Langevin Dynamics With General Kinetic Energies

Abstract

We study Langevin dynamics with a kinetic energy different from the standard, quadratic one in order to accelerate the sampling of Boltzmann-Gibbs distributions. In particular, this kinetic energy can be non-globally Lipschitz, which raises issues for the stability of discretizations of the associated Langevin dynamics. We first prove the exponential convergence of the law of the continuous process to the Boltzmann-Gibbs measure by a hypocoercive approach, and characterize the asymptotic variance of empirical averages over trajectories. We next develop numerical schemes which are stable and of weak order two, by considering splitting strategies where the discretizations of the fluctuation/dissipation are corrected by a Metropolis procedure. We use the newly developped schemes for two applications: optimizing the shape of the kinetic energy for the so-called adaptively restrained Langevin dynamics (which considers perturbations of standard quadratic kinetic energies vanishing around the origin); and reducing the metastability of some toy models using non-globally Lipschitz kinetic energies. arXiv:1609.02891v5 [cond-mat.stat-mech]

Simulating Coulomb and Log-Gases with Hybrid Monte Carlo Algorithms

“…This requires using small time steps, hence long computations. In the present note, we explore for the first time the usage of HMC for general Coulomb gases in the context of random matrices, in the spirit of [71], the difficulty being the singularity of the interaction. This method has the advantage of sampling the exact invariant measure (1.1), while allowing to choose large time steps, which reduces the overall computational cost [27].…”

“…where (B t ) t≥0 is a standard Brownian motion on E, and γ N > 0 is an arbitrary parameter which plays the role of a friction, and which may depend a priori on N and (X t ) t≥0 , even if we do not use this possibility here. In addition, H N and β N are as in (1.1), while U N plays the role of a generalized kinetic energy [71]. This dynamics admits the following generator:…”

“…When U N (y) = 1 2 |y| 2 then Y t = dX t /dt, and in this case X t and Y t can be interpreted respectively as the position and the velocity of a system of N points in S at time t. In this case we say that U N is the kinetic energy. For simplicity, we specialize in what follows to this "physical" or "kinetic" case and refer to [71] for more possibilities.…”

Hybrid Monte Carlo methods for sampling probability measures on submanifolds

“…This is why this algorithm may be interesting in practice: it also yields some dynamical information, while sampling exactly the stationary measure for (44). In the case when the proposal is rejected, Step (iv) implies that momenta are reversed, so that the trajectory "goes backward" in this case; see [22,Remark 2.13] for further discussions, and [34] for an analysis of the weak error of the method. Remark 5.…”

“…Hybrid Monte Carlo 1 (HMC) provides such proposals, which are obtained from the composition of one step of integrators of constrained Hamiltonian dynamics, which are reversible up to momentum reversal, such as RATTLE [3] (this property is proved for sufficiently small timesteps, for example in the monographs [14,21]). It is in fact possible to resort to generalized HMC (GHMC) [17] and other variants of HMC based on partial resampling of momenta (see [23] and [22,Section 3.3.5.4]), geodesic integrators [19], the use of non-quadratic kinetic energies [34], etc.…”