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REVIEW 4 major objections 6 minor 31 references

Accelerating multigrid with streaming chiral SVD for Wilson fermions in lattice QCD

T0 review · 4 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper claims that replacing the conventional multigrid test-vector basis with a chiral SVD truncation improves Wilson-fermion solver convergence, cutting fine-level iterations by a factor of 1.7 near the critical mass.

desk verdict Useful Wilson-multigrid setup improvement with a genuine streaming contribution; the greedy rank-k streaming update needs a robustness check, but the measured gains are credible. read the letter →

arxiv 2505.14399 v1 pith:E6AZWWTH submitted 2025-05-20 hep-lat cs.NAmath.NA

Travis Whyte , Andreas Stathopoulos , Eloy Romero This is my paper · ORCID
classification hep-latcs.NAmath.NA
keywords latticeQCDWilsonfermionsmultigridsingularvaluedecompositionstreamingSVDpreconditionercriticalslowingdownchiralsplitting
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper tries to establish a better way to build the multigrid preconditioner used to solve the Wilson–Dirac equation in lattice QCD. Instead of using all the smoothed test vectors that adaptive multigrid produces, the authors sample a larger basis, split each vector into its two chiral components, and keep only the top k left singular vectors of each domain's SVD. This truncated basis forms prolongation and restriction operators that retain more near-null-space information without enlarging the coarse-grid matrices. A streaming variant that updates the SVD incrementally avoids storing all test vectors, which matters on memory-limited GPUs, and the paper reports that it matches the full SVD's convergence and reduces fine-level iterations by a factor of 1.7 near the critical quark mass.

What carries the argument

The central object is the chiral SVD truncation of the multigrid test-vector basis. In adaptive multigrid for Wilson fermions, the smoother produces m smooth test vectors; the new setup computes the singular value decomposition of the chirally split test vectors restricted to each spacetime domain and keeps the top k left singular vectors (Algorithm 2). Because the SVD delivers an optimal rank-k approximation of the sampled near-null space, the prolongator P = U_k carries the dominant smooth modes without the cost of a coarse matrix built from all m vectors. The streaming machinery is the incremental SVD (iCSVD, Algorithm 3): each batch of r new test vectors is appended to the current rank-k approximation Phi_k = U_k Sigma_k, the SVD is recomputed, and the top k left singular vectors are kept, so only O(n m) storage is needed instead of O(n n_s m).

What would settle it

Compare the iCSVD setup against the full-batch CSVD setup on a gauge configuration whose test-vector singular spectrum decays linearly, not superlinearly, over the first k singular values; if the iteration counts of the resulting solver change with the number of streams or the ordering of the batches, then the streaming rank-k update is discarding information that the full SVD would keep.

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Extended reading notes

Core claim

The paper's central claim is that a rank-k singular value decomposition applied separately to the chiral components of a larger-than-usual test-vector basis produces a multigrid prolongator for Wilson fermions that is more effective than the one built from the conventional basis, at the same coarse-grid size. The numerical experiments on an anisotropic Hadron Spectrum lattice ($32^{3}$ x 256, m_pi approximately 239 MeV) and an isotropic MILC Clover-on-HISQ lattice ($32^{3}$ x 64, m_pi approximately 220 MeV) show that the CSVD setup reduces mean solve time for all tested basis sizes, setup iteration counts, truncation ranks, and lattice volumes ($24^{3}$, $32^{3}$, $40^{3}$). The streaming iCSVD version, which processes test vectors in batches and keeps only the rank-k SVD approximation between batches, reproduces the non-streaming performance at a fraction of the storage, and at m_q approximately m_crit on the MILC configuration it lowers the number of fine-level iterations by about 1.7x compared with conventional multigrid.

Load-bearing premise

The load-bearing premise is that retaining only the top k singular directions after each streaming batch preserves essentially all of the near-null-space information that a full SVD over all sampled test vectors would provide, so the approximate left singular vectors build as good a multigrid preconditioner as the exact ones.

Editorial extensions

If this is right

  • On both tested ensembles, increasing the sampled basis size m while holding the truncation rank k fixed reduces mean solve time down to a saturation point around m = 96 to 128, so the method's benefit comes from sampling more smooth modes without enlarging the coarse grid.
  • The speedup from CSVD is roughly constant across lattice volumes 24^3, 32^3, and 40^3, so the method's advantage is not washed out by critical slowing down as volume grows.
  • The iCSVD setup uses O(n m) storage instead of O(n n_s m) and reduces SVD cost by a factor of about n_s, while reproducing the non-streaming iteration counts in the reported experiments.
  • Near the critical quark mass on the MILC ensemble, moving from conventional multigrid to iCSVD cuts fine-level iterations by roughly a factor of 1.7 and keeps outer iterations stable as the number of streams grows.

Reading between the lines

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

  • If the streaming truncation remains as accurate on configurations with flatter singular spectra, iCSVD should let GPU-based Wilson-fermion solvers sample far more test vectors than their memory permits today, pushing the saturation point seen in the paper's size study to larger m.
  • The same chiral-SVD truncation should transfer to other fermion discretizations, such as domain-wall or overlap formulations, whenever their multigrid setup uses chiral projectors; the paper does not test those cases.
  • The paper's least-squares interpolation connection suggests that tuning the weight matrix Omega away from the identity could yield prolongators that are not simply U_k and might outperform the SVD basis on near-null components that the SVD weights equally.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 6 minor

Summary. This manuscript proposes a modification to the adaptive multigrid setup for the Wilson-Dirac operator in lattice QCD. Instead of using the k smoothed test vectors directly, the method generates m>k test vectors, applies the chiral projectors, and then performs a per-domain SVD of the chiral components, truncating to the top k left singular vectors to build the prolongator and restriction operators. To reduce storage on memory-constrained platforms such as GPUs, the authors also introduce a streaming/incremental chiral SVD (iCSVD) that updates a rank-k basis batch by batch. The methods are tested on an anisotropic Hadron Spectrum ensemble and an isotropic MILC ensemble, including volume scaling and a near-critical-mass case. The central numerical claims are that CSVD reduces mean solve time compared with conventional multigrid, that iCSVD matches CSVD at physical-like masses, and that iCSVD reduces fine-level iterations by a factor of about 1.7 at mq≈mcrit.

Significance. If the streaming variant can be validated, the paper offers a simple and useful upgrade to the multigrid setup for Wilson fermions, with a clear connection to existing aggregation AMG work. The algorithmic presentation is readable, the comparison against conventional multigrid is appropriate, and the numerical experiments cover two different lattice actions, a volume scan, and a difficult near-critical-mass regime. The main weaknesses are the unquantified subspace loss of the streaming update, the absence of setup and total wall-clock timings for iCSVD, and the lack of error bars; these are fixable with additional experiments.

major comments (4)
  1. [§3.1, Eq. (8), Tables 1–2] The central claim for iCSVD rests on the assumption in Eq. (8) that a rank-k truncation after each stream preserves enough near-null-space information for an effective prolongator. Section 3.1 states that streaming SVD accuracy is 'typically higher than what is needed to build multigrid prolongators,' but gives no proof, reference, or sensitivity study. Because the update is greedy, a direction that is not among the top k singular vectors of an early batch can still belong to the top k subspace of the union of all batches, and this loss is never quantified. Tables 1 and 2 show that iCSVD matches or improves on CSVD in iteration counts on the tested ensembles, but iterations are a coarse metric: the K-cycle and outer FGMRES can mask a degraded coarse basis. Please provide a controlled comparison at mq≈mcrit in which CSVD and iCSVD sample the same total number of test vectors, and quantify the distance between the streamed basis and the batch-SVD basis (for example via principal angles or relative residual norms). Without such evidence, the 1.7x fine-iteration claim may not transfer to the memory-constrained GPU setting that motivates the streaming method.
  2. [§4.5, Table 2] The practical acceleration claim is incomplete because no setup or total wall-clock times are reported for iCSVD. All timing figures in Section 4 (Figures 1, 3, 4, 5, and 7) report solve time only, and the text acknowledges that at the largest number of streams the setup phase is costly. Since the title and abstract promise acceleration, please report setup time, solve time, and total time for the near-critical-mass case for every ns value used in Table 2. If the total time is not improved, the contribution should be framed as an iteration-count improvement rather than an end-to-end speedup.
  3. [§4.1–§4.3] The optimal values of m, k, and the number of setup iterations are selected from the same configurations that are later used to report the speedups, and the mean solve times and iteration counts are quoted without error bars or standard deviations. This makes it difficult to assess statistical significance and introduces a selection-bias risk. Please report at least the standard deviation over the ten right-hand sides, and preferably validate the chosen parameters on an independent configuration or discuss the tuning cost explicitly.
  4. [§3.1, Algorithm 3, §4.5] The notation for the streaming method is internally inconsistent. The text defines m=k+r and Algorithm 3 requires m=k+r while also stating 'Initially m=r'; Section 4.5 then says 'm=48 with k=48 and r=48,' which is incompatible with m=k+r. As a result, the reader cannot determine the actual column dimension of Φ in Eq. (8) or the total number of test vectors sampled in the CSVD versus iCSVD columns of Table 2. Please introduce a distinct symbol for the per-stream matrix column count and use it consistently throughout the algorithms and experimental text.
minor comments (6)
  1. [Table 1 caption] The caption reads 'Table 1: .' with an empty placeholder; it should describe the quantities reported.
  2. [Abstract and §5] The abstract says a decrease by a factor of 1.7, while Section 5 says 'speedup of 170%'; as a percentage, 170% would normally mean a factor of 2.7. Please use consistent phrasing, e.g., a 1.7x reduction.
  3. [Introduction] Please proofread for stray spacing in words such as 'di fficulty' and 'su fficiently'.
  4. [Figure 1 caption] In the captions of Figures 1 and 3, 'the bottom and top y-axis displays' should be 'the bottom and top y-axes display'.
  5. [§4.4] The statement that the first domain's singular spectrum is representative of other domains would be more convincing with a measure of domain-to-domain variation; please state how this was checked.
  6. [§4.5] The sentence describing the comparable test-vector counts ('m=48,64 with k=24,48 and r=24,32 for ns=4 streams') should explicitly assign parameters to levels ℓ=0 and ℓ=1 to avoid ambiguity.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the claimed speedups are empirical comparisons measured on independent right-hand sides, with no fitted parameter renamed as a prediction and no load-bearing self-citation chain.

full rationale

The paper's central claim is that a multigrid setup using SVD-truncated chiral test vectors (CSVD) and its streaming variant (iCSVD) reduce solver iteration counts and solve times compared with conventional multigrid. These claims are supported by direct measurements: Tables 1 and 2 report mean iteration counts from linear solves averaged over ten right-hand sides, and Figures 1, 3, 4, 5, and 7 report measured solve times. No equation in the paper derives a predicted speedup from a fitted constant; the SVD truncation is a data-compression step used to build the prolongator, and its effect is assessed empirically against the conventional setup on the same systems. The streaming update in Eq. (8) is a definition of the incremental algorithm, not a circular derivation: the paper does not claim to prove analytically that iCSVD matches CSVD, but instead demonstrates it numerically (e.g., 'It is observed that there is no difference between the performance of the preconditioner created with CSVD and iCSVD methods', Section 4.5). The statement in Section 3.1 that streaming SVD accuracy is 'typically higher than what is needed to build multigrid prolongators' is an unproven assumption and a correctness risk, but it is not a circular step because the subsequent validation is experimental rather than derived from that assumption. There are no load-bearing self-citations: the cited adaptive multigrid [3], rank-k SVD aggregation [15], composite aggregation [14], and incremental SVD [18] are all external prior works, and the authors do not invoke any uniqueness theorem from their own papers. The closest mathematical statement, Eq. (9) in the Summary, is an algebraic identity showing that for a specific choice of least-squares weights the left singular vectors form the optimal prolongator; this is a justification, not an input-output equivalence. The only methodological caveat is that hyperparameters such as m, k, and the number of setup iterations are tuned on the same ensembles used for the final speedup tables, which reduces the strength of the quantitative claims as predictions but does not make them circular: they remain comparative measurements of the proposed algorithm against a baseline. Accordingly, no step in the derivation chain reduces by construction to its own inputs, so the circularity score is 0.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The method introduces no physical entities and no fitted physical constants. Its quantitative claims rest on hand-tuned algorithmic parameters (m, k, ns, r) selected on the same ensembles used for reporting, and on the empirical validity of local coherence and of the streaming SVD accuracy assumption.

free parameters (4)
  • truncation rank k = k0 = 24, k1 = 32, and k = 48 at the coarsest level in the near-critical streaming case
    Chosen by scanning k at fixed m = 96 on the same ensembles used to report results (Section 4.2, Figure 3).
  • oversampled basis size m = m0 = 96, m1 = 128, and m = 48 at the coarsest level in the streaming case
    Selected based on the saturation in mean solve time observed in Figure 1; Tables 1 and 2 use these values.
  • number of streams ns and batch size r = ns swept from 4 to 16, with r = 24 and r = 32 at levels 0 and 1
    The streaming experiments sweep ns and report the best case at ns = 16 in Table 2; r is tied to k in Algorithm 3.
  • number of setup smoother iterations = Ranging from near 0 to 600 in Figures 4 and 5, with no single operational value fixed
    The paper shows CSVD is less sensitive to this parameter than regular multigrid, but the actual choice affects the setup cost and is not pinned down for production use.
assumptions (4)
  • standard math Eckart-Young theorem: the rank-k SVD gives the best rank-k approximation in the 2-norm.
    Used in Section 3 to justify that truncating the test-vector basis to its first k left singular vectors is optimal in the 2-norm for each domain.
  • domain assumption Local coherence: smooth test vectors for the Wilson-Dirac operator approximate the low-lying eigenspace locally on each domain.
    Invoked in Section 2.1 via Refs. [11,12] and underlies the per-domain truncation in Algorithm 2.
  • domain assumption Preserving the gamma5 symmetry via chiral splitting, enforcing the relation gamma5 P = P sigma3, is required for the multigrid correction to work with R = P dagger.
    Stated in Section 2.1; the CSVD method preserves the symmetry by performing the SVD separately on the two chiral components.
  • ad hoc to paper The streamed rank-k SVD subspace is accurate enough for building multigrid prolongators.
    Assumed in Section 3.1 without proof and verified only empirically in Section 4.5; all iCSVD results depend on this assumption.

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Pith. "Pith review of Accelerating multigrid with streaming chiral SVD for Wilson fermions in lattice QCD." pith.science (2026). https://pith.science/paper/E6AZWWTH

@misc{pith2026250514399,
  author       = {Pith},
  title        = {Pith review of: Accelerating multigrid with streaming chiral SVD for Wilson fermions in lattice QCD},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/E6AZWWTH}},
  note         = {Machine review of arXiv:2505.14399}
}
abstract

A modification to the setup algorithm for the multigrid preconditioner of Wilson fermions in lattice QCD is presented. A larger basis of test vectors than that used in conventional multigrid is calculated by the smoother and truncated by singular value decomposition on the chiral components of the test vectors. The truncated basis is used to form the prolongation and restriction matrices of the multigrid hierarchy. This modification of the setup method is demonstrated to increase the convergence of linear solvers on an anisotropic lattice with $m_{\pi} \approx 239$ MeV from the Hadron Spectrum Collaboration and an isotropic lattice with $m_{\pi} \approx 220$ MeV from the MILC Collaboration. The lattice volume dependence of the method is also examined. Increasing the number of test vectors improves speedup up to a point, but storing these vectors becomes impossible in limited memory resources such as GPUs. To address storage cost, we implement a \emph{streaming} singular value decomposition of the basis of test vectors on the chiral components and demonstrate a decrease in the number of fine level iterations by a factor of 1.7 for $m_q \approx m_{crit}$.

Figures

Figures reproduced from arXiv: 2505.14399 by the authors.

Figure 1
Figure 1. The mean solve time as a function of the initial basis size for Configuration A (left) and Configuration B (right). [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. The singular spectrum for the chirally split test vectors on the first domain for level [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. The mean execution time for the system of linear equations when the degree of the truncation is varied using [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: The total number of iterations on ℓ = 0 (upper left), ℓ = 1 (upper right), ℓ = 2 (lower left) and the mean solve time of the system of linear equations for Configuration A. Figures 4 and 5, for configurations A and B, respectively, display the total number of itera￾tio…
Figure 5
Figure 5. Figure 5: As Figure 4 for Configuration B [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: The normalized number of iterations on level [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]
Figure 7
Figure 7. Figure 7: The performance of both MG preconditioners as the lattice volume is increased. [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]
Figure 6
Figure 6. Figure 6: At ℓ = 1, the number of iterations is reduced by approximately 40%. In contrast, ℓ = 2 displays similar performance across all methods. The largest effect is seen on the coarsest level, where the number of normalized iterations increases drastically for the methods uti…

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Works this paper leans on

31 extracted references · 24 canonical work pages

  1. [1]

    Saad, Iterative methods for sparse linear systems, SIAM, 2003

    Y . Saad, Iterative methods for sparse linear systems, SIAM, 2003

    work page 2003

  2. [2]

    T. Kalkreuter, Multigrid methods for propagators in lattice gauge theories, Journal of Computational and Applied Mathematics 63 (1) (1995) 57–68, proceedings of the International Symposium on Mathematical Modelling and Computational Methods Modelling 94. doi:https://doi.org/10.1016/0377-0427(95)00049-6 . URL https://www.sciencedirect.com/science/article/p...

    arXiv 1995

  3. [3]

    Brannick, R

    J. Brannick, R. C. Brower, M. A. Clark, J. C. Osborn, C. Rebbi, Adaptive multigrid algorithm for lattice qcd, Phys. Rev. Lett. 100 (2008) 041601. doi:10.1103/PhysRevLett.100.041601. URL https://link.aps.org/doi/10.1103/PhysRevLett.100.041601

    work page doi:10.1103/physrevlett.100.041601 2008

  4. [4]

    M. A. Clark, B. Joó, A. Strelchenko, M. Cheng, A. Gambhir, R. Brower, Accelerating lattice qcd multigrid on gpus using fine-grained parallelization (2016). arXiv:1612.07873. URL https://arxiv.org/abs/1612.07873

    arXiv 2016

  5. [5]

    R. C. Brower, E. Weinberg, M. A. Clark, A. Strelchenko, Multigrid algorithm for staggered lattice fermions, Phys. Rev. D 97 (2018) 114513. doi:10.1103/PhysRevD.97.114513. URL https://link.aps.org/doi/10.1103/PhysRevD.97.114513

    work page doi:10.1103/physrevd.97.114513 2018

  6. [6]

    R. C. Brower, M. A. Clark, E. Weinberg, D. Howarth, Multigrid for chiral lattice fermions: Domain wall, Phys. Rev. D 102 (2020) 094517. doi:10.1103/PhysRevD.102.094517. URL https://link.aps.org/doi/10.1103/PhysRevD.102.094517

    work page doi:10.1103/physrevd.102.094517 2020

  7. [7]

    Brannick, A

    J. Brannick, A. Frommer, K. Kahl, B. Leder, M. Rottmann, A. Strebel, Multigrid preconditioning for the overlap operator in lattice QCD, Numerische Mathematik 132 (3) (2016) 463–490

    work page 2016

  8. [8]

    Frommer, K

    A. Frommer, K. Kahl, S. Krieg, B. Leder, M. Rottmann, Adaptive aggregation-based domain decomposition multi- grid for the lattice wilson–dirac operator, SIAM journal on scientific computing 36 (4) (2014) A1581–A1608

    work page 2014

Show all 31 references
  1. [9]

    Espinoza-Valverde, A

    J. Espinoza-Valverde, A. Frommer, G. Ramirez-Hidalgo, M. Rottmann, Coarsest-level improvements in multigrid for lattice qcd on large-scale computers, Computer Physics Communications 292 (2023) 108869. doi:https: //doi.org/10.1016/j.cpc.2023.108869. URL https://www.sciencedirec...

    2023

  2. [10]

    Lehner, T

    C. Lehner, T. Wettig, Gauge-equivariant neural networks as preconditioners in lattice QCD, Phys. Rev. D 108 (2023) 034503. doi:10.1103/PhysRevD.108.034503. URL https://link.aps.org/doi/10.1103/PhysRevD.108.034503

    2023 doi

  3. [11]

    Lüscher, Deflation acceleration of lattice qcd simulations, Journal of High Energy Physics 2007 (12) (2007)

    M. Lüscher, Deflation acceleration of lattice qcd simulations, Journal of High Energy Physics 2007 (12) (2007)

    2007

  4. [12]

    URL https://dx.doi.org/10.1088/1126-6708/2007/12/011 15

    doi:10.1088/1126-6708/2007/12/011. URL https://dx.doi.org/10.1088/1126-6708/2007/12/011 15

    2007 doi

  5. [13]

    Lüscher, Local coherence and deflation of the low quark modes in lattice QCD, Journal of High Energy Physics 2007 (07) (2007) 081

    M. Lüscher, Local coherence and deflation of the low quark modes in lattice QCD, Journal of High Energy Physics 2007 (07) (2007) 081. doi:10.1088/1126-6708/2007/07/081. URL https://dx.doi.org/10.1088/1126-6708/2007/07/081

    2007 doi

  6. [14]

    Notay, P

    Y . Notay, P. S. Vassilevski, Recursive krylov-based multigrid cycles, Numerical Linear Algebra with Applications 15 (5) (2008) 473–487. arXiv:https://onlinelibrary.wiley.com/doi/pdf/10.1002/nla.542, doi: https://doi.org/10.1002/nla.542. URL https://onlinelibrary.wiley.com/doi...

    2008 doi

  7. [15]

    D’Ambra, P

    P. D’Ambra, P. S. Vassilevski, Improving solve time of aggregation-based adaptive amg, Numerical Linear Algebra with Applications 26 (6) (2019) e2269

    2019

  8. [16]

    Chow, An aggregation multilevel method using smooth error vectors, SIAM Journal on Scientific Computing 27 (5) (2006) 1727–1741

    E. Chow, An aggregation multilevel method using smooth error vectors, SIAM Journal on Scientific Computing 27 (5) (2006) 1727–1741. arXiv:https://doi.org/10.1137/040608192, doi:10.1137/040608192. URL https://doi.org/10.1137/040608192

    2006 doi

  9. [17]

    Brandt, J

    A. Brandt, J. Brannick, K. Kahl, I. Livshits, Bootstrap amg, SIAM Journal on Scientific Computing 33 (2) (2011) 612–632. arXiv:https://doi.org/10.1137/090752973, doi:10.1137/090752973. URL https://doi.org/10.1137/090752973

    2011 doi

  10. [18]

    Peardon, J

    M. Peardon, J. Bulava, J. Foley, C. Morningstar, J. Dudek, R. G. Edwards, B. Joo, H.-W. Lin, D. G. Richards, K. J. Juge, et al., Novel quark-field creation operator construction for hadronic physics in lattice QCD, Physical Review D—Particles, Fields, Gravitation, and Cosmolog...

    2009

  11. [19]

    M. Brand, Incremental singular value decomposition of uncertain data with missing values, in: Computer Vi- sion—ECCV 2002: 7th European Conference on Computer Vision Copenhagen, Denmark, May 28–31, 2002 Proceedings, Part I 7, Springer, 2002, pp. 707–720

    2002

  12. [20]

    H.-W. Lin, S. D. Cohen, J. Dudek, R. G. Edwards, B. Joó, D. G. Richards, J. Bulava, J. Foley, C. Morningstar, E. Engelson, S. Wallace, K. J. Juge, N. Mathur, M. J. Peardon, S. M. Ryan, First results from 2+1 dynamical quark flavors on an anisotropic lattice: Light-hadron spect...

    2009 doi

  13. [21]

    R. G. Edwards, B. Joo, H.-W. Lin, Tuning for three flavors of anisotropic clover fermions with stout-link smearing, Physical Review D—Particles, Fields, Gravitation, and Cosmology 78 (5) (2008) 054501

    2008

  14. [22]

    Morningstar, M

    C. Morningstar, M. Peardon, Analytic smearing of su(3) link variables in lattice QCD, Physical Review D 69 (5) (2004) 054501

    2004

  15. [23]

    D. J. Wilson, R. A. Briceño, J. J. Dudek, R. G. Edwards, C. E. Thomas, Quark-mass dependence of elastic πk scattering from qcd, Phys. Rev. Lett. 123 (2019) 042002. doi:10.1103/PhysRevLett.123.042002. URL https://link.aps.org/doi/10.1103/PhysRevLett.123.042002

    2019 doi

  16. [24]

    Bazavov, C

    A. Bazavov, C. Bernard, C. DeTar, W. Freeman, S. Gottlieb, U. M. Heller, J. E. Hetrick, J. Laiho, L. Levkova, M. Oktay, J. Osborn, R. L. Sugar, D. Toussaint, R. S. Van de Water, Scaling studies of qcd with the dynamical highly improved staggered quark action, Phys. Rev. D 82 (...

    2010 doi

  17. [25]

    Bazavov, C

    A. Bazavov, C. Bernard, J. Komijani, C. DeTar, L. Levkova, W. Freeman, S. Gottlieb, R. Zhou, U. Heller, J. Hetrick, et al., Lattice qcd ensembles with four flavors of highly improved staggered quarks, Physical Review D—Particles, Fields, Gravitation, and Cosmology 87 (5) (2013) 054505

    2013

  18. [26]

    Hasenfratz, F

    A. Hasenfratz, F. Knechtli, Flavor symmetry and the static potential with hypercubic blocking, Phys. Rev. D 64 (2001) 034504. doi:10.1103/PhysRevD.64.034504. URL https://link.aps.org/doi/10.1103/PhysRevD.64.034504

    2001 doi

  19. [27]

    Gupta, Y .-C

    R. Gupta, Y .-C. Jang, B. Yoon, H.-W. Lin, V . Cirigliano, T. Bhattacharya, Isovector charges of the nucleon from 2 + 1 + 1-flavor lattice qcd, Phys. Rev. D 98 (2018) 034503. doi:10.1103/PhysRevD.98.034503. URL https://link.aps.org/doi/10.1103/PhysRevD.98.034503

    2018 doi

  20. [28]

    Brannick, K

    J. Brannick, K. Kahl, Bootstrap algebraic multigrid for the 2d wilson dirac system, SIAM Journal on Scien- tific Computing 36 (3) (2014) B321–B347. arXiv:https://doi.org/10.1137/130934660, doi:10.1137/ 130934660. URL https://doi.org/10.1137/130934660

    2014 doi

  21. [29]

    R. G. Edwards, B. Joo, The Chroma software system for lattice QCD, Nucl. Phys. Proc. Suppl. 140 (2005) 832, [,832(2004)]. arXiv:hep-lat/0409003, doi:10.1016/j.nuclphysbps.2004.11.254

    2005 arXiv

  22. [30]

    B. Joó, D. D. Kalamkar, K. Vaidyanathan, M. Smelyanskiy, K. Pamnany, V . W. Lee, P. Dubey, W. Watson, Lattice qcd on intel ® xeon phitm coprocessors, in: J. M. Kunkel, T. Ludwig, H. W. Meuer (Eds.), Supercomputing, Springer Berlin Heidelberg, Berlin, Heidelberg, 2013, pp. 40–54

    2013

  23. [31]

    Joó, MG_PROTO: A Multigrid Library for QCD, https://github.com/JeffersonLab/mg_proto/

    B. Joó, MG_PROTO: A Multigrid Library for QCD, https://github.com/JeffersonLab/mg_proto/. 16

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Reviewed August 7, 2026 · model on record in the stance chip above.