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 2020 DFTK has reached maturity, and has been used in three publications in the group. It has been extended to support a variety of features (MPI, threading, arbitrary precision, response properties, Wannier functions, integration with external tools...)
• URL: http://dftk.org
• Contacts: Antoine Levitt, Michael Herbst

6.1.2 gen.parRep

• Keywords: Molecular simulation, MPI, HPC, C++
• Functional Description:

  gen.parRep is the first publicly available implementation of the Generalized Parallel Replica method (BSD 3-Clause license), targeting frequently encountered metastable biochemical systems, such as conformational equilibria or dissociation of protein-ligand complexes.

  It was shown (hal-01832823) that the resulting C++/MPI implementation exhibits a strong linear scalability, providing up to 70 % of the maximum possible speedup on several hundreds of CPUs.

• Release Contributions: The software was modified for reproducibility in its release 1.2.0, see the notes: https://gitlab.inria.fr/parallel-replica/gen.parRep/tags/v1.2.0
• URL: https://gitlab.inria.fr/parallel-replica/gen.parRep
• Publication: hal-01832823
• Authors: Florent Hedin, Tony Lelièvre
• Contacts: Florent Hedin, Tony Lelièvre
• Participants: Florent Hedin, Tony Lelièvre
• Partner: Ecole des Ponts ParisTech

7.1.1 The ground state problem in density functional theory

The team members have continued their study of algorithms for solving the ground-state problem in Kohn-Sham density functional theory, the long-term goal being the construction of robust and efficient numerical methods with guaranteed error bounds.

In 46, E. Cancès, G. Kemlin and A. Levitt have studied the algebraic structure of the self-consistent problems present in mean-field models such as Hartree-Fock and Kohn-Sham density functional theory. They have showed the local convergence of the damped self-consistent iteration and gradient descent, and have compared explicitly their convergence rates, providing insight into the strengths and weaknesses of different methods.

E. Cancès, M. Herbst and A. Levitt have implemented a posteriori error estimators into the DFTK code, and have demonstrated its use in 26. They have provided fully guaranteed error estimators (including discretization, algebraic and machine arithmetic errors), albeit under simplifying assumptions (analytic pseudopotentials, and neglecting explicit electron-electron). Work is underway to extend this formalism to the more realistic case of nonlinear mean-field models.

M. Herbst and A. Levitt have investigated numerically the convergence of self-consistent iterations in extended systems, and the impact of various preconditioners. As is well-known, homogeneous preconditioners are unable to reflect various chemical environments such as interfaces or surfaces. Using the density of states as a proxy for the local dielectric properties of the medium, they have proposed in 53 a simple yet cheap and efficient preconditioner that does not need to be tuned manually for each system. Numerical results have shown a consistent improvement over state-of-the-art methods in a variety of systems, with particularly good performance on metallic surfaces.

In 14, E. Cancès, G. Dusson (CNRS and University of Besançon), Y. Maday (Sorbonne University), B. Stamm (University of Aachen, Germany), and M. Vorhalík (Inria Paris, project-team Serena) have proven a priori error estimates for the perturbation-based post-processing of the plane-wave approximation of Schrödinger equations introduced and tested numerically in previous works of the same authors. They have considered a Schrödinger operator $=-\frac{1}{2}\Delta +V$ in a cubic box with periodic boundary conditions, with a regular-enough potential $V$. The quantities of interest are, on the one hand, the ground-state energy defined as the sum of the lowest $N$ eigenvalues of $H$, and, on the other hand, the ground-state density matrix, that is the spectral projector on the vector space spanned by the associated eigenvectors. Such a problem is central in first-principle molecular simulation, since it corresponds to the so-called linear subproblem in Kohn-Sham density functional theory (DFT). Interpreting the exact eigenpairs of $H$ as perturbations of the numerical eigenpairs obtained by a variational approximation in a plane-wave (i.e. Fourier) basis, they have computed first-order corrections for the eigenfunctions, which are turned into corrections on the ground-state density matrix. This increases the accuracy by a factor proportional to the inverse of the kinetic energy cutoff of both the ground-state energy and the ground-state density matrix in Hilbert–Schmidt norm at a low computational extra-cost. Indeed, the computation of the corrections only requires the computation of the residual of the solution in a larger planewave basis and two Fast Fourier Transforms per eigenvalue.

7.1.2 Response properties

The team members have focused on justifying rigorously and computing efficiently the response properties of molecules and materials.

In 45, E. Cancès, A. Levitt and S. Siraj-Dine have studied the time-dependent response properties of crystals to a uniform electric field. Their results present in a unified framework various regimes described in the physical and mathematical literature (quantum Hall effect in insulators, ballistic conduction and Bloch oscillations in metals, residual conductivity of graphene), and shed light on their range of validity.

M. Herbst and T. Fransson (Heidelberg University, Germany) have used the recently developed adcc code to study the error of the core-valence separation (CVS) approximation in the context of simulating X-ray absorption spectra 25. Using an iterative post-processing procedure they have managed to undo said approximations on obtained computational results and in this way have managed for the first time to study the error of the approximation within the regime of computational parameters employed in practice. Their results show that in the K-edge region of the X-ray spectra errors are negligible and overall only amount to a scalar shift of the simulated spectrum. This shows that the CVS approximation is therefore safe to use for the calculation of core-excited states not lowering the quality of results required for comparison with experimental X-ray absorption spectra.

7.1.3 Various topics

E. Cancès, R. Coyaud and R. Scott (University of Chicago, USA) have pursued their study of van der Waals interactions. In 43, they have extended a method previously introduced by the first and third authors, to compute more terms in the asymptotic expansion of the van der Waals attraction between two hydrogen atoms. These terms are obtained by solving a set of modified Slater–Kirkwood partial differential equations. The accuracy of the method is demonstrated by numerical simulations and comparison with other methods from the literature. It is also shown that the scattering states of the hydrogen atom, that are the states associated with the continuous spectrum of the Hamiltonian, have a major contribution to the C${}_6$ coefficient of the van der Waals expansion.

In 10, R. Benda and E. Cancès, in collaboration with B. Lebental and G. Zucchi (Ecole Polytechnique) have investigated the interaction of polyfluorene and fluorene/carbazole copolymers, bearing various functional groups and side chains, with small to large diameter carbon nanotubes (CNTs) in vacuo, using variable-charge molecular dynamics simulations based on the reactive force field ReaxFF. It is shown that non-covalent functionalization of nanotubes, driven by $\pi -\pi$ interactions, is effective for all the polymers studied, thanks to their conjugated backbone and regardless of the presence of specific functional groups. Both energetic and geometric adsorption features of these polymer/CNT hybrids is analyzed in detail at the scale of each fluorene or carbazole unit. The force field ReaxFF and its available parameterization used for the simulations are validated, thanks to a benchmark and review on higher-level quantum calculations—for simple $\pi -\pi$ interacting compounds made up of polycyclic aromatic molecules adsorbed on a graphene sheet or bilayer graphene. This methodology proves to be a valuable tool for optimal polymer design for nanotube functionalization at no re-parameterization cost and could be adapted to simulate and assist the design of other types of molecular systems.

C. Le Bris has pursued his long term collaboration with P. Rouchon (Ecole des Mines de Paris and Inria/QUANTIC) on the study of high dimensional Lindblad type equations at play in the modelling of open quantum systems. They have co-supervised the M2 internship of L-A. Sellem, that was focused on the simulation of some simple quantum gates, and has investigated several discretization strategies based upon the choice of suitable basis sets. Following up on the topic of his internship, L-A. Sellem has started in September 2020 his PhD work, under the co-supervision of P. Rouchon and C. Le Bris, at the intersection between the QUANTIC and the MATHERIALS project-teams.

7.2.1 Sampling of the configuration space

Various dynamics are used in computational statistical physics to sample the configurations of the system according to the target probability measure at hand and approximate averages as time averages along one realization. It is important to have a good theoretical understanding of the performance of sampling methods in order to choose the optimal parameters in actual numerical simulations, for instance to have a variance as small as possible for time averages along realizations of certain dynamics. A common dynamics is the so-called Langevin dynamics, which corresponds to a Hamiltonian dynamics perturbed by an Ornstein–Uhlenbeck process on the momenta. The generator associated with this stochastic differential equation is hypoelliptic (at best). Proving the longtime convergence of the semigroup requires dedicated tools, under a general strategy known as hypocoercivity, and usually involves prefactors which are difficult to quantitatively estimate. Bounds on the asymptotic variance on the other hand only require bounds on the resolvent of the generator. An approach to directly obtaining estimates on the resolvent of hypocoercive operators is proposed in 40, using Schur complements, rather than obtaining bounds on the resolvent from an exponential decay of the evolution semigroup. The results can be extended, besides Langevin-like dynamics, to the linear Boltzmann equation (which is also the generator of randomized Hybrid Monte Carlo in molecular dynamics). In particular, the dependence of the resolvent bounds on the parameters of the dynamics and on the dimension is made precise, and the relationships with other hypocoercive approaches are highlighted. This work has been performed while Max Fathi (now at Université de Paris, France) was visiting the MATHERIALS project-team.

Other numerical sampling algorithms were also studied. In 58, T. Lelièvre and G. Stoltz, together with W. Zhang (Zuse Institute Berlin, Germany) have proposed new Markov Chain Monte Carlo (MCMC) algorithms to sample probability distributions on submanifolds. Such target distributions are typically encountered when studying systems with molecular constraints (fixed bond angles or bond lengths), or for free energy computations. The newly proposed method generalizes previous algorithms by allowing the use of set-valued maps in the proposal step of the MCMC algorithms. The motivation for this generalization is that the numerical solvers used to project proposed moves to the submanifold of interest may find several solutions. The unbiasedness of these new algorithms is proven thanks to some carefully enforced reversibility property. The interest of the new MCMC algorithms is illustrated on various numerical examples.

For sampling multimodal and anisotropic distributions in Bayesian inference or statistical physics, T. Lelièvre, G. Robin and G. Stoltz, together with G. Pavliotis (Imperial College, United-Kingdom) are studying a new algorithm based on Langevin dynamics, in the framework of G. Robin’s post-doc funded by Inria through the "Programme de Recherche Exploratoire" on "Molecular Dynamics and Learning". They have proposed a new method extending the Metropolis Adjusted Langevin Algorithm (MALA), by introducing a diffusion function allowing a more efficient exploration of the parameter space. This diffusion function is obtained by optimizing the convergence rate of the sampling algorithm, which boils down to solving a spectral problem. The properties of this optimization problem are studied, and a numerical method is developed to compute the corresponding optimal diffusion. Numerical experiments are also performed to evaluate the gain in terms of convergence speed, in comparison to the original MALA with constant diffusion.

Finally, the team has also pursued its endeavour to study and improve free energy biasing techniques, such as adaptive biasing force or metadynamics. The gist of these techniques is to bias the original metastable dynamics used to sample the target probability measure in the configuration space by an approximation of the free energy along well-chosen reaction coordinates. This approximation is built on the fly, using empirical measures over replicas, or occupations measures over the trajectories of the replicas. In 49, V. Ehrlacher and T. Lelièvre, together with P. Monmarché (Sorbonne Université, France) have introduced a modified version of the Adaptive Biasing Force method, in which the free energy is approximated by a sum of tensor products of one-dimensional functions. This enables to handle a larger number of reaction coordinates than the classical algorithm. It is proven that the algorithm is well-defined, and the long-time convergence of the algorithm is studied. Numerical experiments demonstrate that the method is able to capture correlations between reaction coordinates.

7.2.2 Sampling of dynamical properties and rare events

The theory of quasi-stationary distributions has been used by T. Lelièvre and collaborators over the past years in order to rigorously model the exit event from a metastable state by a jump Markov process, and to study this exit event in the small temperature regime. In 39, T. Lelièvre together with M. Baudel and A. Guyader (Sorbonne Université, France) have illustrated how the Hill relation and the notion of quasi-stationary distribution can be used to analyse the error introduced by many algorithms that are used to compute mean reaction times between metastable states for Markov processes. The theoretical findings are illustrated on various examples demonstrating the sharpness of the error analysis as well as the applicability of the study to elliptic diffusions.

In 55, T. Lelièvre, together with D. Le Peutrec (Université d'Orléans, France) and B. Nectoux (Université Clermont Auvergne, France) have considered the first exit point distribution from a bounded domain of the overdamped Langevin dynamics in the small temperature regime, under rather general assumptions on the potential function. This work is a continuation of a previous paper (part 1, see 19) where the exit point distribution is studied starting from the quasi-stationary distribution. The proofs are based on analytical results on the dependency of the exit point distribution on the initial condition, large deviation techniques and results on the genericity of Morse functions.

Quasi-stationary distributions can be seen as the first eigenvector associated with the generator of the stochastic differential equation at hand, on a domain with Dirichlet boundary conditions (which corresponds to absorbing boundary conditions at the level of the underlying stochastic processes). Many results on the quasi-stationary distribution hold for non degenerate stochastic dynamics, the associated generator of which is elliptic. The case of degenerate dynamics is less clear. In 57, T. Lelièvre and M. Ramil, together with J. Reygner (Ecole des Ponts, France), have generalized well-known results on the probabilistic representation of solutions to parabolic equations on bounded domains to the so-called kinetic Fokker Planck equation on bounded domains in positions, with absorbing boundary conditions. Furthermore, a Harnack inequality and a maximum principle are provided for solutions to this kinetic Fokker-Planck equation, as well as the existence of a smooth transition density for the associated absorbed Langevin dynamics. The continuity of this transition density at the boundary is studied as well as the compactness, in various functional spaces, of the associated semigroup. This work is a cornerstone to prove the consistency of various algorithms used to simulate metastable trajectories of the Langevin dynamics, in particular the Parallel Replica algorithm and the Adaptive Multilevel Splitting method.

7.2.3 Nonequilibrium systems and transport properties

Many systems in computational statistical physics are not at equilibrium, but rather in a stationary state. 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 and U. Vaes, together with G. Pavliotis (Imperial College London, United-Kingdom) have studied in 33 the properties of the mobility for generalized Langevin dynamics. These dynamics can be seen as Langevin dynamics with some colored noise, and recast in the quasi-Markovian setting as a stochastic differential equation with additional degrees of freedom modeling the structure of the noise. A hypocoercive analysis in the $H^1$ framework à la Villani allows to obtain sharp bounds on the resolvent of the generator of the dynamics, in various limiting regimes (overdamped regime, Markovian limit corresponding to Langevin dynamics, underdamped or Hamiltonian limit). These bounds can in turn be used to obtain precise asymptotics on the mobility coefficient in these various limiting regimes. These predictions are confirmed by numerical simulations based on a Galerkin discretization of the generator.

Since joining the project-team in November, U. Vaes has also been working with G. Stoltz on variance reduction methods for the computation of mobility coefficients for Langevin dynamics in the underdamped (or Hamiltonian) limit, for two dimensional systems. There is currently no precise numerical or theoretical understanding of the divergent behavior of the mobility as the friction coefficient vanishes. Strong limitations are due to the large computational cost of numerically converging the estimated value of the diffusion coefficient as the friction decreases. U. Vaes and G. Stoltz are studying the use of variance reduction methods to more efficiently compute the mobility.

Finally, G. Stoltz, together with A. Iacobucci and S. Olla (Université Paris-Dauphine, France) have studied in 54 the macroscopic profiles of temperature and angular momentum in the stationary state of chains of rotors under a thermo-mechanical forcing applied at the boundaries. These profiles are solutions of a system of diffusive partial differential equations with boundary conditions determined by the thermo-mechanical forcing. Instead of expensive Monte Carlo simulations of the underlying microscopic physical system, they have performed extensive numerical simulations based on a finite difference method for the system of partial differential equations describing the macroscopic steady state. They have presented a formal derivation of these stationary equations based on a linear response argument and local equilibrium assumptions. This allows to characterize the regime of parameters leading to uphill diffusion, a situation where the energy flows in the direction of the gradient of temperature; and to identify regions of parameters corresponding to a negative thermal conductivity (i.e. a positive linear response to a gradient of temperature). The agreement with previous results obtained by numerical simulation of the microscopic dynamics confirms the validity of the macroscopic equations which were derived.

7.2.4 Coarse-graining and model reduction

In 24, T. Lelièvre, G. Stoltz and Z. Belkacemi have reviewed how machine learning techniques are used in molecular dynamics to extract valuable information from the enormous amounts of data generated by simulation of complex systems. They provide a review of the goals, benefits, and limitations of machine learning techniques for computational studies on atomistic systems, focusing on the construction of empirical force fields from ab-initio databases and the determination of reaction coordinates for free energy computation and enhanced sampling. This work is co-authored with P. Gkeka (Sanofi, France) A. Farimani (Carnegie Mellon University, USA), M. Ceriotti (EPFL, Switzerland), J. Chodera (Memorial Sloane Kettering Cancer Center, USA), A. Dinner (University of Chicago, USA), A. Ferguson (University of Chicago, USA), J.B. Maillet (CEA, France), H. Minoux (Sanofi, France), C. Peter (University of Konstanz, Germany), F. Pietrucci (Sorbonne Université, France), A. Silveira (Memorial Sloane Kettering Cancer Center, USA), A. Tkatchenko (University of Luxembourg, Luxembourg), Z. Trstanova (University of Edinburgh, United Kingdom) and R. Wiewiora (Memorial Sloane Kettering Cancer Center, USA).

7.3.1 Deterministic non-periodic systems

In homogenization theory, members of the project-team have pursued their ongoing systematic study of perturbations of periodic problems (by local and nonlocal defects). This has been done in the context of the PhD work of R. Goudey, who has studied perturbations of periodic systems that do not decay at infinity but become increasingly rare. Another interesting direction studied is that of general algebras for homogenization, introduced by X. Blanc, C. Le Bris and P.-L. Lions two decades ago. Some very illustrative one-dimensional cases have been explored, as well as the case of equations simpler than the diffusion equation, namely elliptic equations with highly oscillatory potentials. A monograph that summarizes the developments completed in the past years, and that combines them with the basic elements of homogenization theory, is currently in writing by C. Le Bris and X. Blanc (half-time on leave at Inria since September 2019 and until August 2021 from Université de Paris).

Also in the context of homogenization theory, O. Gorynina, C. Le Bris and F. Legoll have pursued their work on the question of how to determine the homogenized coefficient of heterogeneous media without explicitly performing an homogenization approach. This work is a follow-up on earlier works over the years by C. Le Bris and F. Legoll in collaboration with K. Li and next S. Lemaire. The 18 months post-doc (January 2019-July 2020) of O. Gorynina has allowed to complete the mathematical study and the numerical improvement of a computational approach initially introduced by R. Cottereau (CNRS Marseille). This approach combines, in the Arlequin framework, the original fine-scale description of the medium (modelled by an oscillatory coefficient) with an effective description (modelled by a constant coefficient) and optimizes upon the coefficient of the effective medium to best fit the response of a purely homogeneous medium. In the limit of asymptotically infinitely fine structures, the approach yields the value of the homogenized coefficient. Various computational improvements have been suggested. They have been collected in a publication currently submitted 52. The theoretical study of the approach is performed in a contribution soon to be submitted.

7.3.2 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). During the year, several research tracks have been pursued in this general direction.

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, with carefully chosen right-hand sides and boundary conditions. The initially proposed version of MsFEM uses as basis functions the solutions to these local problems, posed on each mesh element, with null right-hand sides and with the coarse P1 elements as Dirichlet boundary conditions. Various improvements have next been proposed, such as the oversampling variant, which solves local problems on larger domains and restricts their solutions to the considered element. Despite the fact that the approach has been proposed many years ago, it turns out that not all specific settings are covered by the numerical analyses existing in the literature. Together with C. Le Bris, F. Legoll and A. Lozinski, R. Biezemans (who joined the project-team this year during his M2 internship and is now starting his PhD) has extended the analysis of MsFEM to the case of rectangular meshes and that of coefficients that are not necessarily Hölder continuous. An ongoing research effort is devoted to further improving the error estimates. In addition, R. Biezemans, C. Le Bris, F. Legoll and A. Lozinski have undertaken the precise mathematical study of an existing variant of MsFEM, where the standard basis set is enriched using a singular value decomposition of an operator defined for each edge of the coarse mesh, and the range of which is essentially given by the trace on that edge of the oscillatory solutions (for, say, various right-hand sides). This variant has been recently proposed in the literature, and its appealing numerical performances (basis functions that are locally supported, lack of any resonance error in the regime $H\approx \epsilon$, ...) have motivated the team to better understand it from a theoretical perspective. A convergence analysis of this new variant, together with comprehensive numerical tests, are currently in progress.

Second, within the PhD thesis of A. Lesage, Multiscale Finite Element Methods have been developped for thin heterogeneous plates. The fact that one of the dimension of the domain of interest scales as the typical size of the heterogeneities within the material induces theoretical and practical difficulties that have to be carefully taken into account. In a first step, V. Ehrlacher, F. Legoll and A. Lesage, in collaboration with A. Lebée (ENPC, France), have established strong convergence results (between the oscillatory solution and an adequate two-scale expansion) for problems posed on thin heterogeneous plates. After having considered the case of a diffusion equation, the case of linear elasticity has been studied, both in the so-called membrane case (that is, when the loading lies in the in-plane directions) and in the much more challenging case of bending (that is, when the loading is perpendicular to the in-plane directions). In a second step, several MsFEM variants have been proposed and compared numerically. A priori error bounds have been established, assessing the performances of the approaches, and extensive numerical tests have been performed. All these results are presented in two manuscripts in preparation.

7.4.1 Complex fluids

In 2020, S. Boyaval has pursued his research to improve mathematical models of non-Newtonian fluids for application to large time-space domains, when waves propagate at finite speed. Regarding fluid viscoelasticity (of Maxwell-type), it turns out this can be modeled within the standard framework of classical continuum mechanics, with a symmetric-hyperbolic system of conservation laws unifying the elastodynamics of hyperelastic materials and Newtonian fluid dynamics 42. To that aim, the viscoelastic system is an extension of elastodynamics using a time-dependent structure tensor that produces friction through relaxation. The new framework is versatile and should be consolidated for extension to other material imperfection beyong viscoelasticity.

7.4.2 Cross-diffusion systems

New results were obtained in 2020 by V. Ehrlacher on the analysis and numerical analysis of cross-diffusion systems, in the context of the ANR project COMODO.

In a joint work 44 with C. Cancès (Inria RAPSODI) and L. Monasse (Inria COFFEE), a finite volume numerical scheme for the so-called Maxwell-Stefan model has been developed. The Maxwell-Stefan model is a cross-diffusion system used to describe the evolution of a mixture of gas, and is used in particular in biomedical applications. The numerical method developed is provably convergent and preserves on the discrete level the main mathematical properties of the continuous system, including mass conservation, non-negativity of the solutions and an entropy-entropy dissipation inequality which accunts for the long-time behaviour of the solutions of the system.

In a joint work with J-F. Pietschmann and G. Marino (Chemnitz, Germany) 50, V. Ehrlacher has proved the existence of weak solutions to a multi-species cross-diffusion degenerate Cahn-Hilliard system, including phase separation effects of one species with respect to the other species composing the mixture. The proof of the existence of weak solutions to such a model requires the development of new analysis tools, in particular due to the degenerate nature of the resulting model.

7.4.3 Tensor methods for high-dimensional problems

In a joint work with D. Lombardi and M. Fuente-Ruiz (Inria COMMEDIA), V. Ehrlacher has developed a new algorithm for the computation of a canonical polyadic approximation of a high-order tensor  48. The new method is much more efficient and robust that the standard Alternating Least Square method and gave very encouraging results especially for tensors with high order.

10.4.1 Internal or external Inria responsibilities

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

10.4.2 Internal actions

• C. Le Bris has organized a research day for the students at École des Ponts on October 21st.

International journals

• 8 article AurélienA. Alfonsi, RafaëlR. Coyaud, VirginieV. Ehrlacher and DamianoD. Lombardi. Approximation of Optimal Transport problems with marginal moments constraints Mathematics of Computation 2020
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• 9 article AthmaneA. Bakhta, VirginieV. Ehrlacher and DavidD. Gontier. Numerical reconstruction of the first band(s) in an inverse Hill's problem ESAIM: Control, Optimisation and Calculus of Variations September 2020
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• 10 articleRobertR. Benda, GaëlG. Zucchi, EricE. Cancès and BérengèreB. Lebental. Insights into the $$ - $$ interaction driven non-covalent functionalization of carbon nanotubes of various diameters by conjugated fluorene and carbazole copolymers'Journal of Chemical Physics1526February 2020, 064708
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• 11 articleXavierX. Blanc, MarcM. Josien and ClaudeC. Le Bris. Precised approximations in elliptic homogenization beyond the periodic settingAsymptotic Analysis1162January 2020, 93-137
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• 12 articleEricE. Cancès, Ling-LingL.-L. Cao and GabrielG. Stoltz. A reduced Hartree–Fock model of slice-like defects in the Fermi seaNonlinearity331January 2020, 156-195
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• 13 articleEricE. Cancès, GenevièveG. Dusson, YvonY. Maday, BenjaminB. Stamm and MartinM. Vohralík. Guaranteed a posteriori bounds for eigenvalues and eigenvectors: multiplicities and clustersMathematics of Computation89326July 2020, 2563-2611
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• 14 article EricE. Cancès, GenevièveG. Dusson, YvonY. Maday, BenjaminB. Stamm and MartinM. Vohralík. Post-processing of the planewave approximation of Schrödinger equations. Part I: linear operators IMA Journal of Numerical Analysis September 2020
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• 15 articleEricE. Cancès, VirginieV. Ehrlacher, DavidD. Gontier, AntoineA. Levitt and DamianoD. Lombardi. Numerical quadrature in the Brillouin zone for periodic Schrödinger operatorsNumerische Mathematik144January 2020, 479-526
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• 16 articleEricE. Cancès, VirginieV. Ehrlacher, FrédéricF. Legoll, BenjaminB. Stamm and ShuyangS. Xiang. An embedded corrector problem for homogenization. Part I: TheoryMultiscale Modeling and Simulation: A SIAM Interdisciplinary Journal1832020, 1179-1209
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• 17 articleEricE. Cancès, VirginieV. Ehrlacher, FrédéricF. Legoll, BenjaminB. Stamm and ShuyangS. Xiang. An embedded corrector problem for homogenization. Part II: Algorithms and discretizationJournal of Computational Physics4072020, 109254
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• 18 articleDjalilD. Chafaï, GrégoireG. Ferré and GabrielG. Stoltz. Coulomb gases under constraint: some theoretical and numerical resultsSIAM Journal on Mathematical Analysis531January 2021, 181-220
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• 19 articleGiacomoG. Di Gesù, TonyT. Lelièvre, DorianD. Le Peutrec and BorisB. Nectoux. The exit from a metastable state: concentration of the exit point distribution on the low energy saddle pointsJournal de Mathématiques Pures et Appliquées138June 2020, 242-306
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• 20 article VirginieV. Ehrlacher, LauraL. Grigori, DamianoD. Lombardi and HaoH. Song. Adaptive hierarchical subtensor partitioning for tensor compression SIAM Journal on Scientific Computing 2021
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• 21 article VirginieV. Ehrlacher, DamianoD. Lombardi, OlgaO. Mula and François-XavierF.-X. Vialard. Nonlinear model reduction on metric spaces. Application to one-dimensional conservative PDEs in Wasserstein spaces ESAIM: Mathematical Modelling and Numerical Analysis 2020
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• 22 article GrégoireG. Ferré, MathiasM. Rousset and GabrielG. Stoltz. More on the long time stability of Feynman-Kac semigroups Stochastics and Partial Differential Equations: Analysis and Computations 2020
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• 23 articleGrégoireG. Ferré and GabrielG. Stoltz. Large deviations of empirical measures of diffusions in weighted topologiesElectronic Journal of Probability25October 2020, 121
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• 24 article ParaskeviP. Gkeka, GabrielG. Stoltz, Amir BaratiA. Farimani, ZinebZ. Belkacemi, MicheleM. Ceriotti, JohnJ. Chodera, Aaron R.A. Dinner, AndrewA. Ferguson, Jean-BernardJ.-B. Maillet, HervéH. Minoux, ChristineC. Peter, FabioF. Pietrucci, AnaA. Silveira, AlexandreA. Tkatchenko, ZofiaZ. Trstanova, RafalR. Wiewiora and TonyT. Lelièvre. Machine learning force fields and coarse-grained variables in molecular dynamics: application to materials and biological systems Journal of Chemical Theory and Computation July 2020
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• 25 article Michael F.M. Herbst and ThomasT. Fransson. Quantifying the error of the core-valence separation approximation Journal of Chemical Physics August 2020
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• 26 articleMichael F.M. Herbst, AntoineA. Levitt and EricE. Cancès. A posteriori error estimation for the non-self-consistent Kohn-Sham equationsFaraday Discussions224June 2020, 227-246
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• 27 article Michael F.M. Herbst, MaximilianM. Scheurer, ThomasT. Fransson, Dirk R.D. Rehn and AndreasA. Dreuw. adcc: A versatile toolkit for rapid development of algebraic-diagrammatic construction methods Wiley Interdisciplinary Reviews: Computational Molecular Science January 2020
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• 28 articleThomas J.T. Hudson, FrédéricF. Legoll and TonyT. Lelièvre. Stochastic homogenization of a scalar viscoelastic model exhibiting stress-strain hysteresisESAIM: Mathematical Modelling and Numerical Analysis543May 2020, 879-928
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• 29 articleFrédéricF. Legoll, TonyT. Lelièvre, KeithK. Myerscough and GiovanniG. Samaey. Parareal computation of stochastic differential equations with time-scale separation: a numerical convergence studyComputing and Visualization in Science232020, 9
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• 30 articleBenedictB. Leimkuhler, MatthiasM. Sachs and GabrielG. Stoltz. Hypocoercivity properties of adaptive Langevin dynamicsSIAM Journal on Applied Mathematics803May 2020, 1197–1222
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• 31 articleTonyT. Lelièvre, GiovanniG. Samaey and PrzemysławP. Zieliński. Analysis of a micro-macro acceleration method with minimum relative entropy moment matchingStochastic Processes and their Applications1306June 2020, 3753-3801
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• 32 articleAntoineA. Levitt. Screening in the finite-temperature reduced Hartree-Fock modelArchive for Rational Mechanics and Analysis2382July 2020, 901-927
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• 33 articleGrigorios A.G. Pavliotis, GabrielG. Stoltz and UrbainU. Vaes. Scaling limits for the generalized Langevin equationJournal of Nonlinear Science31January 2021, 8
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• 34 article ZofiaZ. Trstanova, BenB. Leimkuhler and TonyT. Lelièvre. Local and Global Perspectives on Diffusion Maps in the Analysis of Molecular Systems Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 476 2233 January 2020
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• 35 article TingT. Wang, GabrielG. Stoltz and PetrP. Plechac. Convergence of the likelihood ratio method for linear response of non-equilibrium stationary states ESAIM: Mathematical Modelling and Numerical Analysis 2020
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International peer-reviewed conferences

• 36 inproceedings LoucasL. Pillaud-Vivien, FrancisF. Bach, TonyT. Lelièvre, AlessandroA. Rudi and GabrielG. Stoltz. Statistical Estimation of the Poincaré constant and Application to Sampling Multimodal Distributions Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics AISTATS 2020 : 23rd International Conference on Artificial Intelligence and Statistics Palermo / Virtual, Italy August 2020
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Doctoral dissertations and habilitation theses

• 37 thesis SamiS. Siraj-Dine. Dynamics of electrons in 2D materials Université Paris-Est; Inria December 2020
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Reports & preprints

• 38 misc AurélienA. Alfonsi, RafaëlR. Coyaud and VirginieV. Ehrlacher. Constrained overdamped Langevin dynamics for symmetric multimarginal optimal transportation February 2021
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• 39 misc ManonM. Baudel, ArnaudA. Guyader and TonyT. Lelièvre. On the Hill relation and the mean reaction time for metastable processes August 2020
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• 40 misc EtienneE. Bernard, MaxM. Fathi, AntoineA. Levitt and GabrielG. Stoltz. Hypocoercivity with Schur complements December 2020
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• 41 misc SébastienS. Boyaval, SofianeS. Martel and JulienJ. Reygner. Finite-Volume approximation of the invariant measure of a viscous stochastic scalar conservation law January 2021
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• 42 misc SébastienS. Boyaval. Viscoelastic flows of Maxwell fluids with conservation laws July 2020
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• 43 misc EricE. Cancès, RafaëlR. Coyaud and L. RidgwayL. Scott. Van der Waals interactions between two hydrogen atoms: The next orders July 2020
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• 44 misc ClémentC. Cancès, VirginieV. Ehrlacher and LaurentL. Monasse. Finite Volumes for the Stefan-Maxwell Cross-Diffusion System July 2020
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• 45 misc EricE. Cancès, ClotildeC. Fermanian Kammerer, AntoineA. Levitt and SamiS. Siraj-Dine. Coherent electronic transport in periodic crystals November 2020
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• 46 misc EricE. Cancès, GaspardG. Kemlin and AntoineA. Levitt. Convergence analysis of direct minimization and self-consistent iterations October 2020
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• 47 misc Mi-SongM.-S. Dupuy and AntoineA. Levitt. Finite-size effects in response functions of molecular systems February 2021
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• 48 misc VirginieV. Ehrlacher, MariaM. Fuente-Ruiz and DamianoD. Lombardi. CP-TT: using TT-SVD to greedily construct a Canonical Polyadic tensor approximation November 2020
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• 49 misc VirginieV. Ehrlacher, TonyT. Lelièvre and PierreP. Monmarché. Adaptive force biasing algorithms: new convergence results and tensor approximations of the bias July 2020
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• 50 misc VirginieV. Ehrlacher, GretaG. Marino and Jan-FrederikJ.-F. Pietschmann. Existence of weak solutions to a cross-diffusion Cahn-Hilliard type system July 2020
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• 51 misc SolenneS. Gaucher, OlgaO. Klopp and GenevièveG. Robin. Outliers Detection in Networks with Missing Links November 2020
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• 52 misc OlgaO. Gorynina, ClaudeC. Le Bris and FrédéricF. Legoll. Some remarks on a coupling method for the practical computation of homogenized coefficients May 2020
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• 53 misc Michael F.M. Herbst and AntoineA. Levitt. Black-box inhomogeneous preconditioning for self-consistent field iterations in density functional theory September 2020
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• 54 misc AlessandraA. Iacobucci, StefanoS. Olla and GabrielG. Stoltz. Thermo-mechanical transport in rotor chains July 2020
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• 55 misc TonyT. Lelièvre, DorianD. Le Peutrec and BorisB. Nectoux. The exit from a metastable state: concentration of the exit point distribution on the low energy saddle points, part 2 December 2020
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• 56 misc TonyT. Lelièvre, LiseL. Maurin and PierreP. Monmarché. The Adaptive Biasing Force algorithm with non-conservative forces and related topics February 2021
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• 57 misc TonyT. Lelièvre, MouadM. Ramil and JulienJ. Reygner. A probabilistic study of the kinetic Fokker-Planck equation in cylindrical domains October 2020
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• 58 misc TonyT. Lelièvre, GabrielG. Stoltz and WeiW. Zhang. Multiple projection MCMC algorithms on submanifolds March 2020
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• 59 misc MouadM. Ramil. Quasi-stationary distribution for the Langevin process in cylindrical domains, part II: overdamped limit March 2021
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