PaperlayerView exact revision on arXiv

Source-linked AI summary

Effective noise reduction techniques for disconnected loops in Lattice QCD

Gunnar S. Bali, Sara Collins, Andreas Schaefer

arXiv:0910.3970v2hep-lat

TL;DR

All-to-all propagators are prohibitively expensive, so the paper studies unbiased stochastic estimation and noise-reduction methods for disconnected nucleon-structure contributions. It introduces the truncated solver method and combines it with established techniques, typically reducing computer time by an order of magnitude without bias. The study reports Δs = −0.017(21) and finds gauge errors dominant after stochastic variance reduction.

  • Problem

    Direct computation of all-to-all propagators is prohibitively expensive for lattice QCD observables involving disconnected diagrams.

  • Method

    The paper introduces the unbiased truncated solver method and combines it with partitioning, hopping parameter expansion, and low eigenmode deflation.

  • Results

    The truncated solver method typically reduces computer time by an order of magnitude without bias; the study reports Δs = −0.017(21) and gauge errors as the dominant uncertainty after variance reduction.

  • Takeaways & Limitations

    Noise-reduction combinations can make stochastic all-to-all estimates practical while allowing larger stochastic errors once gauge errors dominate.

  • Takeaways & Limitations

    The eigenmode-based estimate can be biased, and the biorthogonal alternative is incompatible with deflating the CG solver.

Abstract

from arXiv · show

Many Lattice QCD observables of phenomenological interest include so-called all-to-all propagators. The computation of these requires prohibitively large computational resources, unless they are estimated stochastically. This is usually done. However, the computational demand can often be further reduced by one order of magnitude by implementing sophisticated unbiased noise reduction techniques. We combine both well known and novel methods that can be applied to a wide range of problems. We concentrate on calculating disconnected contributions to nucleon structure functions, as one realistic benchmark example. In particular we determine the strangeness contributions to the nucleon, <N|ss|N>, and to the spin of the nucleon, Delta s.

1. Introduction

Disconnected diagrams require all-to-all propagators that are prohibitively expensive to compute directly, motivating unbiased stochastic estimates and variance-reduction methods. The paper introduces and tests the truncated solver method alongside existing techniques for nucleon strangeness observables.

  • Computational challenge: All-to-all propagators require resources scaling with lattice volume, making direct evaluation prohibitively expensive.Lattices may contain approximately 10^6 to 10^9 sites, replacing the factor 12V with N random sources can reduce effort substantially.
  • Stochastic estimation: Unbiased stochastic estimation adds statistical error that decreases asymptotically as 1/√N, so methods target its prefactor.The optimal number of estimates depends on the observable and estimator prescription.
  • Noise reduction methods: The paper introduces the truncated solver method and tests it with partitioning, hopping parameter expansion, and low eigenmode deflation.The benchmark observables are the nucleon spin strangeness contribution Δs and scalar strangeness content ⟨N|s̄s|N⟩.
  • Physics benchmarks: Experimentally accessible Δs estimates are consistent with zero over measured x, while the reported lattice result is Δs = −0.017(21).The lattice result may be affected by chiral extrapolation from mPS ≈450 MeV to the physical point.
  • Physics benchmarks: Scalar strangeness is not directly accessible experimentally but is relevant to nuclear-structure models and Higgs–nucleon coupling.The motivation assumes heavy-flavour contributions are strongly suppressed.
  • Paper structure: The article presents notation and stochastic estimators, describes the lattice calculation, reports axial and scalar strangeness results, and concludes with systematic tuning details.The truncated solver method parameters are discussed in Appendix A.

2. Noise reduction techniques

The paper estimates prohibitively large all-to-all propagators stochastically and combines several unbiased variance-reduction methods, including the new truncated solver method, to control computational cost and stochastic noise.

  • Stochastic estimation: Direct evaluation of all-to-all propagators is prohibitively expensive because the propagator has 12V × 12V components.Unbiased stochastic estimation reduces the problem to O(N × 12V) in memory and computer time.
  • Stochastic estimation: The stochastic estimate remains unbiased, while its error decreases approximately as 1/√N and should not dominate the Monte Carlo gauge error.The optimal N depends on the observable and the estimation prescription.
  • Combined methods: The variance-reduction toolkit combines partitioning, hopping parameter expansion, the truncated solver method, and truncated eigenmode acceleration.These methods are combined to minimize stochastic noise at fixed computational effort.
  • Truncated solver method: The truncated solver method uses many cheap approximate solves and fewer precise solves to remove bias while retaining an unbiased estimate.The converged correction can be obtained with an efficient solver, not necessarily the solver used for truncated solutions.
  • Truncated eigenmode acceleration: Truncated eigenmode acceleration can reduce variance and solver iterations through low-mode projection and deflation.Its projection overhead can become significant when combined with the large number of truncated solver estimates.
  • Method limitations: The preferred numbers of stochastic estimates and deflated eigenmodes depend strongly on lattice volume, quark mass, and lattice spacing.Stochastic methods scale more favorably with volume than the deflation space, whose rank can grow as ne ∝ L^4 ∝ Va^4.

3. Evaluation of the disconnected loop

The study evaluates stochastic disconnected-loop estimators on 2+1-flavor lattices, combining truncated solves, hopping-parameter expansion, and eigenmode deflation. These techniques substantially reduce solver costs, with gains depending on quark mass and current structure.

  • Setup: Disconnected loops are estimated stochastically and combined configuration by configuration with nucleon two-point functions computed using point-to-all propagators.The calculation uses Wilson valence quarks on 326, 167, and 152 configurations for the three loop hopping parameters.
  • TSM: The truncated solver method averages many inexpensive truncated solutions and corrects their bias with fewer fully converged solves.Its parameters are tuned through the truncation length nt and the ratio N1/N2.
  • TSM: 5–10 are the approximate fixed-cost gains from TSM alone across representative scalar, vector, tensor, pseudoscalar, and axial currents.Except for Γ = 1, optimized parameters show only mild Γ dependence, allowing one parameter set to retain similar gains.
  • HPE combination: 30 is the largest combined gain at κloop = 0.166 when TSM is combined with HPE.The two methods remove noisy short-range contributions through differently organized separations, so their combined gain is below the product of isolated gains.
  • Mass dependence: 6–19 are the combined TSM+HPE gains at the lightest quark mass, with the range running from Γ = 1 to Γ = γ3γ5.TSM remains largely quark-mass independent, whereas HPE becomes less effective at lighter masses.
  • TEA: Deflating 20 eigenmodes reduces iterations and more than doubles normalized gain factors, but the improvement is attributed to solver acceleration rather than lower stochastic variance.TSM+HPE+TEA reaches break-even with TSM+HPE at a cost equivalent to approximately 100 estimates.

4. Application to ∆qdis and to ⟨N| ¯qq|N⟩dis

The study applies stochastic estimators and noise-reduction methods to disconnected axial and scalar nucleon matrix elements, using finite-time analyses and multiple quark masses. It finds substantial axial-channel efficiency gains and reports a near-zero strange-spin contribution, while scalar results remain less precise and unrenormalized.

  • Matrix-element extraction: The disconnected contributions Δq_dis and ⟨N|q̄q|N⟩_dis are extracted from zero-momentum three-point-to-two-point-function ratios.The scalar and axial channels use different source, sink, and loop projection operators.
  • Lattice setup: Excited-state contributions are expected to be negligible for the symmetric choice t_f = 6 after Wuppertal and APE smearing.The assumption is checked by varying t_f ≥ 4.
  • Lattice setup: The calculation combines time partitioning, TSM, and HPE, costing 6 preconditioned CG solves for the scalar channel and 100 for the axial channel.The loop is inserted at t = 3, approximately 0.38 fm/a.
  • Noise reduction: The scalar channel’s fixed-cost relative-error reductions are small because the correlation with the two-point function dominates the noise.Increasing nucleon sources, potentially with low-mode averaging, is identified as a more effective route for improving total precision.
  • Noise reduction: The axial channel achieves about ten-fold computer-time gains from TSM and HPE relative to time partitioning alone at an equivalent cost of 100 estimates.Its error reductions are close to those obtained for the disconnected loop alone, and the channel is not yet limited by two-point-function accuracy.
  • Final results: Δs = −0.020(11) at the heavier proton mass and −0.017(21) at the lighter mass, with no significant dependence on valence or loop quark mass.The scalar matrix elements are somewhat larger than one; all lattice results are unrenormalized.

5. Conclusions and outlook

The conclusions present TSM as an unbiased, broadly applicable method that reduces stochastic-estimation cost and works effectively with other variance-reduction techniques. In the nucleon benchmark, gauge noise becomes dominant after stochastic reduction, while Δs is near zero and the scalar result remains limited by precision and systematic studies.

  • Conclusions: TSM typically reduces the computer time needed to reach a given stochastic variance by an order of magnitude without introducing bias.Its gain factors remain stable over m_PS from 600 MeV to 300 MeV, and it can be used with any fermionic action.
  • Conclusions: Combining TSM with partitioning, HPE, and TEA further reduces stochastic variance and is straightforward to implement.The methods are presented as applicable beyond the benchmark nucleon-structure calculation.
  • Error balance: After stochastic reduction, gauge errors dominate the uncertainty in the disconnected nucleon-structure calculation.On 2 fm lattices, the scalar matrix element’s stochastic error is overshadowed by gauge error even with only 6 CG solves.
  • Error balance: Increasing nucleon sources can reduce total error more efficiently than increasing improved stochastic estimates when gauge noise dominates.The same strategy is expected to benefit the determination of Δs.
  • Outlook: The measured Δs is approximately zero and agrees with another recent direct calculation, whereas the scalar matrix element is not yet precise enough for a meaningful number.Chiral and infinite-volume behavior still need to be studied.

Appendix A. TSM parameter tuning for the CG and BiCGstab2 solvers

Appendix A tunes TSM’s truncated-solve iteration count and correction-estimate ratio by minimizing variance at fixed cost. It finds stable CG optimization, less smooth BiCGstab2 behavior, and a preference for CG within TSM.

  • Optimization criterion: TSM tuning varies the inexact-solve iteration count n_t and the estimate ratio N_1/N_2 to minimize disconnected-loop variance at fixed cost.The optimization uses a factorized variance expression for large N_1 and N_2.
  • Solver dependence: The dependence of the variance components on n_t is illustrated for CG and BiCGstab2 using 300 stochastic sources on one configuration.The first-term contribution becomes dominant while the correction-term contribution approaches zero as n_t increases.
  • CG tuning: n_t ≈ 18 and N_1/N_2 ≈ 30 are read from the CG optimization procedure for Γ = γ_3γ_5.These values are reported as the optimized parameters in the appendix’s example.
  • CG tuning: Changing the CG ratio from N_1/N_2 = 30 to the equal-cost value ≈35 increases the final variance by only 3%.The authors characterize this increase as statistically insignificant.
  • BiCGstab2 tuning: BiCGstab2 convergence is not a smooth function of n_t, complicating derivative-based optimization.The appendix therefore uses an alternative approach to identify reasonable BiCGstab2 parameters.
  • Solver comparison: CG achieves gains of 4.9 for Γ = 1 and 7.9 for Γ = γ_3γ_5, similar to BiCGstab2’s 4.9 and 7.3 gains.CG is more robust across n_t and is preferred for TSM, although the solvers can be combined.
Loading 0910.3970v2…