Variance reduction for applications in computational statistical physics

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. In the first part of this talk, we propose a new
variance reduction method based on control variates for efficiently
estimating the mobility of Langevin-type dynamics, and we present
numerical experiments illustrating the performance of the approach.

In the second part of this talk, we study an importance sampling
approach for calculating averages with respect to multimodal probability
distributions. Traditional Markov chain Monte Carlo methods to this end,
which are based on time averages along a realization of a Markov process
ergodic with respect to the target probability distribution, are usually
plagued by a large variance due to the metastability of the process. The
estimator we study is based on an ergodic average along a realization of
an overdamped Langevin process for a modified potential. We obtain an
explicit expression for the optimal perturbation potential in dimension
1 and propose a general numerical approach for approximating the optimal
potential in the multi-dimensional setting.