Pith. Sign in

REVIEW

HMC algorithm with multiple time scale integration and mass preconditioning

PaperPDF from arXiv

data files

ReviewThe objections and verdict, as JSON Paper breakdownThe authors' claims, conditions and measurements, as JSON Signed receiptProof this review has not been altered
X Bluesky LinkedIn Reddit Hacker News
C. Urbach, K. Jansen, A. Shindler, U. Wenger

High Energy Physics - Lattice · Submitted 9 Jun 2005 · 42 references

Pith's verdict: A competent, honest methods paper: three-field mass preconditioning with multiple time scales, matched against Lüscher's domain-decomposition HMC at the same beta and kappa — the comparison is real but narrower than the abstract reads.

T0 · AI review

How this review was made: An AI referee (selfhosted/deepseek-v4.1-flash) read the paper on November 11, 2026 and answered a fixed list of questions about it. T0 is the starting level every paper gets. The levels describe how deeply Pith has checked the paper, never the authors.

  1. T0 AI review An AI read the paper and answered a fixed list of questions.
  2. T1 Audit Automatic checks that the references and files the paper relies on exist and match.
  3. T2 Compute The paper's code or calculations re-run with the same results.
  4. T3 Formal statement The claims written out in a formal math language a computer can read.
  5. T4 Proof Those claims proved, and the proof checked by computer.

How Pith checks papers →

· 3 major objections · 9 minor · Ask about this paper · The compiled record · Read the full review →
Challenge this review Comment (0) Read on arXiv ↗
Cite ▾
arXiv IDhep-lat/0506011 (version 2)
Pith IDV7DYZALY, Pith's permanent record of this paper and its review
CitationA. Shindler, C. Urbach, K. Jansen, U. Wenger. "HMC algorithm with multiple time scale integration and mass preconditioning." arXiv:hep-lat/0506011 (2005). https://pith.science/paper/V7DYZALY
Cite this reviewBibTeX for Pith's review of this paper
Claim paper

Why Pith says this

Highlight any text to comment on it.

Referee report

How Pith checks papers → · Full audit of this paper →

The paper

The paper claims

By splitting the two-flavour fermion determinant into a product of mass-shifted ratios and integrating each piece on its own time step, the authors claim a hybrid Monte Carlo lattice QCD simulation that matches a domain-decomposition-pre条件

The reading

The paper asks whether a hybrid Monte Carlo (HMC) simulation of lattice QCD can be pushed to light quark masses without adopting the domain-decomposition preconditioner that had just been reported to give large speedups. Its answer combines two older ideas: rewrite the two-flavour fermion determinant as a product of ratios, each carrying its own unphysical twisted mass $mu_j$, and then integrate each piece of the Hamiltonian on its own time step, chosen so that the operator that is most expensive to invert contributes the smallest force. Tested with Wilson fermions at $beta = 5.6$ on $24^3 times 32$ lattices at pion masses from roughly 380 to 680 MeV, the variant reaches a cost figure of $nu = 0.09(3)$, $0.11(3)$ and $0.23(9)$ at the three simulation points, against $0.69(29)$, $0.50(17)$ and $0.28(9)$ quoted for the domain-decomposition algorithm at the same parameters. That is better at the two heavier points and statistically indistinguishable at the lightest, while a plain single-time-scale leapfrog needs at least ten times more work. The practical payoff the authors claim is an updated ``Berlin Wall'' plot in which simulations at $m_{PS}/m_V approx 0.3$ at a lattice spacing near 0.08 fm become affordable.

What carries the argument

The load-bearing object is the determinant factorisation $det(Q^2) = det(W^+ W^-) det(Q^2) / det(W^+ W^-)$, with $W^pm = Q pm i mu$ so that $W^+ W^- = Q^2 + mu^2$; each ratio becomes one pseudofermion action with its own unphysical mass $mu_j$, and the chain is extended to two or three such fields. Because $W^+ W^-$ is both cheaper to invert than $Q^2$ and, for $|mu| < 1$, has a variation suppressed by roughly $mu^2$, the forces can be ordered by choice of $mu_j$. The second piece of machinery is the recursive Sexton-Weingarten (or leapfrog) integrator, which nests a cheap small-step update inside a more expensive large-step one, preserving reversibility and area preservation, so that each action term is evolved on a time scale matched to the size of its force.

Extended reading notes

Core claim

The central claim is that mass preconditioning and multiple time scale integration, stacked on top of even/odd preconditioning, give a cost figure comparable to the domain-decomposition HMC of the reference it compares against, and roughly an order of magnitude better than a plain leapfrog HMC, in the specific regime tested. The mechanism the authors emphasise is not the usual one of merely reducing condition numbers: they tune the preconditioning masses so that the force contributions of the separate action pieces are strictly hierarchically ordered, with the most expensive operator to invert giving the smallest force, and then assign time steps so that $Delta tau_i |F_i|_{max}$ is roughly constant across levels. Measured ratios at the lightest point show the force from the physical operator reduced by about a factor 42 in the average norm relative to the second preconditioned piece, while the step numbers differ only by about a factor 6. The authors are explicit that the comparison holds only ``in this situation'' and that no attempt was made to tune parameters to their optimum, so the numbers are a first survey rather than a demonstrated optimum.

Load-bearing premise

The comparison rests on the plaquette's integrated autocorrelation time being a fair, transferable measure of how independent successive configurations are, even though the paper itself notes that hadron masses typically have quite different autocorrelation times and that its measured values carry 30 to 50 percent relative errors.

Editorial extensions

If this is right

  • Mass preconditioning plus multiple time scales can be dropped into an existing HMC code and applies to twisted mass, improved Wilson, staggered and in principle overlap Dirac operators, unlike preconditioners tied to a particular operator structure.
  • The observed force hierarchy supplies a practical tuning rule: pick the $mu_j$ so that $Delta tau_i |F_i|_{max}$ is roughly equal across levels, then check the acceptance rate, which depends on $langle exp(-Delta H) rangle$ rather than on force norms alone.
  • With this algorithm the feasibility wall moves to $m_{PS}/m_V approx 0.3$ at $a approx 0.08$ fm and $L approx 2$ fm; under the optimistic $m_{PS}^{-4}$ scaling, even the physical pion-to-rho ratio becomes reachable at that spacing provided the spatial volume is increased.
  • Wilson and staggered fermion simulations in comparable physical situations land within the same order of magnitude in cost once this algorithm is used for the Wilson case.
  • Quark masses of order 20 MeV were simulated without instability, with only a few unexceptional fluctuations in $Delta H$ over the runs.
  • Chronological solver guessing gave only about a 20 percent reduction in matrix-vector multiplications here, with the gain expected to grow as the bare mass is lowered further.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The cost figure $nu$ counts only inversions of the physical operator, so it does not charge for the preconditioning inversions; at the lightest point the preconditioning operators already consume more matrix-vector multiplications than the physical one. A metric weighted by actual solver work per level could therefore narrow the apparent gap to the domain-decomposition algorithm, and the paper's o
  • The authors attribute their smaller plaquette autocorrelation times at the two heavier points partly to the rival algorithm holding a subset of links fixed during molecular dynamics. If that is the right explanation, the advantage is a property of the comparison rather than of mass preconditioning, and a domain-decomposition variant that also updates every link should close it. That is a checkable
  • Because the whole comparison rests on one observable's autocorrelation time, the natural extension is to repeat it for pion and vector correlators and for the topological charge; if the ordering of autocorrelation times differs between the two algorithms, the cost ranking could reverse for the quantities one actually measures.
  • The same ratio-and-nested-integrator construction should transfer to staggered fermions, where the fourth root and the taste structure change which operator is expensive; whether the force hierarchy survives there is an open and testable question.
Original 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.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,296 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.15 · human participants 0.01 · controlled comparison 0.61 · randomized assignment 0.01 · states quantitative result 0.70 · compares with prior results 0.98 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

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.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We present a variant of the HMC algorithm with mass preconditioning (Hasenbusch acceleration) and multiple time scale integration. contribution 0.98 0.81 0.13 asserted 0.97
s1 We have tested this variant for standard Wilson fermions at beta=5.6 and at pion masses ranging from 380 MeV to 680 MeV. method 0.85 0.96 0.04 asserted 0.99
s2 We show that in this situation its performance is comparable to the recently proposed HMC variant with domain decomposition as preconditioner. result 0.99 0.86 0.43 asserted 1.00
s3 We give an update of the ``Berlin Wall'' figure, comparing the performance of our variant of the HMC algorithm to other published performance data. contribution 0.73 0.87 0.09 asserted 0.86
s4 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. interpretation 0.89 0.29 0.22 asserted 1.00

What kind of paper the classifier reports this is: method or tool 0.95 · evidence from simulation or computation 0.99.

What was examined, and what was not

sources read
  • arXiv listing abstract v2
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
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. As a further observation, one sees from figure 2 or from the ratios quoted above that the norm of F1 is almost identical to the norm of F, which is the case for both the average and the maximal values. From these investigations we think one can conclude the following: in the first place it is possible to tune the value of µ1 (and possibly µ2) such that the most expensive force contribution of F2 (or F3) to the total force becomes small
Figure 2. Comparison between the fermionic forces of run C (F1 and F2) and a run with κ = 0.15825 without mass preconditioning and multiple time scales (F). The statistical errors are too small to be visible. κ Nmeas hPi τint(P) A 0.1575 740 0.57250(3) 6(2) B 0.1580 1020 0.57339(3) 7(2) C 0.15825 905 0.57384(4) 10(4)
Figure 3. Monte Carlo histories of the deviation ∆P of the average plaquette from its mean value and of ∆H, both for simulation point C. Therefore we conclude that the norm of the forces can indeed serve as a first criterion to tune the time scales and the preconditioning masses, by looking for a situation in which ∆τikFikmax is a constant independent of i. But, it cannot be the only criterion. Finally, the acceptance rate is deter￾mined by hexp(−∆H)i, which depends in a more complicated way on the forces, see e.g. ref. [27]. It is well known that simulations with the HMC algorithm in…
Figure 4. Computer resources needed to generate 1000 independent configurations of size 243 × 40 at a lattice spacing of about 0.08 fm in units of Tflops · years as a function of mPS/mV. In (a) we compare our results represented by squares to the results of ref. [9] represented by circles. The lines are functions proportional to (mPS/mV) −4 (dashed) and (mPS/mV) −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. 4-7 [12] (solid line) by extrapolating our data with (mPS/mV) −4 (dashed) a

Figures reproduced from arXiv:hep-lat/0506011 by the authors.

References
How later papers used it

56 later papers cite it

Evidence: substantial. 44 of the citing papers share no author with it, and 30 of those carry a confident label from Pith's citation classifier.

Pith read 23 of the 71 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

Uses its method or tool 13 of 39 papers with this label · 5 by the same authors

By the same authors

Compares with it 1 paper
How the literature uses it

47 arXiv papers cite this in 59 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 20 sentences without a confident label

Uses its method or tool 32 papers
Compares with it 1 paper
Cites it as background 4 papers
  • Simulations that include two degenerate quark flavours with a mass approaching or at the physical value typically make use of a hierarchy of two, three or more mass preconditioning terms [ref] .

    Applying polynomial filtering to mass preconditioned Hybrid Monte Carlo · Introduction · 0.82

  • Historically, simulations at physical pion masses and reasonable lattice spacings and volumes seemed to be infeasible up until the early 2000s [ref] , but various algorithmic advancements [ref] over the years have made those simulations possible.

    Parallel Tempered Metadynamics · Introduction · 0.83

  • This property makes the HMC in its variants [ref] and with its improvements [ref] the workhorse for lattice Quantum Chromodynamics (QCD) simulations with dynamical fermions.

    Reversibility Violation in the Hybrid Monte Carlo Algorithm · Introduction · 0.83

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.