An Accelerated Conjugate Gradient Algorithm to Compute Low-Lying Eigenvalues --- a Study for the Dirac Operator in SU(2) Lattice QCD

Abstract

The low-lying eigenvalues of a (sparse) hermitian matrix can be computed with controlled numerical errors by a conjugate gradient (CG) method. This CG algorithm is accelerated by alternating it with exact diagonalisations in the subspace spanned by the numerically computed eigenvectors. We study this combined algorithm in case of the Dirac operator with (dynamical) Wilson fermions in four-dimensional $\SUtwo$ gauge fields. The algorithm is numerically very stable and can be parallelized in an efficient way. On lattices of sizes $4^4-16^4$ an acceleration of the pure CG method by a factor of~$4-8$ is found.

25 pages, uuencoded tar-compressed .ps-file

References in corpus (2)

• A New Approach to the Problem of Dynamical Quarks in Numerical Simulations of Lattice QCD
• Universal fluctuations in spectra of the lattice Dirac operator

Cited by in corpus (104)

• Non-perturbative O(a) improvement of lattice QCD
• Locality properties of Neuberger's lattice Dirac operator
• Effective noise reduction techniques for disconnected loops in Lattice QCD
• O(a) improvement of lattice QCD with two flavors of Wilson quarks
• A practical implementation of the Overlap-Dirac operator
• Efficient $\mathcal{O}(N^2)$ approach to solve the Bethe-Salpeter equation for excitonic bound states
• Quenched Lattice QCD with Domain Wall Fermions and the Chiral Limit
• The chiral transition and U(1)_A symmetry restoration from lattice QCD using Domain Wall Fermions
• The QCD chiral transition, $\ua$ symmetry and the Dirac spectrum using domain wall fermions
• Numerical techniques for lattice QCD in the $\epsilon$--regime
• Staggered eigenvalue mimicry
• A study of chiral symmetry in quenched QCD using the Overlap-Dirac operator
• Stability of lattice QCD simulations and the thermodynamic limit
• Simple Glass Models and their Quantum Annealing
• Quark spectra, topology and random matrix theory
• Lattice QCD in the epsilon-regime and random matrix theory
• A Study of Practical Implementations of the Overlap-Dirac Operator in Four Dimensions
• Microscopic Origin of \boldmath{$U_A(1)$} Symmetry Violation in the High Temperature Phase of QCD
• First-order transitions and the performance of quantum algorithms in random optimization problems
• Improving meson two-point functions in lattice QCD
• Spectral flow, condensate and topology in lattice QCD
• Scaling tests of the improved Kogut-Susskind quark action
• A Polynomial Hybrid Monte Carlo Algorithm
• A variant approach to the overlap action
• Scale-invariance of parity-invariant three-dimensional QED
• Finite-size scaling of the quark condensate in quenched lattice QCD
• Non-equilibrium study of the Chiral Magnetic Effect from real-time simulations with dynamical fermions
• Low-lying fermion modes, topology and light hadrons in quenched QCD
• Chiral Condensate in the Deconfined Phase of Quenched Gauge Theories
• No evidence for bilinear condensate in parity-invariant three-dimensional QED with massless fermions
• Physics issues in simulations with dynamical overlap fermions
• Short distance current correlators: Comparing lattice simulations to the instanton liquid
• Axial Correlation Functions in the epsilon-Regime: a Numerical Study with Overlap Fermions
• Chiral symmetry breaking at large N_c
• The hermitian Wilson-Dirac operator in smooth SU(2) instanton backgrounds
• A New Method for Determining $F_{\pi }$ on the Lattice
• Approach to the Continuum Limit of the Quenched Hermitian Wilson-Dirac Operator
• Inherently Global Nature of Topological Charge Fluctuations in QCD
• Center Vortices and Chiral Symmetry Breaking in SU(2) Lattice Gauge Theory
• Local Chirality of Low-Lying Dirac Eigenmodes and the Instanton Liquid Model
• Topology, fermionic zero modes and flavor singlet correlators in finite temperature QCD
• Eigenvalue spectra of QCD and the fate of $U_A(1)$ breaking towards the chiral limit
• Accelerated Overlap Fermions
• Witten-Veneziano Relation, Quenched QCD, and Overlap Fermions
• Dynamical simulation of N=1 supersymmetric Yang-Mills theory with domain wall fermions
• The PHMC algorithm for simulations of dynamical fermions: I -- description and properties
• Numerical Methods for the QCD Overlap Operator IV: Hybrid Monte Carlo
• Filtered overlap: speedup, locality, kernel non-normality and Z_A~1
• Topological susceptibility from the overlap
• An Efficient Algorithm for QCD with Light Dynamical Quarks
• Domain Wall Fermions with Exact Chiral Symmetry
• Low-lying Eigenvalues of the QCD Dirac Operator at Finite Temperature
• Chirality Correlation within Dirac Eigenvectors from Domain Wall Fermions
• Staggered Fermions and Gauge Field Topology
• A Quark-Antiquark Condensate in Three-Dimensional QCD
• Hadron masses in QCD with one quark flavour
• Are Topological Charge Fluctuations in QCD Instanton Dominated?
• Probing the Region of Massless Quarks in Quenched Lattice QCD using Wilson Fermions
• Topology and chiral symmetry breaking in SU(N) gauge theories
• Light hadron and diquark spectroscopy in quenched QCD with overlap quarks on a large lattice
• Eigenvalue Decomposition of Meson Correlators
• The PHMC algorithm for simulations of dynamical fermions: II - Performance analysis
• Ghost sector and geometry in minimal Landau gauge: further constraining the infinite-volume limit
• The topological objects near the chiral crossover transition in QCD
• Quenched Results for Light Quark Physics with Overlap Fermions
• Scale-invariance and scale-breaking in parity-invariant three-dimensional QCD
• Exact Zero-Modes of the Compact QED Dirac Operator
• Small eigenvalues of the staggered Dirac operator in the adjoint representation and Random Matrix Theory
• Dynamical fermions as a global correction
• On Witten's global anomaly for higher SU(2) representations
• The topological structures in strongly coupled QGP with chiral fermions on the lattice
• Towards a semi-classical description of QCD vacuum around $T_c$
• Numerical study of lattice index theorem usingimproved cooling and overlap fermions
• Low-lying eigenmodes of the Wilson-Dirac operator and correlations with topological objects
• Adjoint modes as probes of gauge field structure
• QED${}_3$-inspired three-dimensional conformal lattice gauge theory without fine-tuning
• Manifestations of the axial anomaly in finite temperature QCD
• Divergent chiral condensate in the quenched Schwinger model
• Quenched divergences in the deconfined phase of SU(2) gauge theory
• Iterative methods for overlap and twisted mass fermions
• Monopole and vortex content of a meron pair
• Residual Chiral Symmetry Breaking in Domain-Wall Fermions
• Low-Lying Eigenvalues of the Wilson-Dirac Operator
• Global anomalies in Chiral Lattice Gauge Theory
• Reducing Residual-Mass Effects for Domain-Wall Fermions
• Ultraviolet filtering of lattice configurations and applications to Monte Carlo dynamics
• Solving the Bethe-Salpeter equation on massively parallel architectures
• SU(2) vortex configuration in Laplacian Center Gauge
• Spectrum of the Hermitian Wilson Dirac operator
• QED2 as a testbed for interpolations between quenched and full QCD
• Comparing iterative methods for overlap and twisted mass fermions
• Topological structure of the SU(3) vacuum and exceptional eigenmodes of the improved Wilson-Dirac operator
• Analysis of systematic errors in the calculation of renormalization constants of the topological susceptibility on the lattice
• Interaction dependence of composite fermion effective masses
• Understanding the approach to thermalization from the eigenspectrum of non-Abelian gauge theories
• Chiral symmetry breaking from Ginsparg-Wilson fermions
• A curious behavior of three-dimensional lattice Dirac operators coupled to monopole background
• Analyzing the spectrum of general, non-hermitian Dirac operators
• Index theorem with Minimally Doubled Fermions in four space-time dimensions
• Probing QCD instantons using jet correlation observables in proton-proton collisions at the LHC
• Numerical experiments using deflation with the HISQ action
• Numerical determination of partial spectrum of Hermitian matrices using a Lanczos method with selective reorthogonalization
• Overlap Dirac Operator, Eigenvalues and Random Matrix Theory
• Imprints of $U_A(1)$ chiral anomaly and disorder in the Dirac eigenspectrum of QCD at finite temperature