[Paper Review] Improvement of algorithms for dynamical overlap fermions

JLQCD Collaboration, Hideo Matsufuru+8 more|ArXiv.org|Oct 3, 2006

Quantum Chromodynamics and Particle Interactions7 references6 citations

TL;DR

This paper improves Hybrid Monte Carlo (HMC) algorithms for dynamical overlap fermions by combining a 5D conjugate gradient solver with multi-time step integration and noisy Metropolis testing, achieving a 3-fold performance gain over standard methods on Blue Gene systems. The approach reduces numerical cost while maintaining accuracy by suppressing near-zero modes via extra Wilson fermions and optimizing solver precision.

ABSTRACT

We investigate the algorithms for dynamical overlap fermions aiming at improving the performance for large-scale simulations. We look for the best combination of Hybrid Monte Carlo options and iterative quark solvers with respect to the numerical costs. Our main target is a $N_f=2$ simulation with overlap fermion on a $16^3 imes 32$ lattice at lattice spacing around 0.12 fm.

Motivation & Objective

• Improve the performance of dynamical overlap fermion simulations for large-scale lattice QCD.
• Reduce the computational cost of inverting the overlap Dirac operator, the most expensive part of HMC simulations.
• Optimize HMC algorithm parameters, including time steps and solver precision, to enhance efficiency.
• Enable large-volume simulations with two light quarks and small quark masses on modern supercomputers.
• Ensure stability and accuracy by suppressing near-zero eigenvalues of the overlap operator through extra Wilson fermions.

Proposed method

• Employ a 5D conjugate gradient solver with partial fraction approximation to compute the sign function in the overlap Dirac operator.
• Implement multi-time step integration, using different time steps for gauge, $S_{PF1}$, $S_{PF2}$, and $S_E$ terms based on force magnitude hierarchy.
• Apply mass preconditioning with a heavier quark mass $m'$ to accelerate convergence of the fermion solver.
• Use extra Wilson fermions with twisted mass $\mu=0.2$ to suppress near-zero modes of $H_W$, eliminating the need for reflection/refraction checks.
• Introduce a noisy Metropolis test to correct for inaccuracies in the 5D solver, accepting new configurations with probability $P = \min\{1, e^{-dS}\}$.
• Optimize solver parameters including number of poles $N$, $m'$, and step size ratios to balance accuracy and performance.

Experimental results

Research questions

• RQ1 What is the optimal combination of HMC parameters and iterative solvers for minimizing computational cost in dynamical overlap fermion simulations?
• RQ2 How does the inclusion of extra Wilson fermions affect the stability and efficiency of the HMC algorithm?
• RQ3 Can the 5D conjugate gradient solver outperform standard and relaxed CG solvers in terms of speed and accuracy?
• RQ4 What is the impact of multi-time step integration on the stability and performance of HMC with hierarchical force terms?
• RQ5 To what extent can the noisy Metropolis test compensate for inaccuracies in the 5D solver while maintaining physical accuracy?

Key findings

• The 5D conjugate gradient solver with $N=20$ poles is 2–3 times faster than the relaxed CG method for the same precision.
• The multi-time step approach with $\Delta\tau_{(PF2)}/\Delta\tau_{(PF1)}=5$ and $\Delta\tau_{(PF1)}/\Delta\tau_{(G,E)}=6$ significantly improves efficiency.
• The noisy Metropolis test allows the use of a less precise 5D solver while maintaining physical accuracy, reducing the need for high-precision inversions.
• The combination of 5D solver, multi-time steps, and noisy Metropolis testing achieves a 3-fold speedup compared to standard nested CG on Blue Gene (512-node).
• The extra Wilson fermion term with $\mu=0.2$ effectively suppresses near-zero modes of $H_W$, eliminating the need for reflection/refraction procedures.
• Performance on Blue Gene (512-node) shows a trajectory time of 22 minutes with the improved algorithm, down from 112 minutes with standard nested CG.