Cost of the Generalised Hybrid Monte Carlo Algorithm for Free Field Theory

Abstract

We study analytically the computational cost of the Generalised Hybrid Monte Carlo (GHMC) algorithm for free field theory. We calculate the Metropolis acceptance probability for leapfrog and higher-order discretisations of the Molecular Dynamics (MD) equations of motion. We show how to calculate autocorrelation functions of arbitrary polynomial operators, and use these to optimise the GHMC momentum mixing angle, the trajectory length, and the integration stepsize for the special cases of linear and quadratic operators. We show that long trajectories are optimal for GHMC, and that standard HMC is more efficient than algorithms based on Second Order Langevin Monte Carlo (L2MC), sometimes known as Kramers Equation. We show that contrary to naive expectations HMC and L2MC have the same volume dependence, but their dynamical critical exponents are z = 1 and z = 3/2 respectively.

54 pages, 3 figures

References in corpus (4)

• Instability in the Molecular Dynamics Step of Hybrid Monte Carlo in Dynamical Fermion Lattice QCD Simulations
• Parallel Tempering in Lattice QCD with O(a)-Improved Wilson Fermions
• Efficiencies and optimization of HMC algorithms in pure gauge theory
• Cost of Generalised HMC Algorithms for Free Field Theory

Cited by in corpus (22)

• High density QCD on a Lefschetz thimble?
• Critical slowing down and error analysis in lattice QCD simulations
• Lattice QCD without topology barriers
• 50 Years of Quantum Chromodynamics
• Randomized Hamiltonian Monte Carlo
• Geometric integrators and the Hamiltonian Monte Carlo method
• Revisiting the Hybrid Quantum Monte Carlo Method for Hubbard and Electron-Phonon Models
• Modified Hamiltonian Monte Carlo for Bayesian inference
• Exploring the HMC trajectory-length dependence of autocorrelation times in lattice QCD
• Fast and scalable quantum Monte Carlo simulations of electron-phonon models
• Extra Chance Generalized Hybrid Monte Carlo
• Adaptive multi-stage integrators for optimal energy conservation in molecular simulations
• On fits to correlated and auto-correlated data
• Metropolis Integration Schemes for Self-Adjoint Diffusions
• Non-renormalizability of the HMC algorithm
• Non-Hertz-Millis scaling of the antiferromagnetic quantum critical metal via scalable Hybrid Monte Carlo
• Investigation of New Methods for Numerical Stochastic Perturbation Theory in ${\phi }^4$ Theory
• Asymptotics of Fixed Point Distributions for Inexact Monte Carlo Algorithms
• Numerical stochastic perturbation theory applied to the twisted Eguchi-Kawai model
• A flexible class of exact Hubbard-Stratonovich transformations
• Perturbative study of large $N$ principal chiral model with twisted reduction
• On Strongly Coupled Matrix Theory and Stochastic Quantization: A New Approach to Holographic Dualities