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329 results on '"MATHematics for MatERIALS (MATHERIALS)"'

1. Numerical quadrature in the Brillouin zone for periodic Schrödinger operators

Author

David Gontier, Antoine Levitt, Eric Cancès, Damiano Lombardi, Virginie Ehrlacher, Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC), MATHematics for MatERIALS (MATHERIALS), École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Université Paris sciences et lettres (PSL), Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Numerical simulation of biological flows (REO), Sorbonne Université (SU)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique ( CERMICS ), École des Ponts ParisTech ( ENPC ), MATHematics for MatERIALS ( MATHERIALS ), École des Ponts ParisTech ( ENPC ) -École des Ponts ParisTech ( ENPC ) -Inria de Paris, Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National de Recherche en Informatique et en Automatique ( Inria ), Numerical simulation of biological flows ( REO ), Laboratoire Jacques-Louis Lions ( LJLL ), Université Pierre et Marie Curie - Paris 6 ( UPMC ) -Université Paris Diderot - Paris 7 ( UPD7 ) -Centre National de la Recherche Scientifique ( CNRS ) -Université Pierre et Marie Curie - Paris 6 ( UPMC ) -Université Paris Diderot - Paris 7 ( UPD7 ) -Centre National de la Recherche Scientifique ( CNRS ) -Inria de Paris, Université Pierre et Marie Curie - Paris 6 ( UPMC ) -Université Paris Diderot - Paris 7 ( UPD7 ) -Centre National de la Recherche Scientifique ( CNRS ), PSL Research University (PSL), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), and École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC)

Subjects
Computation, [ PHYS.COND.CM-MS ] Physics [physics]/Condensed Matter [cond-mat]/Materials Science [cond-mat.mtrl-sci], FOS: Physical sciences, 010103 numerical & computational mathematics, Classification of discontinuities, 01 natural sciences, symbols.namesake, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Mathematics - Numerical Analysis, 0101 mathematics, Mathematics, Condensed Matter - Materials Science, Applied Mathematics, Numerical analysis, Mathematical analysis, Materials Science (cond-mat.mtrl-sci), [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA], Numerical Analysis (math.NA), Numerical integration, Quadrature (mathematics), 010101 applied mathematics, Brillouin zone, Computational Mathematics, symbols, [PHYS.COND.CM-MS]Physics [physics]/Condensed Matter [cond-mat]/Materials Science [cond-mat.mtrl-sci], Ground state, Schrödinger's cat, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

International audience; As a consequence of Bloch's theorem, the numerical computation of the fermionic ground state density matrices and energies of periodic Schrodinger operators involves integrals over the Brillouin zone. These integrals are difficult to compute numerically in metals due to discontinuities in the integrand. We perform an error analysis of several widely-used quadrature rules and smearing methods for Brillouin zone integration. We precisely identify the assumptions implicit in these methods and rigorously prove error bounds. Numerical results for two-dimensional periodic systems are also provided. Our results shed light on the properties of these numerical schemes, and provide guidance as to the appropriate choice of numerical parameters.

Published
2020
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2. On correctors for linear elliptic homogenization in the presence of local defects

Author

C. Le Bris, Xavier Blanc, Pierre-Louis Lions, Sorbonne Université (SU), Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC), MATHematics for MatERIALS (MATHERIALS), École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), Collège de France - Chaire Équations aux dérivées partielles et applications, Collège de France (CdF (institution)), Laboratoire Jacques-Louis Lions ( LJLL ), Université Pierre et Marie Curie - Paris 6 ( UPMC ) -Université Paris Diderot - Paris 7 ( UPD7 ) -Centre National de la Recherche Scientifique ( CNRS ), Sorbonne Universités, Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique ( CERMICS ), École des Ponts ParisTech ( ENPC ), MATHematics for MatERIALS ( MATHERIALS ), École des Ponts ParisTech ( ENPC ) -École des Ponts ParisTech ( ENPC ) -Inria de Paris, Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National de Recherche en Informatique et en Automatique ( Inria ), Collège de France ( CdF ), CEntre de REcherches en MAthématiques de la DEcision ( CEREMADE ), Université Paris-Dauphine-Centre National de la Recherche Scientifique ( CNRS ), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), Chaire Équations aux dérivées partielles et applications, Collège de France (CDF), Collège de France (CdF), and Université Paris Dauphine-PSL-Centre National de la Recherche Scientifique (CNRS)

Subjects
Homogenization, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Computer Science::Computational Complexity, 01 natural sciences, Homogenization (chemistry), 010101 applied mathematics, Elliptic curve, Mathematics - Analysis of PDEs, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], AMS subject classification: 35J15, 35J70, 35B27, 74Q15, 76M50, FOS: Mathematics, Lp estimates, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Defects, 0101 mathematics, Elliptic PDE, Analysis, Analysis of PDEs (math.AP), Mathematics
Abstract

We consider the corrector equation associated, in homogenization theory , to a linear second-order elliptic equation in divergence form --$\partial$i(aij$\partial$ju) = f , when the diffusion coefficient is a locally perturbed periodic coefficient. The question under study is the existence (and uniqueness) of the corrector, strictly sublinear at infinity, with gradient in L r if the local perturbation is itself L r , r < +$\infty$. The present work follows up on our works [7, 8, 9], providing an alternative, more general and versatile approach , based on an a priori estimate, for this well-posedness result. Equations in non-divergence form such as --aij$\partial$iju = f are also considered, along with various extensions. The case of general advection-diffusion equations --aij$\partial$iju + bj$\partial$ju = f is postponed until our future work [10]. An appendix contains a corrigendum to our earlier publication [9]., Communications in Partial Differential Equations, Taylor \& Francis, A Para{\^i}tre

Published
2018

3. Central Limit Theorem for stationary Fleming–Viot particle systems in finite spaces

Author

Loucas Pillaud-Vivien, Tony Lelièvre, Julien Reygner, Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC), MATHematics for MatERIALS (MATHERIALS), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC), Statistical Machine Learning and Parsimony (SIERRA), Département d'informatique de l'École normale supérieure (DI-ENS), École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Centre National de la Recherche Scientifique (CNRS)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria), ANR-17-CE40-0030,EFI,Entropie, flots, inégalités(2017), European Project: 614492,EC:FP7:ERC,ERC-2013-CoG,MSMATH(2014), Département d'informatique - ENS Paris (DI-ENS), Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche en Informatique et en Automatique (Inria)-École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche en Informatique et en Automatique (Inria)-École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Inria de Paris, École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), École normale supérieure - Paris (ENS-PSL), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Paris (ENS-PSL), Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique ( CERMICS ), École des Ponts ParisTech ( ENPC ), MATHematics for MatERIALS ( MATHERIALS ), Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique ( CERMICS ), École des Ponts ParisTech ( ENPC ) -École des Ponts ParisTech ( ENPC ), Statistical Machine Learning and Parsimony ( SIERRA ), Département d'informatique de l'École normale supérieure ( DI-ENS ), École normale supérieure - Paris ( ENS Paris ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Centre National de la Recherche Scientifique ( CNRS ) -École normale supérieure - Paris ( ENS Paris ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Centre National de la Recherche Scientifique ( CNRS ) -Centre National de la Recherche Scientifique ( CNRS ) -Inria de Paris, Institut National de Recherche en Informatique et en Automatique ( Inria ), ANR-17-CE40-0030,EFI,Entropy, flows, inequalities, and European Project : 614492,EC:FP7:ERC,ERC-2013-CoG,MSMATH ( 2014 )

Subjects
Statistics and Probability, Pure mathematics, Stationary distribution, Distribution (number theory), Markov chain, 010102 general mathematics, Probability (math.PR), Space (mathematics), 01 natural sciences, Fleming–Viot particle system, 1991 Mathematics Subject Classification. 60F05, 60J27, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 010104 statistics & probability, Mathematics::Probability, Law of large numbers, FOS: Mathematics, Irreducibility, Infinitesimal generator, 0101 mathematics, [ MATH.MATH-PR ] Mathematics [math]/Probability [math.PR], Mathematics - Probability, Central Limit Theorem, Mathematics, Central limit theorem
Abstract

International audience We consider the Fleming–Viot particle system associated with a continuous-time Markov chain in a finite space. Assuming irreducibility, it is known that the particle system possesses a unique stationary distribution, under which its empirical measure converges to the quasistationary distribution of the Markov chain. We complement this Law of Large Numbers with a Central Limit Theorem. Our proof essentially relies on elementary computations on the infinitesimal generator of the Fleming–Viot particle system, and involves the so-called π-return process in the expression of the asymptotic variance. Our work can be seen as an infinite-time version, in the setting of finite space Markov chains, of results by Del Moral and Miclo [ESAIM: Probab. Statist., 2003] and Cérou, Delyon, Guyader and Rousset [arXiv:1611.00515, arXiv:1709.06771].

Published
2018

4. Reduced basis approximation and a posteriori error bounds for 4D-Var data assimilation

Author

Mark Kärcher, Sébastien Boyaval, Karen Veroy, Martin A. Grepl, Rheinisch-Westfälische Technische Hochschule Aachen University (RWTH), Laboratoire d'Hydraulique Saint-Venant / Saint-Venant laboratory for Hydraulics (LHSV), École des Ponts ParisTech (ENPC)-Centre d'Etudes et d'Expertise sur les Risques, l'Environnement, la Mobilité et l'Aménagement (Cerema)-EDF R&D (EDF R&D), EDF (EDF)-EDF (EDF), MATHematics for MatERIALS (MATHERIALS), Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Rheinisch-Westfälische Technische Hochschule Aachen ( RWTH ), Laboratoire d'Hydraulique Saint-Venant / Saint-Venant laboratory for Hydraulics ( Saint-Venant ), École des Ponts ParisTech ( ENPC ) -Centre d'Etudes et d'Expertise sur les Risques, l'Environnement, la Mobilité et l'Aménagement ( Cerema ) -EDF R&D ( EDF R&D ), EDF ( EDF ) -EDF ( EDF ), MATHematics for MatERIALS ( MATHERIALS ), Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique ( CERMICS ), École des Ponts ParisTech ( ENPC ) -École des Ponts ParisTech ( ENPC ) -Inria de Paris, Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National de Recherche en Informatique et en Automatique ( Inria ), Rheinisch-Westfälische Technische Hochschule Aachen (RWTH), Laboratoire d'Hydraulique Saint-Venant / Saint-Venant laboratory for Hydraulics (Saint-Venant), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), and École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC)

Subjects
Reduced-order models, Control and Optimization, Optimization problem, 010504 meteorology & atmospheric sciences, Aerospace Engineering, Weak-constraint, 010103 numerical & computational mathematics, 01 natural sciences, Strong-constraint, Data assimilation, FOS: Mathematics, Parameter estimation, Initial value problem, Applied mathematics, Variational data assimilation, 0101 mathematics, Electrical and Electronic Engineering, Mathematics - Optimization and Control, 0105 earth and related environmental sciences, Civil and Structural Engineering, Mathematics, 65K10, Basis (linear algebra), Estimation theory, Mechanical Engineering, Function (mathematics), [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA], A posteriori error estimation, Optimization and Control (math.OC), 4D-Var, A priori and a posteriori, Errors-in-variables models, PDE-constrained optimization, Reduced basis method, Software, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

International audience; We propose a certified reduced basis approach for the strong-and weak-constraint four-dimensional variational (4D-Var) data assimilation problem for a parametrized PDE model. While the standard strong-constraint 4D-Var approach uses the given observational data to estimate only the unknown initial condition of the model, the weak-constraint 4D-Var formulation additionally provides an estimate for the model error and thus can deal with imperfect models. Since the model error is a distributed function in both space and time, the 4D-Var formulation leads to a large-scale optimization problem for every given parameter instance of the PDE model. To solve the problem efficiently, various reduced order approaches have therefore been proposed in the recent past. Here, we employ the reduced basis method to generate reduced order approximations for the state, adjoint, initial condition, and model error. Our main contribution is the development of efficiently computable a posteriori upper bounds for the error of the reduced basis approximation with respect to the underlying high-dimensional 4D-Var problem. Numerical results are conducted to test the validity of our approach.

Published
2018

5. Pathwise estimates for effective dynamics: the case of nonlinear vectorial reaction coordinates

Author

Tony Lelièvre, Wei Zhang, Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique ( CERMICS ), École des Ponts ParisTech ( ENPC ), MATHematics for MatERIALS ( MATHERIALS ), École des Ponts ParisTech ( ENPC ) -École des Ponts ParisTech ( ENPC ) -Inria de Paris, Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National de Recherche en Informatique et en Automatique ( Inria ), Zuse Institute Berlin ( ZIB ), European Project : 614492,EC:FP7:ERC,ERC-2013-CoG,MSMATH ( 2014 ), Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC), MATHematics for MatERIALS (MATHERIALS), École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Zuse Institute Berlin (ZIB), The first author's research was supported by the European Research Council under the European Union's Seventh Framework Programme (FP/2007-2013)/ERC grant agreement 614492. The second author's research was supported by the Einstein Foundation Berlin of the Einstein Center for Mathematics (ECMath) through project CH21, European Project: 614492,EC:FP7:ERC,ERC-2013-CoG,MSMATH(2014), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), and École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC)

Subjects
Work (thermodynamics), Reaction coordinate, General Physics and Astronomy, Conditional expectation, 01 natural sciences, Continuation, Approximation error, AMS Subject Classifications: 60J60, 60H10, 82C31, Pathwise estimates, effective dynamics, FOS: Mathematics, Diffusion process, Applied mathematics, 0101 mathematics, Mathematics, 60J60, 60H10, 82C31, Stochastic process, Ecological Modeling, 010102 general mathematics, Probability (math.PR), reaction coordinate, General Chemistry, [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA], Computer Science Applications, 010101 applied mathematics, Nonlinear system, Modeling and Simulation, Effective dynamics, 60J60, 60H10, 82C31 [AMS Subject Classifications], pathwise estimates, time scale separation, Time scale separation, diffusion process, Mathematics - Probability, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

International audience Effective dynamics using conditional expectation was proposed in [F. Legoll and T. Lelièvre, Nonlinearity, 23 (2010), pp. 2131--2163] to approximate the essential dynamics of high-dimensional diffusion processes along a given reaction coordinate. The approximation error of the effective dynamics when it is used to approximate the behavior of the original dynamics has been considered in recent years. As a continuation of the previous work [F. Legoll, T. Lelièvre, and S. Olla, Stochastic Process. Appl., 127 (2017), pp. 2841--2863], in this paper we obtain pathwise estimates of effective dynamics for both nonlinear and vector-valued reaction coordinate functions.

Published
2018

6. Boundary stabilization of one-dimensional cross-diffusion systems in a moving domain: Linearized system

Author

Jean Cauvin-Vila, Virginie Ehrlacher, Amaury Hayat, Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC), MATHematics for MatERIALS (MATHERIALS), École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), The authors acknowledge support from the ANR project COMODO (ANR-19-CE46-0002) which funds the Ph.D of Jean Cauvin-Vila., and ANR-19-CE46-0002,COMODO,Systèmes de diffusion croisée sur des domaines en mouvement(2019)

Subjects
Feedback stabilization, Mathematics - Analysis of PDEs, Boundary control, Backstepping, Applied Mathematics, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Parabolic PDEs, Cross-diffusion systems, Exponential stability, Analysis, Analysis of PDEs (math.AP)
Abstract

We study the boundary stabilization of one-dimensional cross-diffusion systems in a moving domain. We show first exponential stabilization and then finite-time stabilization in arbitrary small-time of the linearized system around uniform equilibria, provided the system has an entropic structure with a symmetric mobility matrix. One example of such systems are the equations describing a Physical Vapor Deposition (PVD) process. This stabilization is achieved with respect to both the volume fractions and the thickness of the domain. The feedback control is derived using the backstepping technique, adapted to the context of a time-dependent domain. In particular, the norm of the backward backstepping transform is carefully estimated with respect to time., Comment: 37 pages, no figure

Published
2023

7. A viscoelastic flow model of Maxwell-type with a symmetric-hyperbolic formulation

Author

Boyaval, Sébastien, Laboratoire d'Hydraulique Saint-Venant / Saint-Venant laboratory for Hydraulics (LHSV), École des Ponts ParisTech (ENPC)-Centre d'Etudes et d'Expertise sur les Risques, l'Environnement, la Mobilité et l'Aménagement (Cerema)-EDF R&D (EDF R&D), EDF (EDF)-EDF (EDF), MATHematics for MatERIALS (MATHERIALS), Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), and ANR-15-CE01-0013,SEDIFLO,Modélisation et simulation du transport solide en rivières(2015)

Subjects
Mechanics of Materials, FOS: Mathematics, General Materials Science, Mathematics - Numerical Analysis, Numerical Analysis (math.NA), [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

Maxwell models for viscoelastic flows are famous for their potential to unify elastic motions of solids with viscous motions of liquids in the continuum mechanics perspective. But the usual Maxwell models allow one to define well motions mostly for one-dimensional flows only. To define unequivocal multi-dimensional viscoelastic flows (as solutions to well-posed initial-value problems) we advocated in [ESAIM:M2AN 55 (2021) 807-831] an upper-convected Maxwell model for compressible flows with a symmetrichyperbolic formulation. Here, that model is derived again, with new details.

Published
2023

8. Practical Error Bounds for Properties in Plane-Wave Electronic Structure Calculations

Author

Eric Cancès, Geneviève Dusson, Gaspard Kemlin, Antoine Levitt, Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC), MATHematics for MatERIALS (MATHERIALS), École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Laboratoire de Mathématiques de Besançon (UMR 6623) (LMB), Université de Bourgogne (UB)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Centre National de la Recherche Scientifique (CNRS), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC), Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), and Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)

Subjects
Condensed Matter - Materials Science, Condensed Matter::Materials Science, Computational Mathematics, Applied Mathematics, FOS: Mathematics, [PHYS.COND.CM-MS]Physics [physics]/Condensed Matter [cond-mat]/Materials Science [cond-mat.mtrl-sci], Materials Science (cond-mat.mtrl-sci), FOS: Physical sciences, Numerical Analysis (math.NA), Mathematics - Numerical Analysis, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

International audience; We propose accurate computable error bounds for quantities of interest in plane-wave electronic structure calculations, in particular ground-state density matrices and energies, and interatomic forces. These bounds are based on an estimation of the error in terms of the residual of the solved equations, which is then efficiently approximated with computable terms. After providing coarse bounds based on an analysis of the inverse Jacobian, we improve on these bounds by solving a linear problem in a small dimension that involves a Schur complement. We numerically show how accurate these bounds are on a few representative materials, namely silicon, gallium arsenide and titanium dioxide.

Published
2022

9. Reduced basis method for non-symmetric eigenvalue problems: application to the multigroup neutron diffusion equations

Author

Conjungo Taumhas, Yonah, Dusson, Geneviève, Ehrlacher, Virginie, Lelièvre, Tony, Madiot, François, Service d’Études des Réacteurs et de Mathématiques Appliquées (SERMA), Département de Modélisation des Systèmes et Structures (DM2S), CEA-Direction des Energies (ex-Direction de l'Energie Nucléaire) (CEA-DES (ex-DEN)), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay-CEA-Direction des Energies (ex-Direction de l'Energie Nucléaire) (CEA-DES (ex-DEN)), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay, Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC), Laboratoire de Mathématiques de Besançon (UMR 6623) (LMB), Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC), MATHematics for MatERIALS (MATHERIALS), École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), ANR-15-IDEX-0003,BFC,ISITE ' BFC(2015), and European Project: 810367,EMC2(2019)

Subjects
FOS: Mathematics, Numerical Analysis (math.NA), Mathematics - Numerical Analysis, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

In this article, we propose a reduced basis method for parametrized non-symmetric eigenvalue problems arising in the loading pattern optimization of a nuclear core in neutronics. To this end, we derive a posteriori error estimates for the eigenvalue and left and right eigenvectors. The practical computation of these estimators requires the estimation of a constant called prefactor, which we can express as the spectral norm of some operator. We provide some elements of theoretical analysis which illustrate the link between the expression of the prefactor we obtain here and its well-known expression in the case of symmetric eigenvalue problems, either using the notion of numerical range of the operator, or via a perturbative analysis. Lastly, we propose a practical method in order to estimate this prefactor which yields interesting numerical results on actual test cases. We provide detailed numerical simulations on two-dimensional examples including a multigroup neutron diffusion equation.

Published
2023

10. Stationary solutions and large time asymptotics to a cross-diffusion-Cahn-Hilliard system

Author

Cauvin-Vila, Jean, Ehrlacher, Virginie, Marino, Greta, Pietschmann, Jan-Frederik, École des Ponts ParisTech (ENPC), MATHematics for MatERIALS (MATHERIALS), Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut für Mathematik [Augsburg], Universität Augsburg [Augsburg], The authors acknowledge support from project COMODO (ANR-19-CE46-0002).GM acknowledges the support of Deutsche Forschungsgemeinschaft (DFG) via grant GZ: MA 10100/1-1 project number 496629752. GM is member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilitae le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). JFP thanks the DFG for support via the Research Unit FOR 5387 POPULAR, Project No. 461909888., and ANR-19-CE46-0002,COMODO,Systèmes de diffusion croisée sur des domaines en mouvement(2019)

Subjects
Mathematics - Analysis of PDEs, Cahn-Hilliard, Cross-diffusion, FOS: Mathematics, Degenerate Ginzburg-Landau, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Mathematics - Numerical Analysis, Numerical Analysis (math.NA), Finite volume, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Analysis of PDEs (math.AP)
Abstract

We study some properties of a multi-species degenerate Ginzburg-Landau energy and its relation to a cross-diffusion Cahn-Hilliard system. The model is motivated by multicomponent mixtures where crossdiffusion effects between the different species are taken into account, and where only one species does separate from the others. Using a comparison argument, we obtain strict bounds on the minimizers from which we can derive first-order optimality conditions, revealing a link with the single-species energy, and providing enough regularity to qualify the minimizers as stationary solutions of the evolution system. We also discuss convexity properties of the energy as well as long time asymptotics of the time-dependent problem. Lastly, we introduce a structure-preserving finite volume scheme for the time-dependent problem and present several numerical experiments in one and two spatial dimensions.

Published
2023

11. Nonlinear reduced basis using mixture Wasserstein barycenters: application to an eigenvalue problem inspired from quantum chemistry

Author

Dalery, Maxime, Dusson, Geneviève, Ehrlacher, Virginie, Lozinski, Alexei, Laboratoire de Mathématiques de Besançon (UMR 6623) (LMB), Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC), Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC), MATHematics for MatERIALS (MATHERIALS), École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), ANR-15-IDEX-0003,BFC,ISITE ' BFC(2015), ANR-19-CE46-0002,COMODO,Systèmes de diffusion croisée sur des domaines en mouvement(2019), and European Project: 810367,EMC2(2019)

Subjects
Wasserstein barycenter, Eigenvalue problem, Optimal Transport, Reduced Basis, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

The aim of this article is to propose a new reduced-order modelling approach for parametric eigenvalue problems arising in electronic structure calculations. Namely, we develop nonlinear reduced basis techniques for the approximation of parametric eigenvalue problems inspired from quantum chemistry applications. More precisely, we consider here a onedimensional model which is a toy model for the computation of the electronic ground state wavefunction of a system of electrons within a molecule, solution to the many-body electronic Schrödinger equation, where the varying parameters are the positions of the nuclei in the molecule. We estimate the decay rate of the Kolmogorov n-width of the set of solutions for this parametric problem in several settings, including the standard L2-norm as well as with distances based on optimal transport. The fact that the latter decays much faster than in the traditional L2-norm setting motivates us to propose a practical nonlinear reduced basis method, which is based on an offline greedy algorithm, and an efficient stochastic energy minimization in the online phase. We finally provide numerical results illustrating the capabilities of the method and good approximation properties, both in the offline and the online phase.

Published
2023

12. Autoencoders for dimensionality reduction in molecular dynamics: collective variable dimension, biasing and transition states

Author

Belkacemi, Zineb, Bianciotto, Marc, Minoux, Herve, Lelievre, Tony, Stoltz, Gabriel, Gkeka, Paraskevi, Sanofi-Aventis R&D, SANOFI Recherche, Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC), MATHematics for MatERIALS (MATHERIALS), École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), and European Project: 810367,EMC2(2019)

Subjects
[PHYS.PHYS.PHYS-COMP-PH]Physics [physics]/Physics [physics]/Computational Physics [physics.comp-ph], FOS: Physical sciences, Computational Physics (physics.comp-ph), Physics - Computational Physics
Abstract

International audience The heat shock protein 90 (Hsp90) is a molecular chaperone that controls the folding and activation of client proteins using the free energy of ATP hydrolysis. The Hsp90 active site is in its N-terminal domain (NTD). Our goal is to characterize the dynamics of NTD using an autoencoder-learned collective variable (CV) in conjunction with adaptive biasing force (ABF) Langevin dynamics. Using dihedral analysis, we cluster all available experimental Hsp90 NTD structures into distinct native states. We then perform unbiased molecular dynamics (MD) simulations to construct a dataset that represents each state and use this dataset to train an autoencoder. Two autoencoder architectures are considered, with one and two hidden layers respectively, and bottlenecks of dimension $k$ ranging from 1 to 10. We demonstrate that the addition of an extra hidden layer does not significantly improve the performance, while it leads to complicated CVs that increases the computational cost of biased MD calculations. In addition, a 2D bottleneck can provide enough information of the different states, while the optimal bottleneck dimension is five. For the 2D bottleneck, the two-dimensional CV is directly used in biased MD simulations. For the 5D bottleneck, we perform an analysis of the latent CV space and identify the pair of CV coordinates that best separates the states of Hsp90. Interestingly, selecting a 2D CV out of the 5D CV space leads to better results than directly learning a 2D CV, and allows to observe transitions between native states when running free energy biased dynamics.

Published
2023

13. A sparse approximation of the Lieb functional with moment constraints

Author

Ehrlacher, Virginie, Nenna, Luca, Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC), MATHematics for MatERIALS (MATHERIALS), École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Laboratoire de Mathématiques d'Orsay (LMO), Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), ANR-11-LABX-0056,LMH,LabEx Mathématique Hadamard(2011), and European Project: 810367,EMC2(2019)

Subjects
Mathematics - Spectral Theory, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], Optimization and Control (math.OC), FOS: Mathematics, FOS: Physical sciences, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Mathematical Physics (math-ph), Spectral Theory (math.SP), Mathematics - Optimization and Control, Mathematical Physics, [MATH.MATH-SP]Mathematics [math]/Spectral Theory [math.SP]
Abstract

The aim of this paper is to present new sparsity results about the so-called Lieb functional, which is a key quantity in Density Functional Theory for electronic structure calculations for molecules. The Lieb functional was actually shown by Lieb to be a convexification of the so-called L\'evy-Lieb functional. Given an electronic density for a system of $N$ electrons, which may be seen as a probability density on $\mathbb{R}^3$, the value of the Lieb functional for this density is defined as the solution of a quantum multi-marginal optimal transport problem, which reads as a minimization problem defined on the set of trace-class operators acting on the space of electronic wave-functions that are anti-symmetric $L^2$ functions of $\mathbb{R}^{3N}$, with partial trace equal to the prescribed electronic density. We introduce a relaxation of this quantum optimal transport problem where the full partial trace constraint is replaced by a finite number of moment constraints on the partial trace of the set of operators. We show that, under mild assumptions on the electronic density, there exist sparse minimizers to the resulting moment constrained approximation of the Lieb (MCAL) functional that read as operators with rank at most equal to the number of moment constraints. We also prove under appropriate assumptions on the set of moment functions that the value of the MCAL functional converges to the value of the exact Lieb functional as the number of moments go to infinity. We also prove some rates of convergence on the associated approximation of the ground state energy. We finally study the mathematical properties of the associated dual problem.

Published
2023

14. Mathematical foundations for the Parallel Replica algorithm applied to the underdamped Langevin dynamics

Author

Mouad Ramil, Tony Lelièvre, Julien Reygner, Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC), MATHematics for MatERIALS (MATHERIALS), École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), ANR-19-CE40-0010,QuAMProcs,Analyse Quantitative de Processus Metastables(2019), ANR-17-CE40-0030,EFI,Entropie, flots, inégalités(2017), ANR-21-CE40-0006,SINEQ,Simulation de dynamiques stochastiques hors d'équilibre(2021), and European Project: 810367,EMC2(2019)

Subjects
Statistics/statistical methods, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Molecular, General Materials Science, Mathematics, Simulation
Abstract

View-only version available at https://rdcu.be/cSBls. International audience Molecular dynamics (MD) methods are used to sample the time evolution of complex molecular systems, namely the transition events between configuration states. When these states are metastable, these transitions correspond to rare events and occur over very long-time scale, thus rendering the use of MD methods inefficient.The Parallel Replica algorithm, was designed to efficiently sample these rare events relying on a parallelization in time computation of the trajectory. Its mathematical formalization was carried out in the literature using the notion of quasi-stationary distribution (QSD), which can be seen as the long-time distribution of the dynamics trapped inside a state. Its existence was only proven recently for the underdamped Langevin dynamics (ULD) involved in the sampling of thermostated molecular systems.In this research letter we shall state the existence of a QSD for the ULD, for which we canprovide a spectral interpretation. The existence of this QSD is then used to justify the application of Parallel Replica algorithm to the ULD. In addition, we shall extend known results regarding the behaviour of the ULD trajectories when the friction coefficient goes to infinity (overdamped limit). This allows us to provide an explicit expression of the overdamped limit of the previous QSD of the ULD. The findings of this study remain valid for ULD with non-conservative forces. They shall help for a better understanding of the mathematical framework underlying the Parallel Replica algorithm or related algorithms involving the QSD to the ULD. It also extends our knowledge on the overdamped limiting behaviour of the ULD.

Published
2022

15. Fixing the flux: A dual approach to computing transport coefficients

Author

Blassel, Noé, Stoltz, Gabriel, Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC), MATHematics for MatERIALS (MATHERIALS), École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), ANR-21-CE40-0006,SINEQ,Simulation de dynamiques stochastiques hors d'équilibre(2021), ANR-19-CE40-0010,QuAMProcs,Analyse Quantitative de Processus Metastables(2019), and European Project: 810367,EMC2(2019)

Subjects
Statistical Mechanics (cond-mat.stat-mech), FOS: Mathematics, FOS: Physical sciences, Mathematics - Numerical Analysis, Numerical Analysis (math.NA), [PHYS.COND.CM-SM]Physics [physics]/Condensed Matter [cond-mat]/Statistical Mechanics [cond-mat.stat-mech], Condensed Matter - Statistical Mechanics, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

We present a method to compute transport coefficients in molecular dynamics. Transport coefficients quantify the linear dependencies of fluxes in non-equilibrium systems subject to small external forcings. Whereas standard non-equilibrium approaches fix the forcing and measure the average flux induced in the system driven out of equilibrium, a dual philosophy consists in fixing the value of the flux, and measuring the average magnitude of the forcing needed to induce it. A deterministic version of this approach, named Norton dynamics, was studied in the 1980s by Evans and Morris. In this work, we introduce a stochastic version of this method, first developing a general formal theory for a broad class of diffusion processes, and then specializing it to underdamped Langevin dynamics, which are commonly used for molecular dynamics simulations. We provide numerical evidence that the stochastic Norton method provides an equivalent measure of the linear response, and in fact demonstrate that this equivalence extends well beyond the linear response regime. This work raises many intriguing questions, both from the theoretical and the numerical perspectives.

Published
2023

16. Estimation of statistics of transitions and Hill relation for Langevin dynamics

Author

Lelièvre, Tony, Ramil, Mouad, Reygner, Julien, Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC), MATHematics for MatERIALS (MATHERIALS), École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Department of Mathematical Sciences [Seoul], Seoul National University [Seoul] (SNU), ANR-19-CE40-0010,QuAMProcs,Analyse Quantitative de Processus Metastables(2019), ANR-17-CE40-0030,EFI,Entropie, flots, inégalités(2017), and European Project: 810367,EMC2(2019)

Subjects
[MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Transition path, 2020 Mathematics Subject Classification. 60J20, 60J70, Probability (math.PR), FOS: Mathematics, [PHYS.COND.CM-SM]Physics [physics]/Condensed Matter [cond-mat]/Statistical Mechanics [cond-mat.stat-mech], Hill relation, Langevin dynamics, Mathematics - Probability, Invariant measure
Abstract

In molecular dynamics, statistics of transitions, such as the mean transition time, are macroscopic observables which provide important dynamical information on the underlying microscopic stochastic process. A direct estimation using simulations of microscopic trajectories over long time scales is typically computationally intractable in metastable situations. To overcome this issue, several numerical methods rely on a potential-theoretic identity, sometimes attributed to Hill in the computational statistical physics litterature, which expresses statistics of transitions in terms of the invariant measure of the sequence of configurations by which the underlying process enters metastable sets. The use of this identity then allows to replace the long time simulation problem with a rare event sampling problem, for which efficient algorithms are available. In this article, we rigorously analyse such a method for molecular systems modelled by the Langevin dynamics. Our main contributions are twofold. First, we prove the Hill relation in the fairly general context of positive Harris recurrent chains, and show that this formula applies to the Langevin dynamics. Second, we provide an explicit expression of the invariant measure involved in the Hill relation, and describe an elementary exact simulation procedure. Overall, this yields a simple and complete numerical method to estimate statistics of transitions., Comment: Accepted version

Published
2023

17. A GKP qubit protected by dissipation in a high-impedance superconducting circuit driven by a microwave frequency comb

Author

Sellem, Lev-Arcady, Sarlette, Alain, Leghtas, Zaki, Mirrahimi, Mazyar, Rouchon, Pierre, Campagne-Ibarcq, Philippe, MATHematics for MatERIALS (MATHERIALS), Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), QUANTum Information Circuits (QUANTIC), Mines Paris - PSL (École nationale supérieure des mines de Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Sorbonne Université (SU)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire de physique de l'ENS - ENS Paris (LPENS), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)-Département de Physique de l'ENS-PSL, École normale supérieure - Paris (ENS-PSL), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-École normale supérieure - Paris (ENS-PSL), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)-Département de Physique de l'ENS-PSL, Université Paris sciences et lettres (PSL), Centre Automatique et Systèmes (CAS), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), Laboratoire de physique de l'ENS - ENS Paris (LPENS), ANR-18-CE47-0005,HAMROQS,Modèles réduits haute précision pour systèmes quantiques ouverts(2018), ANR-20-CE47-0007,SYNCAMIL,Synthèse d'hamiltonians non locaux pour la protection d'information quantique(2020), ANR-22-PETQ-0006,NISQ2LSQ,From NISQ to LSQ: Bosonic and LDPC codes(2022), European Project: 884762,Q-Feedback, European Project: 101042304,HORIZON.1.1 - European Research Council (ERC),ERC-2021-STG - ERC STARTING GRANTS,DANCINGFOOL(2022), and European Project: 851740,H2020-EU.1.1. - EXCELLENT SCIENCE - European Research Council (ERC),10.3030/851740,ECLIPSE(2020)

Subjects
Quantum Physics, [PHYS.QPHY]Physics [physics]/Quantum Physics [quant-ph], Optimization and Control (math.OC), FOS: Mathematics, FOS: Physical sciences, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Quantum Physics (quant-ph), Mathematics - Optimization and Control
Abstract

We propose a novel approach to generate, protect and control GKP qubits. It employs a microwave frequency comb parametrically modulating a Josephson circuit to enforce a dissipative dynamics of a high impedance circuit mode, autonomously stabilizing the finite-energy GKP code. The encoded GKP qubit is robustly protected against all dominant decoherence channels plaguing superconducting circuits but quasi-particle poisoning. In particular, noise from ancillary modes leveraged for dissipation engineering does not propagate at the logical level. In a state-of-the-art experimental setup, we estimate that the encoded qubit lifetime could extend two orders of magnitude beyond the break-even point, with substantial margin for improvement through progress in fabrication and control electronics. Qubit initialization, readout and control via Clifford gates can be performed while maintaining the code stabilization, paving the way toward the assembly of GKP qubits in a fault-tolerant quantum computing architecture., 61 pages, 20 figures, 1 table

Published
2023

18. Simple derivation of moiré-scale continuous models for twisted bilayer graphene

Author

Éric Cancès, Louis Garrigue, David Gontier, Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC), MATHematics for MatERIALS (MATHERIALS), École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), Département de Mathématiques et Applications - ENS Paris (DMA), École normale supérieure - Paris (ENS-PSL), This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement EMC2 No 810367) and from the Simons foundation (Targeted Grant Moiré Materials Magic), and European Project: 810367,EMC2(2019)

Subjects
Condensed Matter - Mesoscale and Nanoscale Physics, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], Mathematical Physics
Abstract

International audience We provide a formal derivation of a reduced model for twisted bilayer graphene (TBG) from Density Functional Theory. Our 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 $w_{AA}$ and $w_{AB}$ of the BM model from first principles. The resulting numerical values, namely $w_{AA} = w_{AB} ≃ 126$ meV for the experimental interlayer mean distance are in good agreement with the empirical values $w_{AA} = w_{AB} = 110$ meV obtained by fitting to experimental data. We also show that if the BM parameters are set to $w_{AA} = w_{AB} ≃ 126$ meV, the BM model is an accurate approximation of our reduced model

Published
2023

19. Approximation volumes finis structure-compatible de systèmes de diffusion croisée couplés par une interface libre

Author

Cancès, Clément, Cauvin-Vila, Jean, Chainais-Hillairet, Claire, Ehrlacher, Virginie, Reliable numerical approximations of dissipative systems (RAPSODI ), Laboratoire Paul Painlevé (LPP), Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Inria Lille - Nord Europe, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), MATHematics for MatERIALS (MATHERIALS), Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC)-Inria de Paris, and ANR-19-CE46-0002,COMODO,Systèmes de diffusion croisée sur des domaines en mouvement(2019)

Subjects
Mathematics - Analysis of PDEs, Cut-cell method, Finite volume scheme, FOS: Mathematics, Cross-diffusion system, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Numerical Analysis (math.NA), Mathematics - Numerical Analysis, Free energy dissipation, Vapor deposition, Moving interface, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Analysis of PDEs (math.AP)
Abstract

We propose a two-point flux approximation finite-volume scheme for the approximation of two cross-diffusion systems coupled by a free interface to account for vapor deposition. The moving interface is addressed with a cut-cell approach, where the mesh is locally deformed around the interface. The scheme preserves the structure of the continuous system, namely: mass conservation, nonnegativity, volume-filling constraints and decay of the free energy. Numerical results illustrate the properties of the scheme., Typo corrected (sign in the flux)

Published
2023

20. Computing Surface Reaction Rates by Adaptive Multilevel Splitting Combined with Machine Learning and Ab Initio Molecular Dynamics

Author

Pigeon, Thomas, Stoltz, Gabriel, Corral-Valero, Manuel, Anciaux-Sedrakian, Ani, Moreaud, Maxime, Lelièvre, Tony, Raybaud, Pascal, IFP Energies nouvelles (IFPEN), Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC), MATHematics for MatERIALS (MATHERIALS), École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), and European Project: 810367,EMC2(2019)

Subjects
Chemical Physics (physics.chem-ph), [CHIM.THEO]Chemical Sciences/Theoretical and/or physical chemistry, Statistical Mechanics (cond-mat.stat-mech), Physics - Chemical Physics, FOS: Physical sciences, Condensed Matter - Statistical Mechanics
Abstract

Computing accurate rate constants for catalytic events occurring at the surface of a given material represents a challenging task with multiple potential applications in chemistry. To address this question, we propose an approach based on a combination of the rare event sampling method called Adaptive Multilevel Splitting (AMS) and ab initio molecular dynamics (AIMD). The AMS method requires a one dimensional reaction coordinate to index the progress of the transition. Identifying a good reaction coordinate is difficult, especially for high dimensional problems such a those encountered in catalysis. We probe various approaches to build reaction coordinates such as Support Vector Machine and path collective variables. The AMS is implemented so as to communicate with a DFT-plane wave code. A relevant case study in catalysis: the change of conformation and the dissociation of a water molecule chemisorbed on the (100) $\gamma$-alumina surface is used to evaluate our approach. The calculated rate constants and transition mechanisms are discussed and compared to those obtained by a conventional static approach based on the Eyring-Polanyi equation with harmonic approximation. It is revealed that the AMS method may provide rate constants which are smaller than the static approach by up to two orders of magnitude due to entropic effects involved in the chemisorbed water., Comment: 33 pages, 9 figures

Published
2023

21. Extending the regime of linear response with synthetic forcings

Author

Spacek, Renato, Stoltz, Gabriel, MATHematics for MatERIALS (MATHERIALS), Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), École des Ponts ParisTech (ENPC), ANR-19-CE40-0010,QuAMProcs,Analyse Quantitative de Processus Metastables(2019), ANR-21-CE40-0006,SINEQ,Simulation de dynamiques stochastiques hors d'équilibre(2021), and European Project: 810367,EMC2(2019)

Subjects
MSC codes. 82C31, 65C40, 65N06, 46N30, 82C05, Statistical Mechanics (cond-mat.stat-mech), Variance reduction, Nonequilibrium molecular dynamics, FOS: Physical sciences, 82C31, 65C40, 65N06, 46N30, 82C05, Numerical Analysis (math.NA), Langevin dynamics, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], FOS: Mathematics, Mathematics - Numerical Analysis, [PHYS.COND.CM-SM]Physics [physics]/Condensed Matter [cond-mat]/Statistical Mechanics [cond-mat.stat-mech], Linear response, Condensed Matter - Statistical Mechanics, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

Transport coefficients, such as the mobility, thermal conductivity and shear viscosity, are quantities of prime interest in statistical physics. At the macroscopic level, transport coefficients relate an external forcing of magnitude $\eta$, with $\eta \ll 1$, acting on the system to an average response expressed through some steady-state flux. In practice, steady-state averages involved in the linear response are computed as time averages over a realization of some stochastic differential equation. Variance reduction techniques are of paramount interest in this context, as the linear response is scaled by a factor of $1/\eta$, leading to large statistical error. One way to limit the increase in the variance is to allow for larger values of $\eta$ by increasing the range of values of the forcing for which the nonlinear part of the response is sufficiently small. In theory, one can add an extra forcing to the physical perturbation of the system, called synthetic forcing, as long as this extra forcing preserves the invariant measure of the reference system. The aim is to find synthetic perturbations allowing to reduce the nonlinear part of the response as much as possible. We present a mathematical framework for quantifying the quality of synthetic forcings, in the context of linear response theory, and discuss various possible choices for them. Our findings are illustrated with numerical results in low-dimensional systems.

Published
2023

22. Low temperature asymptotics for Quasi-Stationary Distributions in a bounded domain

Author

Tony Lelièvre, Francis Nier, MATHematics for MatERIALS (MATHERIALS), Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC)-Inria Paris-Rocquencourt, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), École des Ponts ParisTech (ENPC), Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), MicMac, ANR-11-LABX-0020,LEBESGUE,Centre de Mathématiques Henri Lebesgue : fondements, interactions, applications et Formation(2011), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), MATHematics for MatERIALS ( MATHERIALS ), Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique ( CERMICS ), École des Ponts ParisTech ( ENPC ) -École des Ponts ParisTech ( ENPC ) -Inria Paris-Rocquencourt, Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National de Recherche en Informatique et en Automatique ( Inria ), École des Ponts ParisTech ( ENPC ), Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), and Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS )

Subjects
Diffusion equation, 01 natural sciences, Domain (mathematical analysis), Dirichlet distribution, 58J10, 58J32, symbols.namesake, Mathematics - Analysis of PDEs, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], 0103 physical sciences, FOS: Mathematics, 81Q20, 60J65, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 35P20, 82C80, Statistical physics, Limit (mathematics), 0101 mathematics, Langevin dynamics, quasistationary distributions, Morse theory, low temperature asymptotics and semiclassical asymptotics, Physics, Numerical Analysis, 010304 chemical physics, Applied Mathematics, 010102 general mathematics, Distribution (mathematics), Bounded function, symbols, Witten Laplacian, Analysis, 60J70, Analysis of PDEs (math.AP)
Abstract

International audience; We analyze the low temperature asymptotics of the quasi-stationary distribution associated with the overdamped Langevin dynamics (a.k.a. the Einstein-Smoluchowski diffusion equation) in a bounded domain. This analysis is useful to rigorously prove the consistency of an algorithm used in molecular dynamics (the hyperdynamics), in the small temperature regime. More precisely, we show that the algorithm is exact in terms of state-to-state dynamics up to exponentially small factor in the limit of small temperature. The proof is based on the asymptotic spectral analysis of associated Dirichlet and Neumann realizations of Witten Laplacians. In order to cover a reasonably large range of applications, the usual assumptions that the energy landscape is a Morse function has been relaxed as much as possible.

Published
2015

23. SOTT: Greedy Approximation of a Tensor as a Sum of Tensor Trains

Author

Ehrlacher, Virginie, Ruiz, Maria Fuente, Lombardi, Damiano, Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC), MATHematics for MatERIALS (MATHERIALS), École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), COmputational Mathematics for bio-MEDIcal Applications (COMMEDIA), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), ANR-18-CE46-0001,ADAPT,Méthodes tensorielles parallèles dynamiques et adaptatives(2018), ANR-19-CE46-0002,COMODO,Systèmes de diffusion croisée sur des domaines en mouvement(2019), European Project: 810367,EMC2(2019), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), and École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC)

Subjects
65D15, Computational Mathematics, Tensor methods, Applied Mathematics, Canonical Polyadic, Tensor Train AMS subject classifications. 65F99, [INFO.INFO-MO]Computer Science [cs]/Modeling and Simulation, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

International audience In the present work, a method is proposed in order 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 proposed, 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.

Published
2022

24. A Wasserstein-type metric for generic mixture models, including location-scatter and group invariant measures

Author

Dusson, Geneviève, Ehrlacher, Virginie, Nouaime, Nathalie, Laboratoire de Mathématiques de Besançon (UMR 6623) (LMB), Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC), Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC), MATHematics for MatERIALS (MATHERIALS), École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), CEA- Saclay (CEA), Commissariat à l'énergie atomique et aux énergies alternatives (CEA), This project has received funding from the European Re-search Council (ERC) under the European Union’s Horizon 2020 research and inno-vation programme (grant agreement EMC2 No 810367). This work was supported by the French ‘Investissements d’Avenir’ program, project Agence Nationale de la Recherche (ISITE-BFC) (contract ANR-15-IDEX-0003). GD was also supported by the Ecole des Ponts-ParisTech. VE acknowledges support from the ANR project COMODO (ANR-19-CE46-0002)., ANR-15-IDEX-0003,BFC,ISITE ' BFC(2015), ANR-19-CE46-0002,COMODO,Systèmes de diffusion croisée sur des domaines en mouvement(2019), and European Project: 810367,EMC2(2019)

Subjects
Optimization and Control (math.OC), Probability (math.PR), FOS: Mathematics, Optimal transport, Mixture, Wasserstein distance, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], MSC codes. 65D05,65K10,41A05,41A63,46G99,46T12,60B05,47N50, Mathematics - Optimization and Control, Wasserstein barycenters MSC codes. 65D05, Mathematics - Probability
Abstract

In this article, we study Wasserstein-type metrics and corresponding barycenters for mixtures of a chosen subset of probability measures called atoms hereafter. In particular, this works extends what was proposed by Delon and Desolneux [A Wasserstein-Type Distance in the Space of Gaussian Mixture Models. SIAM J. Imaging Sci. 13, 936-970 (2020)] for mixtures of gaussian measures to other mixtures. We first prove in a general setting that for a set of atoms equipped with a metric that defines a geodesic space, the set of mixtures based on this set of atoms is also geodesic space for the defined modified Wasserstein metric. We then focus on two particular cases of sets of atoms: (i) the set of location-scatter atoms and (ii) the set of measures that are invariant with respect to some symmetry group. Both cases are particularly relevant for various applications among which electronic structure calculations. Along the way, we also prove some sparsity and symmetry properties of optimal transport plans between measures that are invariant under some well-chosen symmetries.

Published
2023

25. Stability and decoherence rates of a GKP qubit protected by dissipation

Author

Sellem, Lev-Arcady, Robin, Rémi, Campagne-Ibarcq, Philippe, Rouchon, Pierre, MATHematics for MatERIALS (MATHERIALS), Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), QUANTum Information Circuits (QUANTIC), Mines Paris - PSL (École nationale supérieure des mines de Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Sorbonne Université (SU)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire de physique de l'ENS - ENS Paris (LPENS), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)-Département de Physique de l'ENS-PSL, École normale supérieure - Paris (ENS-PSL), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-École normale supérieure - Paris (ENS-PSL), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)-Département de Physique de l'ENS-PSL, Université Paris sciences et lettres (PSL), Centre Automatique et Systèmes (CAS), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), Laboratoire de physique de l'ENS - ENS Paris (LPENS), and European Project: 884762,Q-Feedback

Subjects
Quantum Physics, [PHYS.QPHY]Physics [physics]/Quantum Physics [quant-ph], Optimization and Control (math.OC), FOS: Mathematics, FOS: Physical sciences, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Quantum Physics (quant-ph), Mathematics - Optimization and Control
Abstract

We analyze an experimentally accessible Lindblad master equation for a quantum harmonic oscillator. It approximately stabilizes finite-energy periodic grid states called Gottesman-Kitaev-Preskill (GKP) states, that can be used to encode and protect a logical qubit. We give explicit upper bounds for the energy of the solutions of the Lindblad master equation. Using three periodic observables to define the Bloch sphere coordinates of a logical qubit, we show that their dynamics is governed by a diffusion partial differential equation on a 2D-torus with a Witten Laplacian. We show that the evolution of these logical coordinates is exponentially slow even in presence of small diffusive noise processes along the two quadratures of the phase space. Numerical simulations indicate similar results for other physically relevant noise processes., Comment: 16 pages, 1 figure. This work has been accepted to IFAC for publication under a Creative Commons Licence CC-BY-NC-ND

Published
2023
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26. A variance reduction strategy for numerical random homogenization based on the equivalent inclusion method

Author

Brisard, Sebastien, Bertin, Michael, Legoll, Frederic, Modélisation et expérimentation multi-échelle pour les solides hétérogènes (Multi-échelle), Laboratoire Navier (NAVIER UMR 8205), École des Ponts ParisTech (ENPC)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel-École des Ponts ParisTech (ENPC)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel, Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC), MATHematics for MatERIALS (MATHERIALS), École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), and This work has benefited from the French government grant ANR-11-LABX-022-01 (Labex MMCD, Multi-Scale Modelling & Experimentation of Materials for Sustainable Construction) managed by ANR within the frame of the national program Investments for the Future.

Subjects
Computational Engineering, Finance, and Science (cs.CE), FOS: Computer and information sciences, Homogenization, Conductivity, Variance reduction, [SPI.MECA.MEMA]Engineering Sciences [physics]/Mechanics [physics.med-ph]/Mechanics of materials [physics.class-ph], Full-field simulations, Computer Science - Computational Engineering, Finance, and Science, Surrogate model
Abstract

Using the equivalent inclusion method (a method strongly related to the Hashin-Shtrikman variational principle) as a surrogate model, we propose a variance reduction strategy for the numerical homogenization of random composites made of inclusions (or rather inhomogeneities) embedded in a homogeneous matrix. The efficiency of this strategy is demonstrated within the framework of two-dimensional, linear conductivity. Significant computational gains vs full-field simulations are obtained even for high contrast values. We also show that our strategy allows to investigate the influence of parameters of the microstructure on the macroscopic response. Our strategy readily extends to three-dimensional problems and to linear elasticity. Attention is paid to the computational cost of the surrogate model. In particular, an inexpensive approximation of the so-called influence tensors (that are used to compute the surrogate model) is proposed.

Published
2023
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27. Probabilistic formulation of Miner's rule and application to structural fatigue

Author

Cartiaux, Francois-Baptiste, Ehrlacher, Alain, Legoll, Frederic, Libal, Alex, Reygner, Julien, OSMOS GROUP [Paris], OSMOS, Matériaux et Structures Architecturés (MSA), Laboratoire Navier (NAVIER UMR 8205), École des Ponts ParisTech (ENPC)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel-École des Ponts ParisTech (ENPC)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel, Modélisation et expérimentation multi-échelle pour les solides hétérogènes (Multi-échelle), MATHematics for MatERIALS (MATHERIALS), Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), and École des Ponts ParisTech (ENPC)

Subjects
Computational Engineering, Finance, and Science (cs.CE), FOS: Computer and information sciences, [SPI.MECA.MSMECA]Engineering Sciences [physics]/Mechanics [physics.med-ph]/Materials and structures in mechanics [physics.class-ph], Computer Science - Computational Engineering, Finance, and Science
Abstract

The standard stress-based approach to fatigue is based on the use of S-N curves. They are obtained by applying cyclic loading of constant amplitude $S$ to identical and standardised specimens until they fail. The S-N curves actually depend on a reference probability $p$: for a given cycle amplitude $S$, they provide the number of cycles at which a proportion $p$ of specimens have failed. Based on the S-N curves, Miner's rule is next used to predict the number of cycles to failure of a specimen subjected to cyclic loading with variable amplitude. In this article, we present a probabilistic formulation of Miner's rule, which is based on the introduction of the notion of health of a specimen. We show the consistency of that new formulation with the standard approaches, thereby providing a precise probabilistic interpretation of these. Explicit formulas are derived in the case of the Weibull--Basquin model. We next turn to the case of a complete mechanical structure: taking into account size effects, and using the weakest link principle, we establish formulas for the survival probability of the structure. We illustrate our results by numerical simulations on a I-steel beam, for which we compute survival probabilities and density of failure point. We also show how to efficiently approximate these quantities using the Laplace method.

Published
2023
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28. Embedded corrector problems for homogenization in linear elasticity

Author

Ehrlacher, Virginie, Legoll, Frederic, Stamm, Benjamin, Xiang, Shuyang, Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC), MATHematics for MatERIALS (MATHERIALS), École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Laboratoire Navier (NAVIER UMR 8205), École des Ponts ParisTech (ENPC)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel, Center for Computational Engineering Science [Aachen] (CCSE), Rheinisch-Westfälische Technische Hochschule Aachen University (RWTH), and ANR-11-LABX-0022,MMCD,Modélisation et Expérimentation Multi-Echelles des Matériaux pour la Cons-truction Durable(2011)

Subjects
Mathematics - Analysis of PDEs, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Mathematics - Numerical Analysis, Numerical Analysis (math.NA), Analysis of PDEs (math.AP)
Abstract

In this article, we extend the study of embedded corrector problems, that we have previously introduced in the context of the homogenization of scalar diffusive equations, to the context of homogenized elastic properties of materials. This extension is not trivial and requires mathematical arguments specific to the elasticity case. Starting from a linear elasticity model with highly-oscillatory coefficients, we introduce several effective approximations of the homogenized tensor. These approximations are based on the solution to an embedded corrector problem, where a finite-size domain made of the linear elastic heterogeneous material is embedded in a linear elastic homogeneous infinite medium, the constant elasticity tensor of which has to be appropriately determined. The approximations we provide are proven to converge to the homogenized elasticity tensor when the size of the embedded domain tends to infinity. Some particular attention is devoted to the case of isotropic materials.

Published
2023
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29. Photoionization and core resonances from range-separated time-dependent density-functional theory for open-shell states: Example of the lithium atom

Author

Julien Toulouse, Karno Schwinn, Felipe Zapata, Antoine Levitt, Éric Cancès, Eleonora Luppi, Laboratoire de chimie théorique (LCT), Institut de Chimie du CNRS (INC)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Institut Universitaire de France (IUF), Ministère de l'Education nationale, de l’Enseignement supérieur et de la Recherche (M.E.N.E.S.R.), Department of Physics, Lund University [Lund], Laboratoire de Mathématiques d'Orsay (LMO), Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC), MATHematics for MatERIALS (MATHERIALS), École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), CNRS Emergence@INC2021 program (project OPTLHYB), and European Project: 810367,EMC2(2019)

Subjects
Chemical Physics (physics.chem-ph), [CHIM.THEO]Chemical Sciences/Theoretical and/or physical chemistry, [PHYS.PHYS.PHYS-COMP-PH]Physics [physics]/Physics [physics]/Computational Physics [physics.comp-ph], Physics - Chemical Physics, FOS: Physical sciences, General Physics and Astronomy, [PHYS.PHYS.PHYS-CHEM-PH]Physics [physics]/Physics [physics]/Chemical Physics [physics.chem-ph], Computational Physics (physics.comp-ph), Physical and Theoretical Chemistry, Physics - Computational Physics
Abstract

We consider the calculations of photoionization spectra and core resonances of open-shell systems using range-separated time-dependent density-functional theory. Specifically, we use the time-dependent range-separated hybrid (TDRSH) scheme, combining a long-range Hartree-Fock (HF) exchange potential and kernel with a short-range potential and kernel from a local density-functional approximation, and the time-dependent locally range-separated hybrid (TDLRSH) scheme, which uses a local range-separation parameter. To efficiently perform the calculations, we formulate a spin-unrestricted linear-response Sternheimer approach in a non-orthogonal B-spline basis set and using appropriate frequency-dependent boundary conditions. We illustrate this approach on the Li atom, which suggests that TDRSH and TDLRSH are adequate simple methods for estimating single-electron photoionization spectra of open-shell systems., arXiv admin note: substantial text overlap with arXiv:2203.05951

Published
2022

30. The Ensemble Kalman Filter in the Near-Gaussian Setting

Author

Carrillo, Jose Antonio, Hoffmann, Franca, Stuart, Andrew M., Vaes, Urbain, University of Oxford, California Institute of Technology (CALTECH), MATHematics for MatERIALS (MATHERIALS), Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), École des Ponts ParisTech (ENPC), JAC was supported by the Advanced Grant Nonlocal-CPD (Nonlocal PDEs for Complex Particle Dynamics: Phase Transitions, Patterns and Synchronization) of the European Research Council Executive Agency (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 883363). JAC was also partially supported by EPSRC grants EP/T022132/1 and EP/V051121/1. FH is supported by start-up funds at the California Institute of Technology. FH was also supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) via project 390685813 - GZ 2047/1 - HCM. The work of AMS is supported by the Office of Naval Research award N00014-17-1-2079 and by a Department of Defense Vannevar Bush Faculty Fellowship. UV is partially supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 810367), and by the Agence Nationale de la Recherche under grant ANR-21-CE40-0006 (SINEQ)., ANR-21-CE40-0006,SINEQ,Simulation de dynamiques stochastiques hors d'équilibre(2021), European Project: 883363,H2020-EU.1.1,Nonlocal-CPD(2020), European Project: 810367,EMC2(2019), Vaes, Urbain, Simulation de dynamiques stochastiques hors d'équilibre - - SINEQ2021 - ANR-21-CE40-0006 - AAPG2021 - VALID, New mathematical tools to uncover the complex dynamics of many-body systems - Nonlocal-CPD - - H2020-EU.1.12020-10-01 - 2025-09-30 - 883363 - VALID, and Extreme-Scale Mathematically-based Computational Chemistry - EMC2 - 2019-09-01 - 2026-02-26 - 810367 - VALID

Subjects
[MATH] Mathematics [math], [MATH]Mathematics [math]
Abstract

The ensemble Kalman filter is widely used in applications because, for high dimensional filtering problems, it has a robustness that is not shared for example 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. To address this issue we provide the first analysis of the accuracy of the ensemble Kalman filter beyond the Gaussian setting. Our analysis is developed for the mean field ensemble Kalman filter. We rewrite this filter in terms of maps on probability measures, and then we prove that these maps are locally Lipschitz in an appropriate weighted total variation metric. Using these stability estimates we 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. Finally, we provide a generalization of these results to the Gaussian projected filter, which can be viewed as a mean field description of the unscented Kalman filter.

Published
2022

31. Combining machine-learned and empirical force fields with the parareal algorithm: application to the diffusion of atomistic defects

Author

Gorynina, Olga, Legoll, Frederic, Lelièvre, Tony, Perez, Danny, Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC), MATHematics for MatERIALS (MATHERIALS), École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Modélisation et expérimentation multi-échelle pour les solides hétérogènes (Multi-échelle), Laboratoire Navier (NAVIER UMR 8205), École des Ponts ParisTech (ENPC)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel-École des Ponts ParisTech (ENPC)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel, Theoretical Division [LANL], Los Alamos National Laboratory (LANL), This project has received funding from the Agence Nationale de la Recherche (ANR, France) and the European High-Performance Computing Joint Undertaking (JU) under grant agreement 955701 (project Time-X). The JU receives support from the European Union’s Horizon 2020 research and innovation programme and Belgium, France, Germany, Switzerland. This work was also partially funded by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant 810367, project EMC2)., ANR-15-CE23-0019,CINE-PARA,Méthodes de parallélisation pour cinétiques complexes(2015), European Project: 810367,EMC2(2019), and European Project: 955701,Time-X

Subjects
FOS: Mathematics, Numerical Analysis (math.NA), Mathematics - Numerical Analysis, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

We numerically investigate an adaptive version of the parareal algorithm in the context of molecular dynamics. This adaptive variant has been originally introduced in [F. Legoll, T. Lelievre and U. Sharma, SISC 2022]. We focus here on test cases of physical interest where the dynamics of the system is modelled by the Langevin equation and is simulated using the molecular dynamics software LAMMPS. In this work, 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. We consider a self-interstitial atom in a tungsten lattice and compute the average residence time of the system in metastable states. Our numerical results demonstrate significant computational gains using the adaptive parareal algorithm in comparison to a sequential integration of the Langevin dynamics. We also identify a large regime of numerical parameters for which statistical accuracy is reached without being a consequence of trajectorial accuracy.

Published
2022

32. Using Witten Laplacians to locate index-1 saddle points

Author

Lelièvre, Tony, Parpas, Panos, Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC), MATHematics for MatERIALS (MATHERIALS), École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Department of Computing [Imperial College London], Imperial College London, Leverhulme TrustJ.P. Morgan A.I. Faculty AwardsEPSRC through grant number EP/W003317/1, ANR-19-CE40-0010,QuAMProcs,Analyse Quantitative de Processus Metastables(2019), and European Project: 810367,EMC2(2019)

Subjects
MSC codes. 65C35, 37N30, 65K10, FOS: Physical sciences, Numerical Analysis (math.NA), Computational Physics (physics.comp-ph), Dimer method, Optimization and Control (math.OC), 65C35, 37N30, 65K10, Saddle point search, FOS: Mathematics, Mathematics - Numerical Analysis, Witten Laplacian, Mathematics - Optimization and Control, Physics - Computational Physics, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

We introduce a new stochastic algorithm to locate the index-1 saddle points of a function $V:\mathbb R^d \to \mathbb R$, with $d$ possibly large. This 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 (associated with $V$) on $1$-forms and (ii) a probabilistic representation of a partial differential equation involving this differential operator. Numerical examples on simple molecular systems illustrate the efficacy of the proposed approach.

Published
2022

33. Frequency and field-dependent response of confined electrolytes from Brownian dynamics simulations

Author

Hoang Ngoc Minh, Thê, Stoltz, Gabriel, Rotenberg, Benjamin, PHysicochimie des Electrolytes et Nanosystèmes InterfaciauX (PHENIX), Institut de Chimie du CNRS (INC)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC), MATHematics for MatERIALS (MATHERIALS), École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Réseau sur le stockage électrochimique de l'énergie (RS2E), Aix Marseille Université (AMU)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Université de Toulouse (UT)-Université de Haute-Alsace (UHA) Mulhouse - Colmar (Université de Haute-Alsace (UHA))-Collège de France (CdF (institution))-Université de Picardie Jules Verne (UPJV)-Ecole Nationale Supérieure de Chimie de Paris - Chimie ParisTech-PSL (ENSCP), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Université de Pau et des Pays de l'Adour (UPPA)-Institut de Chimie du CNRS (INC)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université de Toulouse (UT)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA)-Université Grenoble Alpes (UGA)-Nantes Université (Nantes Univ)-Université de Montpellier (UM)-Ecole Nationale Supérieure de Chimie de Montpellier (ENSCM), Université de Montpellier (UM), Institut des Sciences du Calcul et des Données (ISCD) of Sorbonne University (IDEX SUPER Grant No. 11-IDEX-0004)., ANR-21-CE40-0006,SINEQ,Simulation de dynamiques stochastiques hors d'équilibre(2021), European Project: ERC-2019-CoG-863473,SENSES, and European Project: ERC-2018-SYG-810367,EMC2

Subjects
Chemical Physics (physics.chem-ph), [PHYS]Physics [physics], Physics - Chemical Physics, General Physics and Astronomy, Soft Condensed Matter (cond-mat.soft), FOS: Physical sciences, [CHIM]Chemical Sciences, Physical and Theoretical Chemistry, Computational Physics (physics.comp-ph), Condensed Matter - Soft Condensed Matter, Physics - Computational Physics
Abstract

Using Brownian dynamics simulations, we investigate the effects of confinement, adsorption on surfaces and ion-ion interactions on the response of confined electrolyte solutions to oscillating electric fields in the direction perpendicular to the confining walls. Nonequilibrium simulations allow to characterize the transitions between linear and nonlinear regimes when varying the magnitude and frequency of the applied field but the linear response, characterized by the frequency-dependent conductivity, is more efficiently predicted from the equilibrium current fluctuations. To that end, we (rederive and) use the Green-Kubo relation appropriate for overdamped dynamics, which differs from the standard one for Newtonian or underdamped Langevin dynamics. This expression highlights the contributions of the underlying Brownian fluctuations and of the interactions of the particles between them and with external potentials. While already known in the literature, this relation has rarely been used to date, beyond the static limit to determine the effective diffusion coefficient or the DC conductivity. The frequency-dependent conductivity always decays from a bulk-like behavior at high frequency to a vanishing conductivity at low frequency due to the confinement of the charge carriers by the walls. We discuss the characteristic features of the crossover between the two regimes, most importantly how the crossover frequency depends on the confining distance and the salt concentration, and the fact that adsorption on the walls may lead to significant changes both at high- and low-frequencies. Conversely, our results illustrate the possibility to obtain information on diffusion between walls, charge relaxation and adsorption by analyzing the frequency-dependent conductivity., 17 pages, 16 figures, contains appendix

Published
2022

34. Non-intrusive implementation of a wide variety of Multiscale Finite Element Methods

Author

Biezemans, Rutger A., Bris, Claude Le, Legoll, Frédéric, Lozinski, Alexei, Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC), MATHematics for MatERIALS (MATHERIALS), École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Modélisation et expérimentation multi-échelle pour les solides hétérogènes (Multi-échelle), Laboratoire Navier (NAVIER UMR 8205), École des Ponts ParisTech (ENPC)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel-École des Ponts ParisTech (ENPC)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel, Laboratoire de Mathématiques de Besançon (UMR 6623) (LMB), Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), and Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)

Subjects
Finite element methods, Non-intrusive implementation, FOS: Mathematics, Mathematics - Numerical Analysis, Numerical Analysis (math.NA), [MATH]Mathematics [math], Partial differential equations, Multiscale problems, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

Multiscale Finite Element Methods (MsFEMs) are now well-established finite element type approaches dedicated to multiscale problems. They first compute local, oscillatory, problem-dependent basis functions that generate a suitable discretization space, and next perform a Galerkin approximation of the problem on that space. We investigate here how these approaches can be implemented in a non-intrusive way, in order to facilitate their dissemination within industrial codes or non-academic environments. We develop an abstract framework that covers a wide variety of MsFEMs for linear second-order partial differential equations. Non-intrusive MsFEM approaches are developed within the full generality of this framework, which may moreover be beneficial to steering software development and improving the theoretical understanding and analysis of MsFEMs., Comment: 46 pages, 4 figures

Published
2022

35. Homogenization of some periodic Hamilton-Jacobi equations with defects

Author

Achdou, Yves, Bris, Claude Le, Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC), MATHematics for MatERIALS (MATHERIALS), École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), and ANR-22-CE40-0010,COSS,COntrôle sur des Structures Stratifiées(2022)

Subjects
Homogenization, Mathematics - Analysis of PDEs, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Defect, Hamilton-Jacobi equation, Analysis of PDEs (math.AP)
Abstract

We study homogenization for a class of stationnary Hamilton-Jacobi equations in which the Hamiltonian is obtained by perturbing near the origin an otherwise periodic Hamiltonian. We prove that the limiting problem consists of a Hamilton-Jacobi equation outside the origin, with the same effective Hamiltonian as in periodic homogenization, supplemented at the origin with an effective Dirichlet condition that keeps track of the perturbation. Various comments and extensions are discussed.

Published
2022

36. Error estimates and variance reduction for nonequilibrium stochastic dynamics

Author

Stoltz, Gabriel, Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC), MATHematics for MatERIALS (MATHERIALS), École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), ANR-21-CE40-0006,SINEQ,Simulation de dynamiques stochastiques hors d'équilibre(2021), ANR-19-CE40-0010,QuAMProcs,Analyse Quantitative de Processus Metastables(2019), and European Project: 810367,EMC2(2019)

Subjects
Statistical Mechanics (cond-mat.stat-mech), FOS: Mathematics, FOS: Physical sciences, Numerical Analysis (math.NA), Mathematics - Numerical Analysis, [PHYS.COND.CM-SM]Physics [physics]/Condensed Matter [cond-mat]/Statistical Mechanics [cond-mat.stat-mech], Condensed Matter - Statistical Mechanics
Abstract

Equilibrium properties in statistical physics are obtained by computing averages with respect to Boltzmann-Gibbs measures, sampled in practice using ergodic dynamics such as the Langevin dynamics. Some quantities however cannot be computed by simply sampling the Boltzmann-Gibbs measure, in particular transport coefficients, which relate the current of some physical quantity of interest to the forcing needed to induce it. For instance, a temperature difference induces an energy current, the proportionality factor between these two quantities being the thermal conductivity. From an abstract point of view, transport coefficients can also be considered as some form of sensitivity analysis with respect to an added forcing to the baseline dynamics. There are various numerical techniques to estimate transport coefficients, which all suffer from large errors, in particular large statistical errors. This contribution reviews the most popular methods, namely the Green-Kubo approach where the transport coefficient is expressed as some time-integrated correlation function, and the approach based on longtime averages of the stochastic dynamics perturbed by an external driving (so-called nonequilibrium molecular dynamics). In each case, the various sources of errors are made precise, in particular the bias related to the time discretization of the underlying continuous dynamics, and the variance of the associated Monte Carlo estimators. Some recent alternative techniques to estimate transport coefficients are also discussed.

Published
2022

37. Complete Positivity Violation in Higher-order Quantum Adiabatic Elimination

Author

Tokieda, Masaaki, Elouard, Cyril, Sarlette, Alain, Rouchon, Pierre, MATHematics for MatERIALS (MATHERIALS), Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Traitement optimal de l'information avec des dispositifs quantiques (QINFO), Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-École normale supérieure de Lyon (ENS de Lyon)-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon-Université Grenoble Alpes (UGA)-Inria Lyon, Institut National de Recherche en Informatique et en Automatique (Inria), QUANTum Information Circuits (QUANTIC), Mines Paris - PSL (École nationale supérieure des mines de Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Sorbonne Université (SU)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire de physique de l'ENS - ENS Paris (LPENS), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)-Département de Physique de l'ENS-PSL, École normale supérieure - Paris (ENS-PSL), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-École normale supérieure - Paris (ENS-PSL), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)-Département de Physique de l'ENS-PSL, Université Paris sciences et lettres (PSL), Centre Automatique et Systèmes (CAS), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), ANR-18-CE47-0005,HAMROQS,Modèles réduits haute précision pour systèmes quantiques ouverts(2018), and European Project: 884762,Q-Feedback

Subjects
Lindblad master equation, Quantum Physics, Quantum information, FOS: Physical sciences, Completely positive map, Jaynes-Cummings Hamiltonian, Open quantum systems, [PHYS.QPHY]Physics [physics]/Quantum Physics [quant-ph], [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], Adiabatic elimination, Qubit, Quantum harmonic oscillator, Quantum Physics (quant-ph), Invariant manifold
Abstract

When a composite Lindblad system consists of weakly coupled sub-systems with fast and slow timescales, the description of slow dynamics can be simplified by discarding fast degrees of freedom. This model reduction technique is called adiabatic elimination. While second-order perturbative expansion with respect to the timescale separation has revealed that the evolution of a reduced state is completely positive, this paper presents an example exhibiting complete positivity violation in the fourth-order expansion. Despite the non-uniqueness of slow dynamics parametrization, we prove that complete positivity cannot be ensured in any parametrization. The violation stems from correlation in the initial state., 6 pages, submitted to IFAC for possible publication

Published
2022

38. Ab initio Canonical Sampling based on Variational Inference

Author

Aloïs Castellano, François Bottin, Johann Bouchet, Antoine Levitt, Gabriel Stoltz, DAM Île-de-France (DAM/DIF), Direction des Applications Militaires (DAM), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA), Laboratoire Matière sous Conditions Extrêmes (LMCE), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Direction des Applications Militaires (DAM), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay, Département d'Etudes des Combustibles (DEC), CEA-Direction des Energies (ex-Direction de l'Energie Nucléaire) (CEA-DES (ex-DEN)), Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC), MATHematics for MatERIALS (MATHERIALS), École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC)-Inria de Paris, and Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)

Subjects
Condensed Matter - Materials Science, Materials Science (cond-mat.mtrl-sci), FOS: Physical sciences, [PHYS.COND.CM-SM]Physics [physics]/Condensed Matter [cond-mat]/Statistical Mechanics [cond-mat.stat-mech]
Abstract

International audience; Finite temperature calculations, based on ab initio molecular dynamics (AIMD) simulations, are a powerful tool able to predict material properties that cannot be deduced from ground state calculations. However, the high computational cost of AIMD limits its applicability for large or complex systems. To circumvent this limitation we introduce a new method named Machine Learning Assisted Canonical Sampling (MLACS), which accelerates the sampling of the Born--Oppenheimer potential surface in the canonical ensemble. Based on a self-consistent variational procedure, the method iteratively trains a Machine Learning Interatomic Potential to generate configurations that approximate the canonical distribution of positions associated with the ab initio potential energy. By proving the reliability of the method on anharmonic systems, we show that the method is able to reproduce the results of AIMD with an ab initio accuracy at a fraction of its computational cost.

Published
2022

39. Numerical stability and efficiency of response property calculations in density functional theory

Author

Eric Cancès, Michael F. Herbst, Gaspard Kemlin, Antoine Levitt, Benjamin Stamm, Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC), MATHematics for MatERIALS (MATHERIALS), École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Rheinisch-Westfälische Technische Hochschule Aachen University (RWTH), Institut für Angewandte Analysis und Numerische Simulation [Stuttgart] (IANS), Universität Stuttgart [Stuttgart], This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 810367). M.H. and B.S. acknowledge funding by the Federal Ministry of Education and Research (BMBF) and the Ministry of Culture and Science of the German State of North Rhine-Westphalia (MKW) under the Excellence Strategy of the Federal Government and the Länder., and European Project: 810367,EMC2(2019)

Subjects
Condensed Matter - Materials Science, FOS: Mathematics, [PHYS.COND.CM-MS]Physics [physics]/Condensed Matter [cond-mat]/Materials Science [cond-mat.mtrl-sci], Materials Science (cond-mat.mtrl-sci), FOS: Physical sciences, Statistical and Nonlinear Physics, Mathematics - Numerical Analysis, Numerical Analysis (math.NA), Mathematical Physics, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

International audience Response calculations in density functional theory aim at computing the change in ground-state density induced by an external perturbation. At finite temperature these are usually performed by computing variations of orbitals, which involve the iterative solution of potentially badly-conditioned linear systems, the Sternheimer equations. Since many sets of variations of orbitals yield the same variation of density matrix this involves a choice of gauge. Taking a numerical analysis point of view we present the various gauge choices proposed in the literature in a common framework and study their stability. Beyond existing methods we propose a new approach, based on a Schur complement using extra orbitals from the self-consistent-field calculations, to improve the stability and efficiency of the iterative solution of Sternheimer equations. We show the success of this strategy on nontrivial examples of practical interest, such as Heusler transition metal alloy compounds, where savings of around 40% in the number of required cost-determining Hamiltonian applications have been achieved.

Published
2022

40. Modified-Operator Method for the Calculation of Band Diagrams of Crystalline Materials

Author

Cancès, Eric, Hassan, Muhammad, Vidal, Laurent, École des Ponts ParisTech (ENPC), Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), MATHematics for MatERIALS (MATHERIALS), École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Sorbonne Université (SU), This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No. 810367)., and European Project: 810367,EMC2(2019)

Subjects
Condensed Matter - Materials Science, Numerical Analysis, Materials Science (cond-mat.mtrl-sci), FOS: Physical sciences, Periodic Schrödinger Operators, 65N25, 65F15 (Primary) 65Z05 (Secondary), Numerical Analysis (math.NA), Eigenvalue Problems on the Torus, 2020 Mathematics Subject Classification. 65N25, 65F15, 65Z05, FOS: Mathematics, [PHYS.COND.CM-MS]Physics [physics]/Condensed Matter [cond-mat]/Materials Science [cond-mat.mtrl-sci], Error Analysis, Mathematics - Numerical Analysis, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

In solid state physics, electronic properties of crystalline materials are often inferred from the spectrum of periodic Schr\"odinger 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 this article, we study 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, we introduce here a systematic formulation of this operator modification approach. We derive a priori error estimates for the resulting energy bands and we show that these bands are periodic and can be made arbitrarily smooth (away from band crossings) by adjusting suitable parameters in the operator modification approach. Numerical experiments involving a toy model in 1D, graphene in 2D, and silicon in 3D validate our theoretical results and showcase the efficiency of the operator modification approach., Comment: 40 pages, 12 figures

Published
2022

41. Adaptive force biasing algorithms: New convergence results and tensor approximations of the bias

Author

Virginie Ehrlacher, Tony Lelièvre, Pierre Monmarché, Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC), MATHematics for MatERIALS (MATHERIALS), École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), This work was supported by the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement number 614492 and under the European Union’s Horizon 2020 Research and Innovation Programme, ERC Grant Agreement number 810367, project EMC2. It was also supported by the ANR JCJC project COMODO (ANR-19-CE46-0002) and the ANR Project EFI (ANR-17-CE40-0030) of he French National Research Agency., ANR-19-CE46-0002,COMODO,Systèmes de diffusion croisée sur des domaines en mouvement(2019), ANR-17-CE40-0030,EFI,Entropie, flots, inégalités(2017), European Project: 9110745(1991), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), and École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC)

Subjects
Statistics and Probability, Importance sampling, Probability (math.PR), Monte Carlo methods, Numerical Analysis (math.NA), Molecular dynamics, 65C05, 65N12 [MSC class (2010)], [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 65C05, 65N12, Tensor, FOS: Mathematics, MSC class (2010): 65C05, 65N12, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Mathematics - Numerical Analysis, Free energy, Statistics, Probability and Uncertainty, Mathematics - Probability
Abstract

International audience We analyze and propose variants of the Adaptive Biasing Force method. First, we prove the convergence of a version of the algorithm where the biasing force is estimated using a weighted occupation measure, with an explicit asymptotic variance. Second, we propose a new flavour of the algorithm adapted to high dimensional reaction coordinates, for which the standard approaches suffer from the curse of dimensionality. More precisely, the free energy is approximated by a sum of tensor products of one-dimensional functions. The consistency of the tensor approximation is established. Numerical experiments on 5-dimensional reaction coordinates demonstrate that the method is indeed able to capture correlations between them.

Published
2022

42. On basis set optimisation in quantum chemistry

Author

Cancès, Eric, Dusson, Geneviève, Kemlin, Gaspard, Vidal, Laurent, MATHematics for MatERIALS (MATHERIALS), Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), École des Ponts ParisTech (ENPC), Laboratoire de Mathématiques de Besançon (UMR 6623) (LMB), Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC), This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement EMC2 No 810367). This work was supported by the French ‘Investissements d’Avenir’ program, project Agence Nationale de la Recherche (ISITE-BFC) (contract ANR-15-IDEX-0003)., and ANR-15-IDEX-0003,BFC,ISITE ' BFC(2015)

Subjects
[CHIM.THEO]Chemical Sciences/Theoretical and/or physical chemistry, FOS: Mathematics, Numerical Analysis (math.NA), Mathematics - Numerical Analysis, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

In this article, we propose general criteria to construct optimal atomic centered basis sets in quantum chemistry. We focus 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 are then 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.; Nous proposons dans cet article des critères généraux pour construire des bases atomiques localisées optimales en chimie quantique. Nous nous intéressons en particulier à deux critères: l’un basé sur la matrice densité à un corps du système, l’autre basé sur l’énergie de l’état fondamental du système. Les performances de ces deux critères sont évaluées et comparées sur un problème aux valeurs propres paramétré, correspondant à une version simplifiée du problème de dissociation de l’état fondamental d’une molécule diatomique.

Published
2022

43. Eyring-Kramers exit rates for the overdamped Langevin dynamics: the case with saddle points on the boundary

Author

Lelièvre, Tony, Peutrec, Dorian Le, Nectoux, Boris, Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC), MATHematics for MatERIALS (MATHERIALS), École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut Denis Poisson (IDP), Université d'Orléans (UO)-Université de Tours (UT)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Mathématiques Blaise Pascal (LMBP), Centre National de la Recherche Scientifique (CNRS)-Université Clermont Auvergne (UCA), Leverhulme Trust professorship, ANR-19-CE40-0010,QuAMProcs,Analyse Quantitative de Processus Metastables(2019), ANR-17-CE40-0030,EFI,Entropie, flots, inégalités(2017), European Project: 810367,EMC2(2019), Lelièvre, Tony, Analyse Quantitative de Processus Metastables - - QuAMProcs2019 - ANR-19-CE40-0010 - AAPG2019 - VALID, Entropie, flots, inégalités - - EFI2017 - ANR-17-CE40-0030 - AAPG2017 - VALID, and Extreme-Scale Mathematically-based Computational Chemistry - EMC2 - 2019-09-01 - 2026-02-26 - 810367 - VALID

Subjects
The exit problem, Mathematics - Analysis of PDEs, Overdamped Langevin, Semi-classical analysis, Probability (math.PR), FOS: Mathematics, Eyring-Kramers law, [MATH.MATH-NA] Mathematics [math]/Numerical Analysis [math.NA], Mathematics - Probability, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Analysis of PDEs (math.AP)
Abstract

Let $(X_t)_{t\ge 0}$ be the stochastic process solution to the overdamped Langevin dynamics $$dX_t=-\nabla f(X_t) \, dt +\sqrt h \, dB_t$$ and let $\Omega \subset \mathbb R^d $ be the basin of attraction of a local minimum of $f: \mathbb R^d \to \mathbb R$. Up to a small perturbation of $\Omega$ to make it smooth, we prove that the exit rates of $(X_t)_{t\ge 0}$ from $\Omega$ through each of the saddle points of $f$ on $\partial \Omega$ can be parametrized by the celebrated Eyring-Kramers laws, in the limit $h \to 0$. This result provides 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.

Published
2022

44. Mobility estimation for Langevin dynamics using control variates

Author

Pavliotis, G. A., Stoltz, G., Vaes, U., Imperial College London, Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC), MATHematics for MatERIALS (MATHERIALS), École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), ANR-21-CE40-0006,SINEQ,Simulation de dynamiques stochastiques hors d'équilibre(2021), and European Project: 810367,EMC2(2019)

Subjects
Statistical Mechanics (cond-mat.stat-mech), FOS: Mathematics, FOS: Physical sciences, Mathematical Physics (math-ph), Numerical Analysis (math.NA), Mathematics - Numerical Analysis, [MATH]Mathematics [math], 65C30, 35Q84, 82C31, 82M22, Mathematical Physics, Condensed Matter - Statistical Mechanics
Abstract

The scaling of the mobility of two-dimensional Langevin dynamics in a periodic potential as the friction vanishes is not well understood for non-separable potentials. Theoretical results are lacking, and numerical calculation of the mobility in the underdamped regime is challenging because the computational cost of standard Monte Carlo methods is inversely proportional to the friction coefficient, while deterministic methods are ill-conditioned. In this work, we propose a new variance-reduction method based on control variates for efficiently estimating the mobility of Langevin-type dynamics. We provide bounds on the bias and variance of the proposed estimator, and illustrate its efficacy through numerical experiments, first in simple one-dimensional settings and then for two-dimensional Langevin dynamics. Our results corroborate previous numerical evidence on the scaling of the mobility in the low friction regime for a simple non-separable potential.

Published
2022

45. A robust and efficient line search for self-consistent field iterations

Author

Michael F. Herbst, Antoine Levitt, Rheinisch-Westfälische Technische Hochschule Aachen University (RWTH), MATHematics for MatERIALS (MATHERIALS), Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), École des Ponts ParisTech (ENPC), RWTH Aachen University, Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), and École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC)

Subjects
Condensed Matter - Materials Science, Numerical Analysis, Physics and Astronomy (miscellaneous), Applied Mathematics, MathematicsofComputing_NUMERICALANALYSIS, Materials Science (cond-mat.mtrl-sci), FOS: Physical sciences, Computational Physics (physics.comp-ph), Computer Science Applications, Computational Mathematics, Condensed Matter::Materials Science, Modeling and Simulation, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Physics::Chemical Physics, Physics - Computational Physics, ComputingMethodologies_COMPUTERGRAPHICS
Abstract

International audience; We propose a novel adaptive damping algorithm for the self-consistent field (SCF) iterations of Kohn-Sham density-functional theory, using a backtracking line search to automatically adjust the damping in each SCF step. This line search is based on a theoretically sound, accurate and inexpensive model for the energy as a function of the damping parameter. In contrast to usual damped SCF schemes, the resulting algorithm is fully automatic and does not require the user to select a damping. We successfully apply it to a wide range of challenging systems, including elongated supercells, surfaces and transition-metal alloys.

Published
2022

46. Structure-preserving reduced order model for parametric cross-diffusion systems

Author

Dabaghi, Jad, Ehrlacher, Virginie, Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC), MATHematics for MatERIALS (MATHERIALS), École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), and ANR-19-CE46-0002,COMODO,Systèmes de diffusion croisée sur des domaines en mouvement(2019)

Subjects
Finite volumes, Proper orthogonal decomposition, Cross-diffusion systems, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

In this work, we construct a structure-preserving reduced-order model for the resolution of parametric cross-diffusion systems. Cross-diffusion systems model the evolution of the concentrations or volumic fractions of mixtures composed of different species and often read as nonlinear degenerated parabolic partial differential equations whose numerical resolutions are highly expensive from a computational point of view. We are interested here in cross-diffusion systems which exhibit a so-called entropic structure, in the sense that they can be formally written as gradient flows of a certain entropy functional which is actually a Lyapunov functional of the system. In this work, we propose a new reduced-order modelling method, based on a reduced basis paradigm, in order to accelerate the resolution of parameter-dependent cross-diffusion systems, which preserves, at the level of the reduced-order model, the main mathematical properties of the continuous solution, namely mass conservation, non-negativeness, preservation of the volume-filling property and entropy-entropy dissipation relationship. The theoretical advantages of our approach are confirmed by several numerical experiments.; Dans ce travail, nous construisons un modèle réduit préservant la structure lors de la résolution d’un système de diffusion croisée. Les systèmes de diffusion croisée modélisent l’évolution des concentrations ou fractions volumiques locales de différentes espèces chimiques constituant un mélange et s’interprètent souvent comme des systèmes d’équations aux dérivées partielles non linéaires dégénérées dont la résolution numérique est très coûteuse. Nous nous intéressons dans ce travail à des systèmes de diffusion croisée ayant une structure entropique, dans le sens où ils peuvent s’écrire comment le flot de gradient d’une certaine fonctionnelle d’entropie; cette dernière étant une fonction de Lyapunov pour ce système. Dans ce travail, nous proposons une nouvelle méthode de réduction de modèle, basée sur une approche de type base réduite, afin d’accélérer la résolution de systèmes de diffusion croisée dépendant de paramètres. Cette résolution préserve au niveau réduit les principales propriétés mathématiques imposées par la solution au niveau continu à savoir la conservation de la masse, la positivité de la solution, la contrainte de volume, et la dissipation d’entropie. Les essais numériques confirmes les avantages de notre approche.

Published
2022

47. A priori error analysis of linear and nonlinear periodic Schrödinger equations with analytic potentials

Author

Cancès, Eric, Kemlin, Gaspard, Levitt, Antoine, Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC), MATHematics for MatERIALS (MATHERIALS), École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), and European Project: 810367,EMC2(2019)

Subjects
FOS: Mathematics, Numerical Analysis (math.NA), Mathematics - Numerical Analysis, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

This paper is concerned with the numerical analysis of linear and nonlinear Schr{\"o}dinger equations with analytic potentials. While the regularity of the potential (and the source term when there is one) automatically conveys to the solution in the linear cases, this is no longer true in general in the nonlinear case. We also study the rate of convergence of the planewave (Fourier) discretization method for computing numerical approximations of the solution.

Published
2022

48. Linear elliptic homogenization for a class of highly oscillating non-periodic potentials

Author

Goudey, Rémi, Bris, Claude Le, Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC), MATHematics for MatERIALS (MATHERIALS), École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), and The research of the second author is partially supported by ONR under Grant N00014-20-1-2691 and by EOARD under Grant FA8655-20-1-7043.

Subjects
Mathematics - Analysis of PDEs, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Analysis of PDEs (math.AP)
Abstract

We consider an homogenization problem for the second order elliptic equation $- \Delta u^{\varepsilon} + \dfrac{1}{\varepsilon} V(./\varepsilon) u^{\varepsilon} + \nu u^{\varepsilon} =f$ when the highly oscillatory potential $V$ belongs to a particular class of non-periodic potentials. We show the existence of an adapted corrector and prove the convergence of $u^{\varepsilon}$ to its homogenized limit., Comment: 39 pages, 3 figures

Published
2022

49. A hybrid parareal Monte Carlo algorithm for parabolic problems

Author

Dabaghi, Jad, Maday, Yvon, Zoia, Andrea, Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Commissariat à l'énergie atomique et aux énergies alternatives (CEA), Université Paris-Saclay, Ecole Supérieure d'Ingénieurs Léonard de Vinci (ESILV), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), This project has received funding from the ANR project 'Ciné-Para' (ANR-15-CE23-0019). This work has received funding from the European Research Council (ERC) under theEuropean Union’s Horizon 2020 research and innovation program (grant agreement No 810367- project EMC2) (YM) and from the European High Performance Computing Joint Undertaking (EuroHPC JU) under the European Union’s Horizon 2020 research and innovationprogram (grant agreement No 955701 - project TIME-X) (YM, ANR-15-CE23-0019,CINE-PARA,Méthodes de parallélisation pour cinétiques complexes(2015), European Project: 810367,EMC2(2019), Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC), MATHematics for MatERIALS (MATHERIALS), École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC)-Inria de Paris, and Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)

Subjects
Monte Carlo method, Computational Mathematics, time-dependent problems, predictor-corrector, Applied Mathematics, parareal-in-time algorithm, Computer Science::Numerical Analysis, Galerkin schemes, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

International audience; In this work, we propose a hybrid Monte Carlo/deterministic ``parareal-in-time'' approachdevoted to acceleratingMonte Carlo simulations over massively parallel computing environments for the simulation of time-dependent problems.This parareal approach iterates on two different solvers: a low-cost “coarse” solver based on a very cheap deterministic Galerkin scheme and a “fine” solver based on a high-fidelity Monte Carlo resolution.In a set of benchmark numerical experiments based on atoy model concerning the time-dependent diffusion equation, we compare our hybrid parareal strategy with a standard full Monte Carlo solution.In particular, we show that for a large number of processors, our hybrid strategy significantly reduces the computational time of the simulation while preserving its accuracy. The convergence properties of the proposed Monte Carlo/deterministic parareal strategy are also discussed.

Published
2023

50. Some Remarks on a Coupling Method for the Practical Computation of Homogenized Coefficients

Author

Claude Le Bris, Frédéric Legoll, Olga Gorynina, Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC), MATHematics for MatERIALS (MATHERIALS), École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Modélisation et expérimentation multi-échelle pour les solides hétérogènes (Multi-échelle), Laboratoire Navier (NAVIER UMR 8205), École des Ponts ParisTech (ENPC)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel-École des Ponts ParisTech (ENPC)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel, Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), and École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC)

Subjects
Coupling, Applied Mathematics, Computation, Elliptic pdes, 010103 numerical & computational mathematics, 01 natural sciences, Computational Mathematics, Optimization and Control (math.OC), FOS: Mathematics, Applied mathematics, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Granularity, 0101 mathematics, Mathematics - Optimization and Control, Mathematics
Abstract

International audience; We numerically investigate, and improve upon, a computational approach originally introduced in [Cottereau, IJNME 2013] which aims at evaluating the effective coefficient of a medium modelled by a highly oscillatory coefficient. This computational approach is based on a Arlequin type coupling. It combines 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 the actual heterogeneous medium using a purely homogeneous medium. We present here a mathematical formalization of the approach along with various improvements of the algorithms, in order to obtain a procedure as efficient as possible. Representative numerical results demonstrate the added value of our approach in comparison to the original approach.

Published
2021

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