8.3.1 H2020 projects

9.2.1 Teaching

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

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

• Analyse et calcul scientifique, 30h (R. Biezemans, J. Cauvin-Vila, V. Ehrlacher, E. Letournel, R. Santet, G. Stoltz)
• Équations aux dérivées partielles: approches variationnelles, 15h (R. Biezemans, J. Cauvin-Vila, F. Legoll, R. Santet)
• Probabilités, 24h (N. Blassel)
• Hydraulique numérique, 15h (S. Boyaval)
• Outils mathématiques pour l’ingénieur (E. Cancès: 18h, V. Ehrlacher, E. Letournel, F. Legoll, A. Levitt, G. Stoltz, L. Vidal: 9h)
• Mécanique quantique, 15h (E. Cancès, A. Levitt)
• Méthodes numériques pour les problèmes en grande dimension, 17h30 (V. Ehrlacher)
• Pratique du calcul scientifique, 15h (A. Levitt)
• Initiation au travail en projet, 13h (R. Santet, R. Spacek, L. Vidal)
• PAMS project, 16h (R. Spacek)

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

• Contrôle de systèmes dynamiques et équations aux dérivées partielles, 18h (E. Cancès)
• Statistiques numériques et analyse de données, 26h (S. Darshan)
• Problèmes d'évolution, 36h (V. Ehrlacher, F. Legoll)
• Projets Modéliser Programmer Simuler (T. Lelièvre)
• Statistics and data sciences, 30h (G. Stoltz)
• Projet de physique statistique et mécanique quantique, 10h (G. Stoltz, L. Vidal)

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

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

At other institutions:

• Modal de Mathématiques Appliquées (MAP473D), 15h, Ecole Polytechnique (T. Lelièvre)
• Aléatoire (MAP361), 40h, Ecole Polytechnique (T. Lelièvre)
• Gestion des incertitudes et analyse de risque (MAP568), 20h, Ecole Polytechnique (T. Lelièvre)
• Théorie spectrale et mécanique quantique, 30h, ENS Paris (S. Perrin-Roussel)
• Modélisation, 40h, ENS Paris (S. Perrin-Roussel)
• Numerical Analysis (in Spring and Fall), $2\times 56$h, NYU Paris (U. Vaes)

9.2.2 Supervision

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

• Zineb Belkacemi, funding CIFRE SANOFI, Deciphering protein function with artificial intelligence, co-supervised by T. Lelièvre and G. Stoltz, defended in July
• Raed Blel, funding UM6P, Model-order reduction methods for stochastic problems, co-supervised by V. Ehrlacher and T. Lelièvre, defended in June
• Rémi Goudey, funding CDSN, Homogenization problems in the presence of defects, supervised by C. Le Bris, defended in October
• Gaspard Kemlin, funding ERC Synergy EMC2, Mathematical and numerical analysis for electronic structures, co-supervised by E. Cancès and A. Levitt, defended in December
• Idrissa Niakh, thèse CIFRE EDF, Reduced basis for variational inequalities, co-supervised by V. Ehrlacher and A. Ern (Inria SERENA), defended in December
• Inass Sekkat, funding UM6P, Large scale Bayesian inference, supervised by G. Stoltz, defended in September

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

• Hichem Belbal, thèse CIFRE EDF, Understanding suspended matter measures in Loire river, since September 2022, supervised by S. Boyaval
• Elisa Beteille, thèse CIFRE EDF, Propagation of Urban Flood waves, since November 2021, supervised by S. Boyaval
• Rutger Biezemans, funding DIM Math Innov (Inria), Difficult multiscale problems and non-intrusive approaches, Ecole des Ponts, since October 2020, co-supervised by C. Le Bris and A. Lozinski (University of Besançon)
• 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
• Jean Cauvin-Vila, funding Ecole des Ponts, Cross-diffusion systems on moving boundary domains, since October 2020, co-supervised by V. Ehrlacher and A. Hayat
• Yonah Conjugo-Taumhas, thèse CIFRE CEA, Reduced basis methods for non-symmetric eigenvalue problems, since October 2020, co-supervised by T. Lelièvre and V. Ehrlacher together with G. Dusson (CNRS Besançon) and F. Madiot (CEA)
• 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)
• Maria Fuente-Ruiz, funding INRIA, Parallel algorithms for tensor methods, since September 2020, co-supervised by V. Ehrlacher and D. Lombardi (Inria COMMEDIA)
• Abbas Kabalan, thèse CIFRE SAFRANTech, Reduced-order models for problems with non-parametric geometrical variations, since November 2022, co-supervised by V. Ehrlacher and F. Casenave (SAFRANTech)
• Albéric Lefort, funding CERMICS-ENPC, Multiscale numerical methods for reaction-diffusion equations and related problems, Ecole des Ponts, since November 2022, co-supervised by F. Legoll and C. Le Bris
• Eloïse Letournel, funding DIM Math Innov (Inria), Finite size effects in electronic structure, École des Ponts, since September 2021, supervised by A. Levitt
• Alfred Kirsch, funding Simons foundation, Mathematical and numerical analysis of interacting electrons models, École des Ponts, since September 2021, co-supervised by E. Cancès and D. Gontier (Paris-Dauphine CEREMADE)
• 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)
• Thomas Pigeon, funding Inria, Combining machine learning and quantum computations to discover new catalytic mechanisms, Université de Lyon, since October 2020, co-supervised by P. Raybaud (IFPEN) and T. Lelièvre, together with G. Stoltz and M. Corral-Vallero (IFPEN)
• 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, thèse IFPEN, Fast simulation of CO2/H2 storage in geological bassins, since November 2021, supervised by S. Boyaval
• Jean-Paul Travert, thèse CIFRE EDF, Data assimilation for flood predictions, since November 2022, supervised by S. Boyaval
• Laurent Vidal, funding ERC Synergy EMC2, Model reduction in physics and quantum chemistry, since February 2021, supervisd by E. Cancès and A. Levitt.

9.2.3 Juries

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

• S. Boyaval, PhD of Omar Mokhtari (“Écoulement de solutions de polymères en milieux poreux: impact des effets viscoélastiques à l’échelle du pore sur les propriétés effectives à l’échelle de Darcy”), defended at INP Toulouse in July
• E. Cancès, PhD of Etienne Polack (“ Development of efficient multiscale methods and extrapolation techniques for multiphysics molecular chemistry”), defended at Sorbonne University in January (chair)
• E. Cancès, PhD of Augustin Blanchet (“De la surface au cœur des étoiles: vers une modélisation unifiée de la matière condensée aux plasmas”), defended at the University of Paris Saclay in February (chair)
• E. Cancès, PhD of Michele Nottoli (“Fast and accurate multi- layer polarizable embedding strategies for the static and dynamic modeling of complex systems”), defended at the University of Pisa in February (referee)
• E. Cancès, PhD of Martin Mrovec (“Mathematical Methods of Modelling Electronic Structure of Large Systems”), defended at the University of Ostrava in April (referee)
• V. Ehrlacher, PhD of Kiran Kollepara ("Low-rank and sparse approximations for contact mechanics"), defended at Nantes University in July (referee)
• V. Ehrlacher, PhD of Philip Edel ("Reduced basis method for parameter-dependent linear equations. Application to time-harmonic problems in electromagnetism and in aeroacoustics."), defended at Sorbonne University in October (referee)
• V. Ehrlacher, PhD of Emilie Bourne ("Non-Uniform Numerical Schemes for the Modelling of Turbulence in the 5D GYSELA Code"), defended at CEA Cadarache in December (referee)
• V. Ehrlacher, PhD of Katharina Eichinger ("Problèmes variationnels pour l’interpolation dans l’espace de Wasserstein"), defended at University Paris-Dauphine in December (referee)
• T. Lelièvre, PhD of Aurélien Enfroy ("Contributions à la conception, l'étude et la mise en œuvre de méthodes de Monte Carlo par chaîne de Markov appliquées à l'inférence bayésienne"), defended at Institut Polytechnique de Paris in July (chair)
• T. Lelièvre, PhD of Fiona Desplats ("Development of a hybrid method coupling deterministic and stochastic neutronic calculations: Applications to heterogeneous PWR and SFR calculations"), defended at Université Grenoble Alpes in October (referee)
• T. Lelièvre, PhD of Loris Felardos ("Data-free Generation of Molecular Configurations with Normalizing Flows"), defended at Université Grenoble Alpes in December (referee)
• A. Levitt, PhD of Jean Cazalis (“Systèmes quantiques non linéaires en dissociation : l'exemple du graphène”), defended at Université Paris Dauphine in July
• G. Stoltz, PhD of Paul Rohrbach ("Multilevel Monte Carlo simulation of soft matter using coarse-grained models"), defended at the University of Cambridge in October (referee)
• G. Stoltz, PhD of Benjamin Stottrup ("Spectral, scattering, and regularity properties related to various functional and differential equations"), defended at Aalborg University in April (referee)
• G. Stoltz, PhD of Lorenzo Campana ("Stochastic modeling of non-spherical particles in turbulence"), defended at Inria Sophia in March (referee)
• G. Stoltz, PhD of Clovis Lapointe ("Modélisation multi-échelles des défauts d’irradiation dans les métaux cubiques centrés"), defended at INSTN Saclay in February (referee)

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

• V. Ehrlacher and T. Lelièvre (referee), HdR of Marie Billaud-Friess ("Contributions for the approximation and model order reduction of partial differential equations"), defended at Nantes University in October

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

• V. Ehrlacher, MCF position at Laboratoire de Mathématiques d'Orsay.
• V. Ehrlacher, member of the 2022 ANR project selection committee CE46 ("Modeling and simulation").
• T. Lelièvre, positions in applied mathematics, Ecole Polytechnique.

International journals

• 8 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, 393-436
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• 9 articleZ.Zineb Belkacemi, P.Paraskevi Gkeka, T.Tony Lelièvre and G.Gabriel Stoltz. Chasing Collective Variables using Autoencoders and biased trajectories.Journal of Chemical Theory and Computation181January 2022, 59-78
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• 10 articleR.Robert Benda, E.Eric Cancès, V.Virginie Ehrlacher and B.Benjamin Stamm. Multi-center decomposition of molecular densities: a mathematical perspective.Journal of Chemical Physics156April 2022, 164107
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• 11 articleE.Etienne Bernard, M.Max Fathi, A.Antoine Levitt and G.Gabriel Stoltz. Hypocoercivity with Schur complements.Annales Henri Lebesgue5May 2022, 523-557
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• 12 articleR. A.Rutger A. 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 Physics2023
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• 13 articleS.Sébastien Boyaval, S.Sofiane Martel and J.Julien Reygner. Finite-Volume approximation of the invariant measure of a viscous stochastic scalar conservation law.IMA Journal of Numerical Analysis423July 2022, 2710-2770
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• 14 articleE.Evan Camrud, D. P.David P. Herzog, G.Gabriel Stoltz and M.Maria Gordina. Weighted $L^2$-contractivity of Langevin dynamics with singular potentials.Nonlinearity352January 2022, 998-1035
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• 15 articleE.Eric Cancès, G.Geneviève Dusson, G.Gaspard Kemlin and A.Antoine Levitt. Practical error bounds for properties in plane-wave electronic structure calculations.SIAM Journal on Scientific Computing445October 2022
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• 16 articleJ. A.Jose Antonio Carrillo, F.Franca Hoffmann, A. M.Andrew M Stuart and U.Urbain Vaes. Consensus Based Sampling.Studies in Applied Mathematics1483January 2022, 1069-1140
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• 17 articleA.Alois Castellano, F.Francois Bottin, J.Johann Bouchet, A.Antoine Levitt and G.Gabriel Stoltz. Ab initio Canonical Sampling based on Variational Inference.Physical Review B10616October 2022, L161110
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• 18 articleL.Ludovic Chamoin and F.Frederic Legoll. An introductory review on a posteriori error estimation in Finite Element computations.SIAM Review2022
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• 19 articleM.-S.Mi-Song Dupuy and A.Antoine Levitt. Finite-size effects in response functions of molecular systems.SMAI Journal of Computational MathematicsDecember 2022
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• 20 articleV.Virginie Ehrlacher, M.Maria Fuente-Ruiz and D.Damiano Lombardi. SoTT: greedy approximation of a tensor as a sum of Tensor Trains.SIAM Journal on Scientific Computing442March 2022
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• 21 articleV.Virginie Ehrlacher, T.Tony Lelièvre and P.Pierre Monmarché. Adaptive force biasing algorithms: new convergence results and tensor approximations of the bias.The Annals of Applied Probability325October 2022
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• 22 articleO.Olga Gorynina, C.Claude Le Bris and F.Frederic Legoll. Mathematical analysis of a coupling method for the practical computation of homogenized coefficients.ESAIM: Control, Optimisation and Calculus of Variations282022, 44
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• 23 articleR.Rémi Goudey. A periodic homogenization problem with defects rare at infinity.Networks & Heterogeneous Media1742022, 547-592
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• 24 articleR.Rémi Goudey. Elliptic homogenization with almost translation-invariant coefficients.Asymptotic AnalysisPre-pressJune 2022, 1-42
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• 25 articleJ.Jérôme Hénin, T.Tony Lelièvre, M.Michael Shirts, O.Omar Valsson and L.Lucie Delemotte. Enhanced sampling methods for molecular dynamics simulations.Living Journal of Computational Molecular Science412022
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• 26 articleM. F.Michael F. Herbst and A.Antoine Levitt. A robust and efficient line search for self-consistent field iterations.Journal of Computational Physics459June 2022
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• 27 articleF.Frédéric Legoll, T.Tony Lelièvre and U.Upanshu Sharma. An adaptive parareal algorithm: application to the simulation of molecular dynamics trajectories.SIAM Journal on Scientific Computing4412022, B146-B176
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• 28 articleF.Frederic Legoll, P.-L.Pierre-Loik Rothé, C.Claude Le Bris and U.Ulrich Hetmaniuk. An MsFEM approach enriched using Legendre polynomials.Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal2022022, 798-834
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• 29 articleT.Tony Lelièvre, D.Dorian Le Peutrec and B.Boris Nectoux. The exit from a metastable state: concentration of the exit point distribution on the low energy saddle points, part 2.Stochastics and Partial Differential Equations: Analysis and Computations1012022, 317–357
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• 30 articleT.Tony Lelièvre, L.Lise Maurin and P.Pierre Monmarché. The Adaptive Biasing Force algorithm with non-conservative forces and related topics.ESAIM: Mathematical Modelling and Numerical Analysis562March 2022, 529-564
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• 31 articleT.Tony Lelièvre, M.Mouad Ramil and J.Julien Reygner. A probabilistic study of the kinetic Fokker-Planck equation in cylindrical domains.Journal of Evolution Equations22382022
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• 32 articleT.Tony Lelièvre, M.Mouad Ramil and J.Julien Reygner. Quasi-stationary distribution for the Langevin process in cylindrical domains, part I: existence, uniqueness and long-time convergence.Stochastic Processes and their Applications144February 2022, 173-201
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• 33 articleG. A.Grigorios A. Pavliotis, A. M.Andrew M. Stuart and U.Urbain Vaes. Derivative-free Bayesian Inversion Using Multiscale Dynamics.SIAM Journal on Applied Dynamical Systems211January 2022, 284-326
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• 34 articleM.Mouad Ramil, T.Tony Lelièvre and J.Julien Reygner. Mathematical foundations for the Parallel Replica algorithm applied to the underdamped Langevin dynamics.MRS CommunicationsJuly 2022
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• 35 articleK.Karno Schwinn, F.Felipe Zapata, A.Antoine Levitt, E.Eric Cancès, E.Eleonora Luppi and J.Julien Toulouse. Photoionization and core resonances from range-separated density-functional theory: General formalism and example of the beryllium atom.Journal of Chemical Physics156222022, 224106
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• 36 articleJ.Julien Toulouse, K.Karno Schwinn, F.Felipe Zapata, A.Antoine Levitt, E.Eric Cancès and E.Eleonora Luppi. Photoionization and core resonances from range-separated time-dependent density-functional theory for open-shell states: Example of the lithium atom.Journal of Chemical Physics157December 2022, 244104
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International peer-reviewed conferences

• 37 inproceedingsI.Idrissa Niakh, A.Alexandre Ern, V.Virginie Ehrlacher and G.Guillaume Drouet. Réduction de modèles pour les inéquations variationnelles.15e colloque national en calcul des structuresHyères-les-Palmiers, FranceMay 2022
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Reports & preprints

• 38 miscY.Yves Achdou and C.Claude Le Bris. Homogenization of some periodic Hamilton-Jacobi equations with defects.November 2022
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• 39 miscB.Beatrice Battisti, T.Tobias Blickhan, G.Guillaume Enchery, V.Virginie Ehrlacher, D.Damiano Lombardi and O.Olga Mula. Wasserstein model reduction approach for parametrized flow problems in porous media.May 2022
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• 40 miscA.Adrien Beguinet, V.Virginie Ehrlacher, R.Roberta Flenghi, M.Maria Fuente-Ruiz, O.Olga Mula and A.Agustin Somacal. Deep learning-based schemes for singularly perturbed convection-diffusion problems.May 2022
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• 41 miscR. 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.December 2022
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• 42 reportR. A.Rutger A. Biezemans, A. B.Arsene Brice Zotsa Ngoufack, C.Chiara Cordier and A.Ayoub Belhachmi. Some solutions for non-homogeneous smoothing.IRMAR, University of Rennes 1May 2022
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• 43 miscS.Sébastien Boyaval. A viscoelastic flow model of maxwell-type with a symmetric-hyperbolic formulation.December 2022
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• 44 miscS.Sébastien Boyaval. About the structural stability of Maxwell fluids: convergence toward elastodynamics.December 2022
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• 45 miscE.Eric Cancès, G.Geneviève Dusson, G.Gaspard Kemlin and L.Laurent Vidal. On basis set optimisation in quantum chemistry.December 2022
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• 46 miscC.Clément Cancès, V.Virginie Ehrlacher and L.Laurent Monasse. Finite Volumes for the Stefan-Maxwell Cross-Diffusion System.2022
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• 47 miscÉ.Éric Cancès, L.Louis Garrigue and D.David Gontier. A simple derivation of moiré-scale continuous models for twisted bilayer graphene.November 2022
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• 48 miscE.Eric Cancès, M.Muhammad Hassan and L.Laurent Vidal. Modified-Operator Method for the Calculation of Band Diagrams of Crystalline Materials.October 2022
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• 49 miscE.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.October 2022
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• 50 miscE.Eric Cancès, G.Gaspard Kemlin and A.Antoine Levitt. A priori error analysis of linear and nonlinear periodic Schrödinger equations with analytic potentials.June 2022
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• 51 miscJ. A.Jose Antonio Carrillo, F.Franca Hoffmann, A. M.Andrew M. Stuart and U.Urbain Vaes. The Ensemble Kalman Filter in the Near-Gaussian Setting.December 2022
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• 52 miscJ.Jean Cauvin-Vila, V.Virginie Ehrlacher and A.Amaury Hayat. Boundary stabilization of one-dimensional cross-diffusion systems in a moving domain: linearized system.January 2023
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• 53 miscJ.Jad Dabaghi, V.Virginie Ehrlacher and C.Christoph Strössner. Computation of the self-diffusion coefficient with low-rank tensor methods: application to the simulation of a cross-diffusion system.April 2022
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• 54 miscJ.Jad Dabaghi, V.Virginie Ehrlacher and C.Christoph Strössner. Tensor approximation of the self-diffusion matrix of tagged particle processes.October 2022
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• 55 miscJ.Jad Dabaghi and V.Virginie Ehrlacher. Structure-preserving reduced order model for parametric cross-diffusion systems.June 2022
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• 56 miscI.Ivan Duchemin, L.Luigi Genovese, E.Eloïse Letournel, A.Antoine Levitt and S.Simon Ruget. Efficient extraction of resonant states in systems with defects.September 2022
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• 57 miscL.Louis Garrigue. Mixed state representability of entropy-density pairs.March 2022
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• 58 miscO.Olga Gorynina, F.Frederic 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.December 2022
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• 59 miscR.Rémi Goudey and C.Claude Le Bris. Linear elliptic homogenization for a class of highly oscillating non-periodic potentials.June 2022
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• 60 miscT.Tony Lelièvre and P.Panos Parpas. Using Witten Laplacians to locate index-1 saddle points.December 2022
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• 61 miscT.Tony Lelièvre, D. L.Dorian Le Peutrec and B.Boris Nectoux. Eyring-Kramers exit rates for the overdamped Langevin dynamics: the case with saddle points on the boundary.July 2022
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• 62 miscT.Tony Lelièvre, M.Mouad Ramil and J.Julien Reygner. Estimation of statistics of transitions and Hill relation for Langevin dynamics.June 2022
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• 63 miscT.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.May 2022
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• 64 miscI.Idrissa Niakh, G.Guillaume Drouet, V.Virginie Ehrlacher and A.Alexandre Ern. Stable model reduction for linear variational inequalities with parameter-dependent constraints.September 2022
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• 65 miscG. A.Grigorios A. Pavliotis, G.Gabriel Stoltz and U.Urbain Vaes. Mobility estimation for Langevin dynamics using control variates.June 2022
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• 66 miscG.Gabriel Stoltz. Error estimates and variance reduction for nonequilibrium stochastic dynamics.November 2022
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• 67 miscM.Masaaki Tokieda, C.Cyril Elouard, A.Alain Sarlette and P.Pierre Rouchon. Complete Positivity Violation in Higher-order Quantum Adiabatic Elimination.November 2022
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