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Please note that Research Lectures 1 and 2 will be taking place in the CDT Room 401 (directions), and the Research Lecture 3 will be taking place in the Huxley Building Room 139. All the three lectures will be between 3-4pm.

Leverhulme Research Lecture 1

Date and Time: Monday, 03 February 2020, 15:00 – 16:00
Venue: CDT Room 401 

Topic: Free energy adaptive biasing methods

We will present some mathematical results on adaptive biasing techniques to sample multimodal distributions which have been introduced in computational statistical physics. These algorithms can be seen as adaptive importance sampling methods, the importance function being the free energy associated with some reaction coordinate. The analysis of convergence and efficiency of these techniques rely on various tools: functional inequalities for PDEs, martingale convergence results applied to stochastic approximation algorithms, etc.

References:
– TL, M. Rousset and G. Stoltz, Long-time convergence of an Adaptive Biasing Force method, Nonlinearity, 21, 2008.
– G. Fort, B. Jourdain, E. Kuhn, TL and G. Stoltz, Convergence of the Wang-Landau algorithm, Mathematics of Computation, 84(295), 2015.
– G. Fort, B. Jourdain, TL and G. Stoltz, Convergence and efficiency of adaptive importance sampling techniques with partial biasing, Journal of Statistical Physics, 171(2), 2018.

Leverhulme Research Lecture 2

Date and Time: Monday, 10 February 2020, 15:00 – 16:00
Venue: CDT Room 401 

Topic: Sampling measures supported on submanifolds

Various applications require the sampling of a probability measure with support a submanifold defined as the level set of some function. Examples in computational statistical physics will be given. We will present recent results on so-called Hybrid Monte Carlo methods, which consists in adding an extra momentum variable to the state of the system, and discretizing the associated Hamiltonian dynamics with some stochastic perturbation in the extra variable. In order to avoid biases in the invariant probability measures sampled by discretizations of these stochastically perturbed Hamiltonian dynamics, a Metropolis rejection procedure can be considered. The so-obtained scheme belongs to the class of generalized Hybrid Monte Carlo (GHMC) algorithms, and we will discuss how to ensure that the sampling method is unbiased in practice.

References:
– TL, M. Rousset and G. Stoltz, Langevin dynamics with constraints and computation of free energy differences, Mathematics of Computation, 81(280), 2012.
– TL, M. Rousset and G. Stoltz, Hybrid Monte Carlo methods for sampling probability measures on submanifolds, Numerische Mathematik, 143(2), 2019.

Leverhulme Research Lecture 3

Date and Time: Monday, 24 February 2020, 15:00 – 16:00
Venue: Huxley Room 139 

Topic: Splitting methods for rare event simulations

The simulation of rare events is a subject of paramount importance in many contexts. The two classical approaches to simulate such events are importance sampling methods, and splitting methods. We will present in this lecture a robust splitting method, called adaptive multilevel splitting. The efficiency of this approach will be illustrated on two examples: the sampling of reactive trajectories in molecular dynamics, and the simulation of neutron transport with applications to radiation protection. Moreover, we will present some recent works on the computation of transition times between metastable states, using splitting methods and the Hill relation.

References:
– F. Cérou, A. Guyader, TL and D. Pommier, A multiple replica approach to simulate reactive trajectories, Journal of Chemical Physics, 134, 2011.
– C.-E. Bréhier, M. Gazeau, L. Goudenège, TL and M. Rousset, Unbiasedness of some generalized Adaptive Multilevel Splitting algorithms, Annals of Applied Probability, 26(6), 2016.
– TL and L.J.S. Lopès, Analysis of the Adaptive Multilevel Splitting method with the alanine di-peptide’s isomerization, Journal of Computational Chemistry, 40(11), 2019.

About Professor Tony Lelièvre
Tony Lelièvre is a world leader in the mathematical analysis of stochastic numerical methods, and their applications to molecular dynamics simulations, in particular. Among his achievements are new mathematical frameworks and algorithms for sampling multimodal measures, for sampling metastable stochastic trajectories, and for coarse-graining high dimensional problems. In terms of mathematical advances, his contributions lie at the interface between probability theory and the analysis of partial differential equations.