1,206 research outputs found
Accelerating Wilson Fermion Matrix Inversions by Means of the Stabilized Biconjugate Gradient Algorithm
The stabilized biconjugate gradient algorithm BiCGStab recently presented by
van der Vorst is applied to the inversion of the lattice fermion operator in
the Wilson formulation of lattice Quantum Chromodynamics. Its computational
efficiency is tested in a comparative study against the conjugate gradient and
minimal residual methods. Both for quenched gauge configurations at beta= 6.0
and gauge configurations with dynamical fermions at beta=5.4, we find BiCGStab
to be superior to the other methods. BiCGStab turns out to be particularly
useful in the chiral regime of small quark masses.Comment: 25 pages, WUB 94-1
SSOR Preconditioning of Improved Actions
We generalize local lexicographic SSOR preconditioning for the
Sheikholeslami-Wohlert improved Wilson fermion action and the truncated perfect
free fermion action. In our test implementation we achieve performance gains as
known from SSOR preconditioning of the standard Wilson fermion action.Comment: 3 pages, Latex, 3 figures, Talk presented at Lattice'9
Linear systems solvers - recent developments and implications for lattice computations
We review the numerical analysis' understanding of Krylov subspace methods
for solving (non-hermitian) systems of equations and discuss its implications
for lattice gauge theory computations using the example of the Wilson fermion
matrix. Our thesis is that mature methods like QMR, BiCGStab or restarted GMRES
are close to optimal for the Wilson fermion matrix. Consequently,
preconditioning appears to be the crucial issue for further improvements.Comment: 7 pages, LaTeX using espcrc2.sty, 2 figures, 9 eps-files, Talk
presented at LATTICE96(algorithms), submitted to Nucl. Phys. B, Proc. Supp
A Parallel SSOR Preconditioner for Lattice QCD
A parallelizable SSOR preconditioning scheme for Krylov subspace iterative
solvers in lattice QCD applications involving Wilson fermions is presented. In
actual Hybrid Monte Carlo and quark propagator calculations it helps to reduce
the number of iterations by a factor of 2 compared to conventional odd-even
preconditioning. This corresponds to a gain in cpu-time of 30\% - 70\% over
odd-even preconditioning.Comment: Talk presented at LATTICE96(algorithms), 3 pages, LaTeX file, 3
epsf-files include
How to compute Green's Functions for entire Mass Trajectories within Krylov Solvers
The availability of efficient Krylov subspace solvers play a vital role for
the solution of a variety of numerical problems in computational science. Here
we consider lattice field theory. We present a new general numerical method to
compute many Green's functions for complex non-singular matrices within one
iteration process. Our procedure applies to matrices of structure , with
proportional to the unit matrix, and can be integrated within any Krylov
subspace solver. We can compute the derivatives of the solution
vector with respect to the parameter and construct the Taylor expansion
of around . We demonstrate the advantages of our method using a minimal
residual solver. Here the procedure requires intermediate vector for each
Green's function to compute. As real life example, we determine a mass
trajectory of the Wilson fermion matrix for lattice QCD. Here we find that we
can obtain Green's functions at all masses at the price of one
inversion at mass .Comment: 11 pages, 2 eps-figures, needs epsf.st
Improving Inversions of the Overlap Operator
We present relaxation and preconditioning techniques which accelerate the
inversion of the overlap operator by a factor of four on small lattices, with
larger gains as the lattice size increases. These improvements can be used in
both propagator calculations and dynamical simulations.Comment: lattice2004(machines
A comparative study of numerical methods for the overlap Dirac operator--a status report
Improvements of various methods to compute the sign function of the hermitian
Wilson-Dirac matrix within the overlap operator are presented. An optimal
partial fraction expansion (PFE) based on a theorem of Zolotarev is given.
Benchmarks show that this PFE together with removal of converged systems within
a multi-shift CG appears to approximate the sign function times a vector most
efficiently. A posteriori error bounds are given.Comment: 3 pages, poster contribution to Lattice2001(algorithms
Numerical Methods for the QCD Overlap Operator: I. Sign-Function and Error Bounds
The numerical and computational aspects of the overlap formalism in lattice
quantum chromodynamics are extremely demanding due to a matrix-vector product
that involves the sign function of the hermitian Wilson matrix. In this paper
we investigate several methods to compute the product of the matrix
sign-function with a vector, in particular Lanczos based methods and partial
fraction expansion methods. Our goal is two-fold: we give realistic comparisons
between known methods together with novel approaches and we present error
bounds which allow to guarantee a given accuracy when terminating the Lanczos
method and the multishift-CG solver, applied within the partial fraction
expansion methods.Comment: 30 pages, 2 figure
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