ANR MEGAS

MEthodes Géométriques et échantillonnage: Application à la Simulation moléculaire

Geometric methods and sampling: applications to molecular simulation

2009 - 2012

Geometric methods and sampling: applications to molecular simulation

2009 - 2012

This project is supported by the Agence Nationale de la Recherche, under grant ANR-09-BLAN-0216-01

Scientific scope :

Today's and tomorrow's materials are developed at a microscopic level, using information at ever smaller time and space scales. In many cases, these materials are described at the atomic level, based upon the physical theories of quantum physics and statistical physics. Recent examples of novel materials include: carbon nanotubes, which could be the building blocks of a whole generation of electronic devices exhibiting new mechanical, thermal and electrical properties ; in silico design of new drugs, where the chemical properties of new molecules are simulated on a computer instead of resorting to expensive synthesis ; surface chemistry in fuel cells.

These examples are only some of the most prominent instances of a now ubiquituous technique: molecular simulation. However, the methods developed originally in this field may also have applications in areas very different in nature, ranging from astronomy to financial mathematics.

The remarkable accuracy of numerical simulations at the microscopic scale has a counterpart: their numerical costs. The typical length and time scales now tractable are of the order of several billions of atoms whose dynamics can be followed for a few nanoseconds. In any case, macroscopic time and length scales are beyond reach, even with the considerable increase in computational power witnessed in the past few years. On the other hand, it is often the case that new physics, and that the identification of new phenomena both arise from the consideration of larger systems simulated on longer times. So, it is no surprise that molecular simulations represent today a huge part of the CPU time used by scientists.

However, this field is mostly unexplored to date by numerical analysts and experts in scientific computing. We believe that molecular simulations are the source of many new and interesting topics in numerical analysis and scientific computing. A mathematical understanding of the (hierarchy of) models and the problems arising in their simulation is one of the key for reducing the time/scale gap with the macroscopic world. A mathematical understanding is also important for validating and assessing the precision of the methods used.

The aim of this project is to investigate various aspects of the mathematical questions raised by molecular simulations, including : computation of free energy differences and improved sampling ; highly oscillating dynamics ; coarse-graining of dynamics ; steady-state non equilibrium sampling ; Quantum Monte-Carlo methods ; discrete to continuum coupling methods.

Partners:

- Molecular and multiscale modeling group at Ecole des Ponts (coordinator : Tony Lelièvre)
- IPSO team-project at INRIA Rennes (person in charge : Philippe Chartier)
- Mathias Rousset at INRIA Lille (person in charge : Mathias Rousset)
- Chris Chipot at CNRS Nancy (person in charge : Chris Chipot)

Some events of the ANR:

- Rencontres EDP/Probas, 10ème journée, vendredi 23 octobre 2009, Institut Henri Poincaré.
- Workshop Hybrid simulations of dynamical systems and applications to molecular dynamics, Institut Henri Poincaré, 27 - 30 septembre 2010.
- Journées thématiques du groupe SMAI - MAS: Stochastic processes and metastability, Ecole des Ponts ParisTech, 21 - 23 septembre 2011.

Journal articles:

- C. Le Bris et F. Legoll, Integrators for highly oscillatory Hamiltonian systems: an homogenization approach, Discrete and Continuous Dynamical Systems - B, vol. 13 (2), 347-373 (2010).
- B. Jourdain, T. Lelièvre et R. Roux, Existence, uniqueness and convergence of a particle approximation for the Adaptive Biasing Force process, Mathematical Modelling and Numerical Analysis, 44, 831-865, (2010).
- X. Blanc, C. Le Bris, F. Legoll et C. Patz, Finite-temperature coarse-graining of one-dimensional models: mathematical analysis and computational approaches, Journal of Nonlinear Science, vol.20 (2), 241-275, (2010).
- B. Dickson, F. Legoll, T. Lelièvre, G. Stoltz et P. Fleurat-Lessard, Free energy calculations: An efficient adaptive biasing potential method, J. Phys. Chem. B, 114, 5823-5830, (2010).
- J.-B. Maillet, E. Bourasseau, L. Soulard, J. Clerouin et G. Stoltz, Constant entropy sampling and release waves of shock compressions, Phys. Rev. E 80, 021135 (2009).
- P. Plechac et M. Rousset, Implicit Mass-Matrix Penalization of Hamiltonian dynamics with application to exact sampling of stiff systems. SIAM MMS, 8(2), (2009).
- C. Chipot, T. Lelièvre et K. Minoukadeh, Potential of mean force calculations: a multiple-walker adaptive biasing force approach, Journal of Chemical Theory and Computation, 6(4), 1008-1017, (2010).
- X. Blanc, F. Legoll, C. Le Bris et T. Lelièvre, Beyond multiscale and multiphysics: Multimaths for model coupling, Networks and Heterogeneous Media, 5(3), 423-460, (2010).
- F. Legoll et T. Lelièvre, Effective dynamics using conditional expectations, Nonlinearity, 23, 2131-2163, (2010).
- F. Cérou, A. Guyader, T. Lelièvre et D. Pommier, A multiple replica approach to simulate reactive trajectories, Journal of Chemical Physics 134, 054108, (2011).
- M. Rousset, On a probabilistic interpretation of shape derivatives of Dirichlet groundstates with application to Fermion nodes. M2AN, 44: 977-995, (2010).
- P. Chartier, J.M. Sanz-Serna and A. Murua, Higher-order averaging, formal series and numerical integration I: B-series, FOCM, Vol. 10, No. 6, (2010).
- T. Lelièvre et K. Minoukadeh Long-time convergence of an Adaptive Biasing Force method : the bi-channel case, à paraître dans Archive for Rational Mechanics and Analysis.
- C. Chipot et T. Lelièvre, Enhanced sampling of multidimensional free-energy landscapes using adaptive biasing forces, à paraître dans SIAM Journal of Applied Mathematics.
- N. Chopin, T. Lelièvre et G. Stoltz, Free energy methods for efficient exploration of mixture posterior densities, à paraître dans Statistics and Computing.
- A. Iacobucci, F. Legoll, S. Olla, G. Stoltz, Thermal conductivity of the Toda lattice with conservative noise, J. Stat. Phys. 140(2), 336-348, 2010.
- M. Dobson, C. Le Bris et F. Legoll, Symplectic schemes for highly oscillatory Hamiltonian systems with varying fast frequencies (Intégrateurs symplectiques pour des systèmes Hamiltoniens hautement oscillants avec fréquences rapides variables), C. R. Acad. Sci. Paris, Série I, vol. 348 (17-18), 1033-1038 (2010).
- T. Lelièvre, M. Rousset et G. Stoltz, Langevin dynamics with constraints and computation of free energy differences, http://hal.archives-ouvertes.fr/hal-00495517 .
- C. Le Bris, T. Lelièvre, M. Luskin et D. Perez, A mathematical formalization of the parallel replica dynamics, http://hal.archives-ouvertes.fr/hal-00596161/fr/ .
- P. Chartier, J.M. Sanz-Serna et A. Murua, Higher-order averaging, formal series and numerical integration II: the quasi-periodic case, submitted.
- M.P. Calvo, P. Chartier, J.M. Sanz-Serna et A. Murua, Numerical experiments with the stroboscopic method, submitted.
- X. Dai, C. Le Bris, F. Legoll et Y. Maday, Symmetric parareal algorithms for Hamiltonian systems, http://hal.archives-ouvertes.fr/hal-00541166/fr/ .
- M. Dobson, C. Le Bris et F. Legoll, Symplectic schemes for highly oscillatory Hamiltonian systems: the homogenization approach beyond the constant frequency case, http://arxiv.org/abs/1008.1030 .
- R. Joubaud and G. Stoltz, Nonequilibrium shear viscosity computations with Langevin dynamics, http://fr.arxiv.org/abs/1106.0633 .
- C. Bernardin and G. Stoltz, Anomalous diffusion for a class of systems with two conserved quantities, http://hal.archives-ouvertes.fr/hal-00593617 .
- E. Bourasseau, J.-B. Maillet, N. Desbiens and G. Stoltz, Microscopic calculations of Hugoniot curves of neat TATB and of its detonation products, http://hal.archives-ouvertes.fr/hal-00595191 .

- F. Legoll et T. Lelièvre, Some remarks on free energy and coarse-graining, in Numerical Analysis and Multiscale Computations, Lect. Notes Comput. Sci. Eng., Vol. 82, Springer 2011, http://hal.archives-ouvertes.fr/hal-00511221 .
- M.P.
Calvo, P. Chartier, J.M. Sanz-Serna et A. Murua, A stroboscopic
numerical method for highly oscillatory problems, in Numerical
Analysis and Multiscale Computations, Lect. Notes Comput. Sci. Eng.,
Vol. 82, Springer 2011, 73-87.
- T. Lelièvre, M. Rousset et G. Stoltz, Free energy computations: A mathematical perspective, Imperical College Press, 2010.