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We propose and analyze an implicit mass-matrix penalization (IMMP) techniquewhich enables efficient and exact sampling of the (Boltzmann/Gibbs) canonicaldistribution associated to Hamiltonian systems with fast degrees of freedom(fDOFs). The penalty parameters enable arbitrary tuning of the timescale forthe selected fDOFs, and the method is interpreted as an interpolation betweenthe exact Hamiltonian dynamics and the dynamics with infinitely slow fDOFs(equivalent to geometrically corrected rigid constraints). This propertytranslates in the associated numerical methods into a tunable trade-off betweenstability and dynamical modification. The penalization is based on an extendedHamiltonian with artificial constraints associated with each fDOF. Byconstruction, the resulting dynamics is statistically exact with respect to thecanonical distribution in position variables.  The algorithms can be easily implemented with standard geometric integratorswith algebraic constraints given by the expected fDOFs, and has no additionalcomplexity in terms of enforcing the constraint and force evaluations. Themethod is demonstrated on a high dimensional system with non-convexinteractions. Prescribing the macroscopic dynamical timescale, it is shown thatthe IMMP method increases the time-step stability region with a gain that growslinearly with the size of the system. The latter property, as well asconsistency of the macroscopic dynamics of the IMMP method is proved rigorouslyfor linear interactions. Finally, when a large stiffness parameter isintroduced, the IMMP method can be tuned to be asymptotically stable,converging towards the heuristically expected Markovian effective dynamics onthe slow manifold.

Exact and non-stiff sampling of highly oscillatory systems: an implicit mass-matrix penalization approach