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Fast Fermion Monte Carlo

Philippe de Forcrand, Tetsuya Takaishi

TL;DR

The paper tackles speeding Hybrid Monte Carlo (HMC) simulations for dynamical fermions by evaluating adaptive step sizing, approximate Hamiltonians, and preconditioning of the fermionic action. It finds adaptive step size offers limited practical gains due to overhead, while approximate Hamiltonians—notably Chebyshev polynomial approximations of the fermion inverse and low-accuracy solver criteria—can significantly reduce per-trajectory cost. It also demonstrates that preconditioning the fermionic action, such as ILU-based even-odd schemes, reduces fermionic forces and enables larger step-sizes with high acceptance, with preliminary results suggesting speedups approaching a factor of ~5. These strategies collectively point to substantial efficiency improvements for HMC in lattice QCD simulations. In particular, $P_n(D)$ Chebyshev schemes and ILU/SE integration offer concrete paths to lower computational cost while preserving accuracy, especially near critical parameters.

Abstract

Three possibilities to speed up the Hybrid Monte Carlo algorithm are investigated. Changing the step-size adaptively brings no practical gain. On the other hand, substantial improvements result from using an approximate Hamiltonian or a preconditioned action.

Fast Fermion Monte Carlo

TL;DR

The paper tackles speeding Hybrid Monte Carlo (HMC) simulations for dynamical fermions by evaluating adaptive step sizing, approximate Hamiltonians, and preconditioning of the fermionic action. It finds adaptive step size offers limited practical gains due to overhead, while approximate Hamiltonians—notably Chebyshev polynomial approximations of the fermion inverse and low-accuracy solver criteria—can significantly reduce per-trajectory cost. It also demonstrates that preconditioning the fermionic action, such as ILU-based even-odd schemes, reduces fermionic forces and enables larger step-sizes with high acceptance, with preliminary results suggesting speedups approaching a factor of ~5. These strategies collectively point to substantial efficiency improvements for HMC in lattice QCD simulations. In particular, Chebyshev schemes and ILU/SE integration offer concrete paths to lower computational cost while preserving accuracy, especially near critical parameters.

Abstract

Three possibilities to speed up the Hybrid Monte Carlo algorithm are investigated. Changing the step-size adaptively brings no practical gain. On the other hand, substantial improvements result from using an approximate Hamiltonian or a preconditioned action.

Paper Structure

This paper contains 6 sections, 10 equations, 2 figures.

Table of Contents

  1. Introduction
  2. HMC with adaptive step size
  3. HMC with approximate Hamiltonian
  4. Chebyshev polynomial approximation to the Hamiltonian
  5. Low accuracy stopping criterion
  6. HMC with preconditioned action

Figures (2)

  • Figure 1: (a) Efficiency vs. step-size $\Delta t$ for polynomials of various degree $n$. For the standard HMC, the number of iterations in the $BiCG\gamma_5$ solver was taken as $n$. The stopping accuracy was set to $||r||=10^{-8}$, where $||r||$ is the residual norm. (b) Efficiency vs. residual norm. The trajectory length is 1(0.5) for a $4^4(8^4)$ lattice.
  • Figure 2: RMS force for various actions.