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Archive : Probability

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Grégoire Ferré99Grégoire FerréZoom (email organizers for the password)<p><b>​Title: </b>Importance sampling and low temperature asymptotics<br><b>Abstract: </b>Studying rare events is a matter of concern in many scientific areas, in particular in statistical physics. Typically, one wishes to compute the probability for a system to reach some state, which does not happen frequently at the time scale of a simulation, in particular at low temperature, when the noise is small. Since estimating these events with Monte Carlo simulation in general has a high variance, one generally resorts to importance sampling for accelerating convergence.<br><br>The purpose of this talk is to present a novel importance sampling method. As others it relies on an underlying Hamilton-Jacobi equation describing the optimal control of system. The originality of our work is to expand this equation around a low temperature particular path called the instanton. I will present heuristically the approach and present a couple of simple applications. If times allows I will present a recent work where I prove variance reduction properties for the estimators.<br></p>10/25/2021 4:15:00 PM10/25/2021 5:20:00 PM
Greg Pavliotis (Imperial College, London)98Greg Pavliotis (Imperial College, London)Zoom​ (email organizers for the password)<p><strong>​</strong><span class="wrap-text"><strong>Title:</strong> Optimal Langevin samplers<br><strong>Abstract: </strong>Sampling from a probability distribution in a high dimensional spaces is a standard problem in computational statistical mechanics, Bayesian statistics and other applications. A standard approach for doing this is by constructing an appropriate Markov process that is ergodic with respect to the measure from which we wish to sample. In this talk we will present a class of sampling schemes based on Langevin-type stochastic differential equations. We will show, in particular, nonreversible Langevin samplers, i.e. stochastic dynamics that do not satisfy detailed balance, have, in general, better properties than their reversible counterparts, in the sense of accelerating convergence to equilibrium and of reducing the asymptotic variance. Numerical schemes for such nonreversible samplers will be discussed and the connection with nonequilibrium statistical mechanics will be made.</span><br></p>10/11/2021 4:15:00 PM10/11/2021 5:05:00 PM
Tony Lelievre (ENPC-CERMICS, Paris)97Tony Lelievre (ENPC-CERMICS, Paris)<table border="0" cellpadding="0" cellspacing="0" width="100%"><tbody><tr><td valign="top" width="350px"><b>​Mode:</b> <a href="https://udel.zoom.us/j/92196084584" target="_blank">Zoom</a>​ (email organizers for the password).<br><br><div><b>Title:</b> Molecular dynamics algorithms: numerical and mathematical analysis</div><div><br></div><div><b>Abstract:</b> The objective of this talk will be to present models used in classical molecular dynamics, and some mathematical questions raised by their simulations. In particular, we will present recent results on the connection between a metastable Markov process with values in a continuous state space (satisfying e.g. the Langevin or overdamped Langevin equation) and a jump Markov process with values in a discrete state space. This is useful to analyze and justify methods which use the jump Markov process underlying a metastable dynamics as a support to efficiently sample the state-to-state dynamics (accelerated dynamics techniques à la A.F. Voter). It also provides a mathematical framework to justify the use of transition state theory and the Eyring-Kramers formula to build kinetic Monte Carlo or Markov state models.</div><div><br></div><div><b>References:</b></div><div>- T. Lelièvre and G. Stoltz, Partial differential equations and stochastic methods in molecular dynamics, Acta Numerica, 25, 2016.</div><div>- G. Di Gesù, T. Lelièvre, D. Le Peutrec and B. Nectoux, Jump Markov models and transition state theory: the Quasi-Stationary Distribution approach, Faraday Discussion, 195, 2016.</div><div>- G. Di Gesù, T. Lelièvre, D. Le Peutrec et B. Nectoux, Sharp asymptotics of the first exit point density, Annals of PDE, 5(1), 2019.<br></div></td></tr></tbody></table><p><br></p>9/20/2021 4:15:00 PM9/20/2021 5:05:00 PM
Gabriel Stoltz: CERMICS, Ecole des Ponts & INRIA Paris93Gabriel Stoltz: CERMICS, Ecole des Ponts & INRIA ParisZoom<br> Title: Longtime convergence of some diffusion processes in molecular dynamics <br> Abstract: I will present an introduction to two paradigmatic diffusion processes in molecular simulation, namely Langevin dynamics and its overdamped limit. I will in particular present results on their longtime convergence, based on Poincare inequalities for overdamped Langevin dynamics, and hypocoercive techniques for Langevin dynamics. I will also discuss the implication of these results in terms of estimates on the asymptotic variance of longtime averages, a key elements to quantify the statistical error in numerical simulations.4/28/2021 5:00:00 PM4/28/2021 6:00:00 PM

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