Exact and Non-stiff Sampling of Highly Oscillatory Systems: an Implicit Mass-matrix Penalization Approach

We propose and analyze an implicit mass-matrix penalization (IMMP) technique which enables efficient and exact sampling of the (Boltzmann/Gibbs) canonical distribution associated with highly oscillatory systems. The penalization is based on an extended Hamiltonian whith artificial constraints associated with each expected fast degree of freedom (fDOF). The penalty parameters enable arbitrary tuning of the timescale for the selected fDOFs. Associated (stochastic) numerical methods are shown to be dynamically consistent when the penalty vanishes with the time-step, and always statistically exact with respect to canonical distributions for any chosen penalty. Moreover, the IMMP method is shown to be asymptotically stable in the infinite stiffness limit, converging towards standard effective dynamics on the slow manifold. It can be easily implemented from standard geometric integrators with algebraic constraints given by the slow manifold, and has no additional complexity in terms of enforcing the constraint and force evaluations. For high dimensional systems, the IMMPmethod enables a tunable slowdown of high frequencies thereby relaxing time-step stability restrictions, while, at the same time, conserving macroscopic features of the system’s dynamics. This property is proved rigorously for a linear harmonic atomic chain, and numerical evidence is given in the case of non-linear interactions.