Hessian-free force-gradient integrators

TL;DR

This work introduces Hessian-free force-gradient integrators for separable Hamiltonian systems to avoid explicit Hessian calculations of the potential. Building on FGIs, the authors replace the Hessian-dependent FG-term with a Hessian-free approximation, deriving explicit schemes up to 11 stages and analyzing their error structure and backward behavior. The methods preserve time-reversibility and volume but are not strictly symplectic, leading to potential linear energy drift that can be controlled via coefficient tuning. Numerical experiments in N-body dynamics, the 2D Schwinger model, and 4D lattice QCD demonstrate significant efficiency gains over exact FGIs and standard non-gradient schemes, highlighting their practical value for computational physics problems with expensive Hessians. The paper lays out a complete classification of Hessian-free FGIs and points to future work on stability analyses, higher-order schemes, and nested integration in lattice field theory contexts.

Abstract

We propose a new framework of Hessian-free force-gradient integrators that do not require the analytical expression of the force-gradient term based on the Hessian of the potential. Due to that the new class of decomposition algorithms for separable Hamiltonian systems with quadratic kinetic energy may be particularly useful when applied to Hamiltonian systems where an evaluation of the Hessian is significantly more expensive than an evaluation of its gradient, e.g. in molecular dynamics simulations of classical systems. Numerical experiments of an N-body problem, as well as applications to the molecular dynamics step in the Hybrid Monte Carlo (HMC) algorithm for lattice simulations of the Schwinger model and Quantum Chromodynamics (QCD) verify these expectations.

Figures (6)

• Figure 1: Outer solar system. Numerical solution using the Hessian-free FGI \ref{['eq:ABADABADABA']} and $h=200$ over a time period of 200,000 days.
• Figure 2: Outer solar system. Numerical verification of the convergence order of Hessian-free FGIs. For each variant of Hessian-free FGIs, the scaling has been measured performing simulations using different step sizes. The plot displays the mean, as well as the standard deviation of the numerical measurement of the convergence order.
• Figure 3: Outer solar system. Global error vs. number of force evaluations for a selected number of Hessian-free FGIs (blue lines) and non-gradient schemes (red lines). The simulation has been performed over a time period of 200,000 days using different number of steps, $\mathrm{nsteps} \in \{1000,1500,\ldots,5000\}$.
• Figure 4: Outer solar system. Relative energy error of the Hessian-free FGI \ref{['eq:ABADABADABA']} using $h=200$ over a time period of 200,000 days.
• Figure 5: Schwinger model. Estimation of the required number of force steps per unit trajectory $n_f/h$ to reach $P_\mathrm{acc} = 90\%$. Evaluations of the FG-term are counted as two force evaluations. The subfigures show a pair-wise comparison to Hessian-free FGIs with coefficients from Tab. \ref{['tab:aFGIs']} (blue squares) with (a) exact FGIs with coefficients from omelyan2003symplectic, (b) Hessian-free FGIs with coefficients from omelyan2003symplectic, (c) exact FGIs with coefficients from Tab. \ref{['tab:aFGIs']} (red circles).

Theorems & Definitions (1)

• proof