Randomized Runge-Kutta-Nyström Methods for Unadjusted Hamiltonian and Kinetic Langevin Monte Carlo

• URL: http://arxiv.org/abs/2310.07399v2
• Date: Thu, 03 Oct 2024 14:31:13 GMT
• Title: Randomized Runge-Kutta-Nyström Methods for Unadjusted Hamiltonian and Kinetic Langevin Monte Carlo
• Authors: Nawaf Bou-Rabee, Tore Selland Kleppe,
• Abstract summary: We introduce $5/2$- and $7/2$-accurate randomized Runge-Kutta-Nystr"om methods for approximating Hamiltonian flows within non-reversible Markov chain Monte Carlo samplers. The numerical experiments demonstrate the superior efficiency of the proposed unadjusted samplers on a variety of well-behaved, high-dimensional target distributions.
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• Abstract: We introduce $5/2$- and $7/2$-order $L^2$-accurate randomized Runge-Kutta-Nystr\"{o}m methods, tailored for approximating Hamiltonian flows within non-reversible Markov chain Monte Carlo samplers, such as unadjusted Hamiltonian Monte Carlo and unadjusted kinetic Langevin Monte Carlo. We establish quantitative $5/2$-order $L^2$-accuracy upper bounds under gradient and Hessian Lipschitz assumptions on the potential energy function. The numerical experiments demonstrate the superior efficiency of the proposed unadjusted samplers on a variety of well-behaved, high-dimensional target distributions.

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