Adaptive Multigrid Algorithm for the Lattice Wilson-Dirac Operator

Babich, Ronald; Brannick, James; Brower, Richard C.; Clark, Michael A.; Manteuffel, Thomas A.; McCormick, Stephen F.; Osborn, James C.; Rebbi, C.

doi:10.1103/physrevlett.105.201602

Cited by 160 publications

(194 citation statements)

References 9 publications

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“…Systems may be considered having non-zero overall momentum, P = 2π L n x , n y , n z , which satisfy periodic boundary conditions if n x , n y , n z are integers. Such 1 The large number of propagators are very efficiently computed using an Adaptive Multi-Grid solver [32,33] for the light quarks on CPUs, and the strange quark propagators are efficiently computed using Graphical Processing Units [34,35]. Table I.…”

Section: Calculating the Finite-volume Spectrummentioning

confidence: 99%

“…The software codes Chroma [43], QUDA [34,35], QPhiX [44], and QOPQDP [32,33] In our basis we included operators that were specifically constructed to resemble both the ππ and KK states one would expect to exist in the absence of meson-meson interactions, as well as fermion bilinears,ψΓψ, which resemble qq-like constructions. As is visible in Figure 6, the low-lying states overlap with both sets of operators.…”

Section: Acknowledgmentsmentioning

confidence: 99%

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Coupledππ,KK¯scattering inP

et al. 2015

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We determine elastic and coupled-channel amplitudes for isospin-1 meson-meson scattering in P -wave, by calculating correlation functions using lattice QCD with light quark masses such that mπ = 236 MeV in a cubic volume of ∼ (4 fm)3 . Variational analyses of large matrices of correlation functions computed using operator constructions resembling ππ, KK and qq, in several moving frames and several lattice irreducible representations, leads to discrete energy spectra from which scattering amplitudes are extracted. In the elastic ππ scattering region we obtain a detailed energydependence for the phase-shift, corresponding to a ρ resonance, and we extend the analysis into the coupled-channel KK region for the first time, finding a small coupling between the channels.The study of hadron spectroscopy from first principles QCD is entering a new stage of development where the relationship between the discrete spectrum of the theory in a finite-volume and the infinite-volume scattering amplitudes is being practically utilized to study resonances. The tool which allows us access to the spectrum is lattice QCD in which the quark and gluon fields are considered on a finite-grid of points with only systematically improvable approximations being made.Predictably it is the simplest resonant scattering channel which has attracted the greatest initial interest [1][2][3][4][5], that of ππ with isospin=1, in which a low-lying elastic vector resonance called the ρ appears. These works have made use of the formalism relating the discrete spectrum at rest and in moving frames to elastic scattering amplitudes, which has been in place for many years [6][7][8][9][10]. Recently the extension to coupled-channels has been presented [11][12][13][14], and the first lattice QCD study of a coupled-channel system, that of πK, ηK in S, P and Dwaves, has appeared [15,16], showing that the energy dependence and resonant content of the scattering matrix for such a system can be extracted from finite volume spectra.To date virtually all determinations of hadron scattering amplitudes in lattice QCD calculations have worked with artificially heavy u, d quark mass values, a choice which leads to heavier than physical pseudoscalar mesons -this reduces the computational cost, allowing calculations in smaller volumes (where m π L remains large), and pushes up in energy the thresholds for multihadron scattering such as ππππ, for which a finite-volume formalism is not yet in place (but see Refs. [17][18][19][20] The Hadron Spectrum Collaboration previously computed ππ scattering using 391 MeV pions [21,22], extracting detailed spectra of QCD eigenstates from variational analysis of two-point correlation functions computed in several moving frames in three different volumes. By obtaining a significant number of energy levels in the elastic scattering region they were able to map out the energy dependence of the scattering amplitude and show that there is a narrow ρ resonance barely above ππ threshold.In this paper we deliver an extension of the work presen...

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Section: Calculating the Finite-volume Spectrummentioning

confidence: 99%

Section: Acknowledgmentsmentioning

confidence: 99%

Coupledππ,KK¯scattering inP

et al. 2015

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“…(The initial hierarchy is constructed using only k r test vectors.) On the finest grid, the vectors, v (1) , . .…”

Section: Numerical Resultsmentioning

confidence: 99%

Bootstrap Algebraic Multigrid for the 2D Wilson Dirac system

Brannick

Kahl

2014

SIAM J. Sci. Comput.

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We develop an algebraic multigrid method for solving the non-Hermitian Wilson discretization of the two-dimensional Dirac equation. The proposed approach uses a bootstrap setup algorithm based on a multigrid eigensolver. It computes test vectors which define the least squares interpolation operators by working mainly on coarse grids, leading to an efficient and integrated self-learning process for defining algebraic multigrid interpolation. The algorithm is motivated by the γ 5 -symmetry of the Dirac equation, which carries over to the Wilson discretization. This discrete γ 5 -symmetry is used to reduce a general Petrov Galerkin bootstrap setup algorithm to a Galerkin method for the Hermitian and indefinite formulation of the Wilson matrix. Kaczmarz relaxation is used as the multigrid smoothing scheme in both the setup and solve phases of the resulting Galerkin algorithm. The overall method is applied to the odd-even reduced Wilson matrix, which also fulfills the discrete γ 5 -symmetry. Extensive numerical results are presented to motivate the design and demonstrate the effectiveness of the proposed approach.

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“…There is a number of variant formulations of highly optimized multigrid solvers, which yield improvements of more than an order of magnitude for operators at the physical value of the light quark mass, as reported in Refs. [1][2][3][4].…”

Section: Introductionmentioning

confidence: 99%

Multigrid accelerated simulations for Twisted Mass fermions

2018

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Abstract. Simulations at physical quark masses are affected by the critical slowing down of the solvers. Multigrid preconditioning has proved to deal effectively with this problem. Multigrid accelerated simulations at the physical value of the pion mass are being performed to generate N f = 2 and N f = 2 + 1 + 1 gauge ensembles using twisted mass fermions. The adaptive aggregation-based domain decomposition multigrid solver, referred to as DD-αAMG method, is employed for these simulations. Our simulation strategy consists of an hybrid approach of different solvers, involving the Conjugate Gradient (CG), multi-mass-shift CG and DD-αAMG solvers. We present an analysis of the multigrid performance during the simulations discussing the stability of the method. This significant speeds up the Hybrid Monte Carlo simulation by more than a factor 4 at physical pion mass compared to the usage of the CG solver.

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Adaptive Multigrid Algorithm for the Lattice Wilson-Dirac Operator

Cited by 160 publications

References 9 publications

Coupledππ,KK¯scattering inP

Coupledππ,KK¯scattering inP

Bootstrap Algebraic Multigrid for the 2D Wilson Dirac system

Multigrid accelerated simulations for Twisted Mass fermions

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