176 research outputs found
Spectrum of the Dirac Operator and Inversion Algorithms with Dynamical Staggered Fermions
Complete spectra of the staggered Dirac operator \Dirac are determined in
four-dimensional gauge fields with and without dynamical fermions.
An attempt is made to relate the performance of multigrid and conjugate
gradient algorithms for propagators with the distribution of the eigenvalues
of~\Dirac.Comment: 3 pages, 1 figure, uuencoded tar-compressed .ps-file, contribution to
LATTICE'94, report# HUB-IEP-94/2
Universal correlations in spectra of the lattice QCD Dirac operator
Recently, Kalkreuter obtained complete Dirac spectra for lattice
gauge theory both for staggered fermions and for Wilson fermions. The lattice
size was as large as . We performed a statistical analysis of these data
and found that the eigenvalue correlations can be described by the Gaussian
Symplectic Ensemble for staggered fermions and by the Gaussian Orthogonal
Ensemble for Wilson fermions. In both cases long range spectral fluctuations
are strongly suppressed: the variance of a sequence of levels containing
eigenvalues on average is given by
( is equal to 4 and 1, respectively) instead of for a
random sequence of levels. Our findings are in agreement with the anti-unitary
symmetry of the lattice Dirac operator for with staggered fermions
which differs from Wilson fermions (with the continuum anti-unitary symmetry).
For , we predict that the eigenvalue correlations are given by the
Gaussian Unitary Ensemble.Comment: Talk present at LATTICE96(chirality in QCD), 3 pages, Late
Multigrid for propagators of staggered fermions in four-dimensional gauge fields
Multigrid (MG) methods for the computation of propagators of staggered
fermions in non-Abelian gauge fields are discussed. MG could work in principle
in arbitrarily disordered systems. The practical variational MG methods tested
so far with a ``Laplacian choice'' for the restriction operator are not
competitive with the conjugate gradient algorithm on lattices up to .
Numerical results are presented for propagators in gauge fields.Comment: 4 pages, 3 figures (one LaTeX-figure, two figures appended as
encapsulated ps files); Contribution to LATTICE '92, requires espcrc2.st
Numerical analysis of the spectrum of the Dirac operator in four-dimensional SU(2) gauge fields
Two numerical algorithms for the computation of eigenvalues of Dirac
operators in lattice gauge theories are described: one is an accelerated
conjugate gradient method, the other one a standard Lanczos method. Results
obtained by Cullum's and Willoughby's variant of the Lanczos method (whose
convergence behaviour is closely linked with the local spectral density) are
presented for euclidean Wilson fermions in quenched and unquenched SU(2) gauge
fields. Complete spectra are determined on lattices up to , and
we derive numerical values for fermionic determinants and results for spectral
densities.Comment: 6 pages, uuencoded tar-compressed ps-file, contribution to the
Proceedings of the International Symposium Ahrenshoop on the Theory of
Elementary Particles, Buckow'95, talk also given at the DESY Workshop 199
Idealized Multigrid Algorithm for Staggered Fermions
An idealized multigrid algorithm for the computation of propagators of
staggered fermions is investigated.
Exemplified in four-dimensional gauge fields, it is shown that the
idealized algorithm preserves criticality under coarsening.
The same is not true when the coarse grid operator is defined by the Galerkin
prescription.
Relaxation times in computations of propagators are small, and critical
slowing is strongly reduced (or eliminated) in the idealized algorithm.
Unfortunately, this algorithm is not practical for production runs, but the
investigations presented here answer important questions of principle.Comment: 11 pages, no figures, DESY 93-046; can be formatted with plain LaTeX
article styl
Some Comments on Multigrid Methods for Computing Propagators
I make three conceptual points regarding multigrid methods for computing
propagators in lattice gauge theory: 1) The class of operators handled by the
algorithm must be stable under coarsening. 2) Problems related by symmetry
should have solution methods related by symmetry. 3) It is crucial to
distinguish the vector space from its dual space . All the existing
algorithms violate one or more of these principles.Comment: 16 pages, LaTeX plus subeqnarray.sty (included at end),
NYU-TH-93/07/0
Spectrum of the Dirac Operator and Multigrid Algorithm with Dynamical Staggered Fermions
Complete spectra of the staggered Dirac operator \Dirac are determined in
quenched four-dimensional gauge fields, and also in the presence of
dynamical fermions.
Periodic as well as antiperiodic boundary conditions are used.
An attempt is made to relate the performance of multigrid (MG) and conjugate
gradient (CG) algorithms for propagators with the distribution of the
eigenvalues of~\Dirac.
The convergence of the CG algorithm is determined only by the condition
number~ and by the lattice size.
Since~'s do not vary significantly when quarks become dynamic,
CG convergence in unquenched fields can be predicted from quenched
simulations.
On the other hand, MG convergence is not affected by~ but depends on
the spectrum in a more subtle way.Comment: 19 pages, 8 figures, HUB-IEP-94/12 and KL-TH 19/94; comes as a
uuencoded tar-compressed .ps-fil
An Accelerated Conjugate Gradient Algorithm to Compute Low-Lying Eigenvalues --- a Study for the Dirac Operator in SU(2) Lattice QCD
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 an acceleration of the pure CG method by a factor
of~ is found.Comment: 25 pages, uuencoded tar-compressed .ps-fil
ISU - Multigrid for computing propagators
The Iteratively Smoothing Unigrid algorithm (ISU), a new multigrid method for
computing propagators in Lattice Gauge Theory, is explained. The main idea is
to compute good (i.e.\ smooth) interpolation operators in an iterative way.
This method shows {\em no critical slowing down} for the 2-dimensional Laplace
equation in an SU(2) gauge field. First results for the Dirac-operator are also
shown.Comment: 3 pages, latex, no figures, Contribution to Lattice 94, uses
espcrc2.sty and fleqn.sty as required for lattice proceeding
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