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An implicit mass-matrix penalization (IMMP) of Hamiltonian dynamics isproposed, and associated dynamical integrators, as well as sampling Monte-Carloschemes, are analyzed for systems with multiple time scales. The penalizationis based on an extended Hamiltonian with artificial constraints associated withsome selected DOFs. The penalty parameters enable arbitrary tuning oftimescales for the selected DOFs. The IMMP dynamics is shown to be aninterpolation between the exact Hamiltonian dynamics and the dynamics withrigid constraints. This property translates in the associated numericalintegrator into a tunable trade-off between stability and dynamicalmodification. Moreover, a penalty that vanishes with the time-step yields ordertwo convergent schemes for the exact dynamics. Moreover, by construction, theresulting dynamics preserves the canonical equilibrium distribution in positionvariables, up to a computable geometric correcting potential, leading toMetropolis-like unbiased sampling algorithms. The algorithms can be implementedwith a simple modification of standard geometric integrators with algebraicconstraints imposed on the selected DOFs, and has no additional complexity interms of enforcing the constraints and force evaluations. The properties of theIMMP method are demonstrated numerically on the $N$-alkane model, showing thatthe time-step stability region of integrators and the sampling efficiency canbe increased with a gain that grows with the size of the system. This featureis mathematically analyzed for a harmonic atomic chain model. When a largestiffness parameter is introduced, the IMMP method is shown to beasymptotically stable and to converge towards the heuristically expectedMarkovian effective dynamics on the slow manifold.

Implicit Mass-matrix Penalization of Hamiltonian Dynamics with Application to Exact Sampling of Stiff Systems