An explicit splitting SAV scheme for the kinetic Langevin dynamics

Abstract

The kinetic Langevin dynamics finds diverse applications in various disciplines such as molecular dynamics and Hamiltonian Monte Carlo sampling. In this paper, a novel splitting scalar auxiliary variable (SSAV) scheme is proposed for the dynamics, where the gradient of the potential $U$ is possibly non-globally Lipschitz continuous with superlinear growth. As an explicit scheme, the SSAV method is efficient, robust and is able to reproduce the energy structure of the original dynamics. By an energy argument, the SSAV scheme is proved to possess an exponential integrability property, which is crucial to establishing the order-one strong convergence without the global monotonicity condition. Moreover, moments of the numerical approximations are shown to have polynomial growth with respect to the time length. This helps us to obtain weak error estimates of order one, with error constants polynomially (not exponentially) depending on the time length. Despite the obtained polynomial growth, the explicit scheme is shown to be computationally effective for the approximation of the invariant distribution of the dynamics with exponential ergodicity. Numerical experiments are presented to confirm the theoretical findings and to show the superiority of the algorithm in sampling.