HMC algorithm with multiple time scale integration and mass preconditioning

TL;DR

The paper develops and tests a variant of the Hybrid Monte Carlo algorithm that combines Hasenbusch mass preconditioning with multiple time scale integration for two-flavor Wilson fermions. By arranging force contributions through tuned preconditioning masses and leveraging distinct integration scales, the method achieves performance comparable to domain-decomposition preconditioned HMC and surpasses plain leap-frog integration, with a notable improvement in scaling toward lighter quark masses. The authors provide a detailed numerical study at beta=5.6 on 24^3×32 lattices, showing stable simulations down to about 20 MeV quark masses, updated Berlin Wall plots, and a clear path toward accessing smaller m_PS/m_V ratios in realistic lattice volumes. The approach is broad in applicability to various lattice Dirac operators and offers a straightforward implementation path, along with suggestions for future enhancements such as Chronological Solver Guess (CSG) and Polynomial HMC (PHMC).

Abstract

We present a variant of the HMC algorithm with mass preconditioning (Hasenbusch acceleration) and multiple time scale integration. We have tested this variant for standard Wilson fermions at beta=5.6 and at pion masses ranging from 380 MeV to 680 MeV. We show that in this situation its performance is comparable to the recently proposed HMC variant with domain decomposition as preconditioner. We give an update of the ``Berlin Wall'' figure, comparing the performance of our variant of the HMC algorithm to other published performance data. Advantages of the HMC algorithm with mass preconditioning and multiple time scale integration are that it is straightforward to implement and can be used in combination with a wide variety of lattice Dirac operators.

Figures (4)

• Figure 1: Average and maximal forces for simulation points $B$ and $C$. The statistical errors are too small to be visible due to the large number of measurements.
• Figure 2: Comparison between the fermionic forces of run $C$ ($F_1$ and $F_2$) and a run with $\kappa=0.15825$ without mass preconditioning and multiple time scales ($F$). The statistical errors are too small to be visible.
• Figure 3: Monte Carlo histories of the deviation $\Delta P$ of the average plaquette from its mean value and of $\Delta H$, both for simulation point $C$.
• Figure 4: Computer resources needed to generate $1000$ independent configurations of size $24^3\times 40$ at a lattice spacing of about $0.08\ \mathrm{fm}$ in units of $\mathrm{Tflops}\cdot \mathrm{years}$ as a function of $m_\mathrm{PS}/m_\mathrm{V}$. In (a) we compare our results represented by squares to the results of ref. Orth:2005kq represented by circles. The lines are functions proportional to $(m_\mathrm{PS}/m_\mathrm{V})^{-4}$ (dashed) and $(m_\mathrm{PS}/m_\mathrm{V})^{-6}$ (solid) with a coefficient such that they cross the data points corresponding to the lightest pseudo scalar mass. In (b) we compare to the formula of eq. \ref{['results:ukawa']}Ukawa:2002pc (solid line) by extrapolating our data with $(m_\mathrm{PS}/m_\mathrm{V})^{-4}$ (dashed) and with $(m_\mathrm{PS}/m_\mathrm{V})^{-6}$ (dotted), respectively. The arrow indicates the physical pion to rho meson mass ratio. Additionally, we add points from staggered simulations as were used for the corresponding plot in ref. Jansen:2003nt. Note that all the cost data were scaled to match a lattice time extend of $T=40$.