[Paper Review] Algorithm Shootout: R versus RHMC

M. A. Clark, Ph. de Forcrand+1 more|ArXiv.org|Sep 30, 2005

Particle physics theoretical and experimental studies3 references4 citations

TL;DR

This paper compares the Rational Hybrid Monte Carlo (RHMC) and Rational (R) algorithms for lattice QCD simulations with chiral fermions and staggered fermions near the chiral phase transition. RHMC eliminates step-size errors inherent in R by using a rational approximation to the fermion determinant and enables exact HMC dynamics via pseudofermion fields, resulting in significantly higher efficiency and accuracy, especially at small quark masses and near phase transitions.

ABSTRACT

We present initial results comparing the RHMC and R algorithms on large lattices with small quark masses using chiral fermions. We also present results concerning staggered fermions near the deconfinement/chiral phase transition. We find that the RHMC algorithm not only eliminates the step-size error of the R algorithm, but is also considerably more efficient. We discuss several possibilities for further improvement to the RHMC algorithm.

Motivation & Objective

• To evaluate the performance and accuracy of the R algorithm versus the exact RHMC algorithm in lattice QCD simulations with chiral fermions.
• To investigate the impact of finite step-size errors on phase transition location and observables near the chiral transition point.
• To determine whether RHMC can replace the R algorithm in simulations requiring non-integer fermion determinants.
• To explore optimization strategies for RHMC, including multi-timescale integration and relaxed solver convergence.
• To compare algorithmic efficiency and systematic error control in both the finite-temperature and low-temperature regimes.

Proposed method

• RHMC reformulates the fermion determinant using a rational approximation to $ \mathcal{M}^{-\alpha} $, enabling exact HMC dynamics with pseudofermion fields.
• The rational function is decomposed into partial fractions $ r(x) = \sum_{k=1}^{n} \alpha_k / (x + \beta_k) $, allowing use of multi-shift conjugate gradient solvers.
• A stochastic accept/reject step corrects for approximation errors, ensuring exactness even with low-degree rational approximations.
• The fermion force is computed as a sum of HMC-like terms: $ S'_{\text{F}} = -\sum_{i=1}^{\bar{n}} \bar{\alpha}_i \phi^\dagger (\mathcal{M} + \bar{\beta}_i)^{-1} \mathcal{M}' (\mathcal{M} + \bar{\beta}_i)^{-1} \phi $.
• For domain wall fermions, a two-stage rational approximation is used: $ r_1(x) \approx x^{1/4} $, $ r_2(x) \approx x^{-1/2} $, enabling heatbath updates.
• Multi-timescale integration and relaxed convergence for small shifts are explored to reduce computational cost per MD step.

Experimental results

Research questions

• RQ1 How do finite step-size errors in the R algorithm affect the location of the chiral phase transition in QCD?
• RQ2 Can RHMC achieve the same physical results as R with significantly larger integration step-sizes?
• RQ3 What is the relative computational cost and efficiency of RHMC versus R in simulations with small quark masses and near the chiral transition?
• RQ4 Can multi-timescale integration or relaxed solver convergence improve RHMC performance without introducing bias?
• RQ5 Is the RHMC algorithm sufficiently accurate and efficient to replace the R algorithm in modern lattice QCD simulations?

Key findings

• The R algorithm exhibits $ O(\delta\tau^2) $ systematic errors due to step-size dependence, which are absent in RHMC.
• RHMC achieves consistency with R only when R uses a very small step-size ($ \delta\tau \leq m_{\text{ud}}/4 $), which is computationally expensive.
• At $ \delta\tau = 0.055 $, RHMC achieves an acceptance rate of $ \approx 84\% $, while R with $ \delta\tau = 0.0019 $ shows no acceptance rate but still has significant errors.
• For $ m_{\text{ud}} = 0.02 $, R with $ \delta\tau = 0.005 $ gives $ \langle \bar{\psi}\psi \rangle = 0.60817(3) $, while RHMC with $ \delta\tau = 0.0185 $ gives $ 0.60809(1) $, showing RHMC's accuracy at larger step-sizes.
• The study suggests that at least a factor of two performance improvement is possible with multi-timescale integration or relaxed solver convergence in RHMC.