# COmputational Statistics and MOlecular Simulation

# 2014 – 2019

# This project is supported by the Agence Nationale de la Recherche, under grant ANR-14-CE23-0012

# Scientific scope:

This project aims at developing numerical techniques dedicated to the sampling of high-dimensional probability measure describing a system of interest. There are two application fields of interest: computational statistical physics (a field also known as molecular simulation), and computational statistics. These two fields share some common history, but it seems that, in view of the quite recent specialization of the scientists and the techniques used in these respective fields, the communication between molecular simulation and computational statistics is not as intense as it should be.

We believe that there are therefore many opportunities in considering both fields at the same time: in particular, the adaption of a successful simulation technique from one field to the other requires first some abstraction process where the features specific to the original field of application are discarded and only the heart of the method is kept. Such a cross-fertilization is however only possible if the techniques developed in a specific field are sufficiently mature: this is why some fundamental studies specific to one of the application fields are still required. Our belief is that the embedding in a more general framework of specific developments in a given field will accelerate and facilitate the diffusion to the other field.

The main topics are the following:

- Free energy based adaptive importance sampling;
- Sampling reactive trajectories;
- Dynamical behavior of Metropolis-Hastings algorithms;
- Introducing non-reversible perturbations to enhance the exploration power of equilibrium dynamics.

# Partners:

- CERMICS, Ecole des Ponts (coordinator: Gabriel Stoltz)
- LTCI, CNRS-Telecom (local coordinator: Gersende Fort)
- ASPI, Inria Rennes (local coordinator: Frédéric Cerou)
- IBPC (local coordinator: Jérôme Hénin)

# Some events related to the project and (co)organized by members of it:

- Kick-off meeting, Paris, March 2nd, 2015
- One-day workshop Interactions between Monte Carlo methods and optimization algorithms (26 November 2015)
- workshop COSMOS (2-5 February 2016)
- Thematic Cycle on Monte-Carlo techniques (October 2015-June 2016); in particular the closing conference
- workshop on Numerical methods for nonequilibrium systems (25-27 April 2017, IHP trimester on Stochastic dynamics out of equilibrium)
- Workshop “Computational Statistics and Molecular Simulation: A Practical Cross-Fertilization” (BIRS-Oaxaca, November 12-16, 2018)

# Softwares:

# Publications of the project (by topics as listed in the proposal):

### Review article

- T. Lelievre and G. Stoltz, Partial differential equations and stochastic methods in molecular dynamics, Acta Numerica
**25**, 681-880 (2016) (HAL)

### Adaptive importance sampling: convergence, algorithmic improvements and popularization

- G. Fort, B. Jourdain, T. Lelievre and G. Stoltz, Self-Healing Umbrella Sampling: Convergence and efficiency,
*Stat. Comput.***27**(1) 147-168 (2017) (HAL) - G. Fort, B. Jourdain, T. Lelievre and G. Stoltz, Convergence and efficiency of adaptive importance sampling techniques with partial biasing,
*J. Stat. Phys.***171**(2), 220-268 (2018) (HAL) - A. Lesage, T. Lelièvre, G. Stoltz and J. Hénin, Smoothed biasing forces yield unbiased free energies with the extended-system adaptive biasing force method,
*J. Phys. Chem. B***121**(15), 3676–3685 (2017) (HAL) - B. Jourdain, T. Lelièvre and P.-A. Zitt, Convergence of metadynamics: discussion of the adiabatic hypothesis,
*arXiv preprint***1904.08677**(2019) (HAL)

### Sampling reactive trajectories

- F. Cerou and A. Guyader, Fluctuation Analysis of Adaptive Multilevel Splitting,
*Annals of Applied Probability,***26**(6), 3319-3380 (2016) (HAL) - F. Cerou, B. Delyon, A. Guyader and M. Rousset, A central limit theorem for Fleming-Viot particle systems, to appear in
*Annales de l’IHP (Probability and Statistics)*(2017) (HAL) - F. Cérou, A. Guyader and M. Rousset, Adaptive Multilevel Splitting: Historical Perspective and Recent Results,
*Chaos***29**, 043108 (2019) (HAL) - F. Cérou, B. Delyon, A. Guyader and M. Rousset, On the asymptotic normality of Adaptive Multilevel Splitting,
*SIAM/ASA Journal on Uncertainty Quantification*,**7**(1), 1–30 (2019) (HAL) - Q. Du and A. Guyader, Variance Estimation in Adaptive Sequential Monte Carlo, arXiv preprint 1909.13602 (2019) (preprint)

### Dynamical behavior of Metropolis-Hastings algorithms

- M. Fathi and G. Stoltz, Improving dynamical properties of stabilized discretizations of overdamped Langevin dynamics,
*Numer. Mathematik***136**(2), 545–602 (2017) (HAL) - A. Durmus and E. Moulines, Non-asymptotic convergence analysis for the Unadjusted Langevin Algorithm,
*The Annals of Applied Probability*,**27**(3), 1551-1587 (2017) (HAL) - A. Durmus, S. Le Corff, E. Moulines, G. O. Roberts, Optimal scaling of the Random Walk Metropolis algorithm under Lp mean differentiability,
*Journal of Applied Probability*,**54**(4), 1233-1260 (2017) (HAL) - T. Lelièvre, M. Rousset and G. Stoltz, Hybrid Monte Carlo methods for sampling probability measures on submanifolds,
*Numer. Math.***143**(2), 379-421 (2019) (HAL) - D. Chafaï, G. Ferré and G. Stoltz, Coulomb gases under constraint: some theoretical and numerical results,
*arXiv preprint***1907.05803**(2019) (HAL) - A. Durmus and E. Moulines, High-dimensional Bayesian inference via the unadjusted Langevin algorithm,
*Bernoulli***25**(4A), 2854-2882 (2019) (HAL)

### Variance reduction for non-reversible dynamics and study of nonequilibrium dynamics in general:

- R. Assaraf, B. Jourdain, T. Lelievre, R. Roux, Computation of sensitivities for the invariant measure of a parameter dependent diffusion,
*Stochastics and Partial Differential Equations: Analysis and Computations,***6**(2)*,*125–183 (2018) (HAL) - G. Stoltz and Z. Trstanova, Langevin dynamics with general kinetic energies,
*Multiscale Model. Simul.*,**16**(2), 777-806 (2018) (HAL) - S. Redon, G. Stoltz and Z. Trstanova, Error Analysis of Modified Langevin Dynamics,
*J. Stat. Phys.***164**(4), 735-771 (2016) (HAL) - A. Iacobucci, S. Olla and G. Stoltz, Convergence rates for nonequilibrium Langevin dynamics,
*Ann. Math. Quebec***43**(1), 73-98 (2019) (HAL) - J. Roussel and G. Stoltz, Spectral methods for Langevin dynamics and associated error estimates,
*M2AN***52**(3), 1051-1083 (2018) (HAL) - G. Stoltz and E. Vanden-Eijnden, Longtime convergence of the Temperature-Accelerated Molecular Dynamics Method,
*Nonlinearity***31**(8), 3748-3769 (2018) (HAL) - G. Ferré and G. Stoltz, Error estimates on ergodic properties of discretized Feynman-Kac semigroups,
*Numer. Math.***143**(2), 261–313 (2019) (HAL) - J. Roussel and G. Stoltz, A perturbative approach to control variates in molecular dynamics,
*Multiscale Model. Simul.***17**(1), 552–591 (2019) (HAL) - G. Ferré, M. Rousset and G. Stoltz, More on the long time stability of Feynman-Kac semigroups,
*arXiv preprint***1807.00390**(2018) (HAL) - G. Ferré and G. Stoltz, Large deviations of the empirical measure of diffusions in fine topologies with applications,
*arXiv preprint***1906.09411**(2019) (HAL) - P. Plechac, G. Stoltz and T. Wang, Convergence of the likelihood ratio method for linear response of non-equilibrium stationary states,
*arxiv preprint***1910.02479**(2019) *Stochastic Dynamics out of Equilibrium,*Springer Proceedings in Mathematics & Statistics, volume 282 [link]

Edited by G. Giacomin, S. Olla, E. Saada, H. Spohn and G. Stoltz, following up on the trimester we organized at Institut Henri Poincare.

### Computational statistics with techniques of molecular dynamics:

- B. Leimkuhler, M. Sachs and G. Stoltz, Hypocoercivity properties of adaptive Langevin dynamics,
*arXiv preprint***1908.09363**(2019) (HAL)

### Model reduction and effective dynamics:

- G. Faure and G. Stoltz, Stable and accurate schemes for smoothed dissipative particle dynamics,
*Appl. Math. Mech.-Engl.*39(1), 83-102 (2018) (HAL) - G. Stoltz, Stable schemes for dissipative particle dynamics with conserved energy,
*J. Comput. Phys.***340***,*451–469 (2017) (HAL) - G. Faure, J. Roussel, J.-B. Maillet and G. Stoltz, Size consistency in Smoothed Dissipative Particle Dynamics,
*Phys. Rev. E***94**043305 (2016) (HAL) - A.-A. Homman, J.-B. Maillet, J. Roussel and G. Stoltz, New parallelizable schemes for integrating the Dissipative Particle Dynamics with Energy Conservation,
*J. Chem. Phys.***144**024112 (2016) (HAL)