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GCS - Geometry, compatibility and structure preservation in computational differential equations

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 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:
– 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 densities and integrating functions. Communications in Pure and Applied Mathematics, 71(12), 2018.




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