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 $A=D-m$, with $m$ proportional to the unit matrix, and can be integrated within any Krylov subspace solver. We can compute the derivatives $x^{(n)}$ of the solution vector $x$ with respect to the parameter $m$ and construct the Taylor expansion of $x$ around $m$. We demonstrate the advantages of our method using a minimal residual solver. Here the procedure requires $1$ 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 $\geq m$ at the price of one inversion at mass $m$.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