The Hamiltonian (or Hybrid) Monte Carlo (HMC) method is a Markov Chain Monte Carlo technique that uses proposal moves computed with the help of an auxiliary Hamiltonian dynamics. It offers the potential of generating proposals that are far away from the current location of the chain and, at the same time, may be accepted with high probability. Unfortunately, the success of the method depends on choosing appropriately the values of two parameters: the duration of the Hamiltonian flow and the time-step of the numerical integrator. For some unfortunate values of the parameters, the method may perform poorly or even not be ergodic. We shall present simple variants of the standard HMC method that may be rigorously proved to be geometrically ergodic regardless of the choice of the parameter values. The use of these variants will be illustrated in several multimodal test problems taken from the statistics and statistical physics literature.

Jesús María Sanz-Serna obtained his PhD degree in mathematics from Universidad de Valladolid in 1977. He is now a professor in Universidad Carlos III de Madrid. Sanz-Serna pioneered the field of geometric integration and wrote the first book on this subject. From 1998 to 2006, he was rector of the University of Valladolid. He received the inaugural Dahlquist Prize from the Society for Industrial and Applied Mathematics in 1995. In 2012, he became one of the inaugural fellows of the American Mathematical Society. His 60th birthday was celebrated at the 2013 International Conference on Scientific Computation and Differential Equations (SciCADE) in Valladolid.