Sampling of probability measures supported on submanifolds
A talk in the Keine Reihe series by
Tony Lelievre from Ecole des Ponts ParisTech
| Abstract: | Various applications require the sampling of probability measures restricted
to submanifolds defined as the level set of some functions, in particular in computational
statistical physics. We will present recent results on so–called Hybrid Monte Carlo meth-
ods, 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 dis-
cretizations 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:
- T. Lelièvre, M. Rousset and G. Stoltz, Langevin dynamics with constraints and computation
of free energy differences, Mathematics of Computation, 81(280), 2012.
- T. Lelièvre, M. Rousset and G. Stoltz, Hybrid Monte Carlo methods for sampling probability
measures on submanifolds, to appear in Numerische Mathematik, 2019.
- E. Zappa, M. Holmes-Cerfon, and J. Goodman. Monte Carlo on manifolds: sampling den-
sities and integrating functions. Communications in Pure and Applied Mathematics, 71(12),
2018. |