Speaker
Description
I shall give a pedagogical introduction to the Hamiltonian/Hybrid Monte Carlo algorithm (HMC) on Riemannian manifolds. I will explain how Hamiltonian systems are most naturally formulated in terms of Hamiltonian vector fields $\hat X$ on symplectic manifolds: the relationship between commutators of Hamiltonian vector fields and Poisson brackets, $[\hat X,\hat Y] = \widehat{\{X,Y\}}$; why a conserved shadow Hamiltonian exists; and why the natural Riemannian volume element $\sqrt{\det g}$ appears automatically. I will show that the symplectic integrator step $e^{\hat T\delta\tau}$ is a geodesic on a Riemannian manifold, and how this corresponds to matrix exponentiation on Lie groups. Finally, time permitting, I will explain how this leads to a practical HMC algorithm on symmetric spaces (and slightly more generally on homogeneous reductive manifolds) such as ${\mathbb CP}^n$ or $\mathbb S^2$ using Hamiltonian reduction.
