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Lattice gauge theory is a fairly young research area in Theoretical Particle Physics. It is of great promise as it offers the framework for an ab-initio treatment of the nonperturbative features of strong interactions. Ever since its adolescence the simulation of quantum chromodynamics has attracted the interest of numerical analysts and there is growing interdisciplinary engage­ ment between theoretical physicists and applied mathematicians to meet the grand challenges of this approach. This volume contains contributions of the interdisciplinary workshop "Nu­ merical Challenges in Lattice Quantum Chromo dynamics" that the Institute of Applied Computer Science (IAI) at Wuppertal University together with the Von-Neumann-Institute-for-Computing (NIC) organized in August 1999. The purpose of the workshop was to offer a platform for the exchange of key ideas between lattice QCD and numerical analysis communities. In this spirit leading experts from both fields have put emphasis to transcend the barriers between the disciplines. The meetings was focused on the following numerical bottleneck problems: A standard topic from the infancy of lattice QCD is the computation of Green's functions, the inverse of the Dirac operator. One has to solve huge sparse linear systems in the limit of small quark masses, corresponding to high condition numbers of the Dirac matrix. Closely related is the determination of flavor-singlet observables which came into focus during the last years.



Overlap Fermions and Matrix Functions

The Overlap Dirac Operator

This introductory presentation describes the Overlap Dirac Operator, why it could be useful in numerical QCD, and how it can be implemented.
Herbert Neuberger

Solution of f(A)x = b with Projection Methods for the Matrix A

In this paper, we expand on an idea for using Krylov subspace information for the matrix A and the vector b. This subspace can be used for the approximate solution of a linear system f (A)x = b, where f is some analytic function. We will make suggestions on how to use this for the case where f is the matrix sign function.
Henk A. van der Vorst

A Numerical Treatment of Neuberger’s Lattice Dirac Operator

We describe in some detail our numerical treatment of Neuberger’s lattice Dirac operator as implemented in a practical application. We discuss the improvements we have found to accelerate the numerical computations and give an estimate of the expense when using this operator in practice.
Pilar Hernández, Karl Jansen, Laurent Lellouche

Fast Methods for Computing the Neuberger Operator

I describe a Lanczos method to compute the Neuberger Operator and a multigrid algorithm for its inversion.
Artan Boriçi

Light Quarks, Lanczos and Multigrid Techniques

On Lanczos-Type Methods for Wilson Fermions

Numerical simulations of lattice gauge theories with fermions rely heavily on the iterative solution of huge sparse linear systems of equations. Due to short recurrences, which mean small memory requirement, Lanczos-type methods (including suitable versions of the conjugate gradient method when applicable) are best suited for this type of problem. The Wilson formulation of the lattice Dirac operator leads to a matrix with special symmetry properties that makes the application of the classical biconjugate gradient (BICG) particularly attractive, but other methods, for example BICGSTAB and BICGSTAB2 have also been widely used. We discuss some of the pros and cons of these methods. In particular, we review the specific simplification of BICG, clarify some details, and discuss general results on the roundoff behavior.
Martin H. Gutknecht

An Algebraic Multilevel Preconditioner for Symmetric Positive Definite and Indefinite Problems

We present a preconditioning method for the iterative solution of large sparse systems of equations. The preconditioner is based on ideas both from ILU preconditioning and from multigrid. The resulting preconditioning technique requires the matrix only. A multilevel structure is obtained by constructing a maximal independent set of the graph of a reduced matrix. The computation of a Schur complement approximation is based on a Galerkin approach with a matrix dependent prolongation and restriction. The resulting preconditioner has a transparant modular structure similar to the algorithmic structure of a multigrid V-cycle. The method is applied to symmetric positive definite and indefinite Helmholtz problems. The multilevel preconditioner is compared with standard ILU preconditioning methods.
Arnold Reusken

On Algebraic Multilevel Preconditioning

Algebraic multilevel preconditioners are based on a block incomplete factorization process applied to the system matrix partitioned in hierarchical form. They have as key ingredient a technique to derive a (sparse) approximation to the Schur complement resulting from the elimination of the fine grid nodes. Once such a relevant approximation is found, it is relatively easy to set up efficient two-and multi-level schemes. Compared with more standard (classical or algebraic) multi-grid methods, an obvious advantage of this approach is that it does not require smoothing, i.e. the convergence properties do not critically depend on the good interaction between a smoother and the coarse grid correction. As a consequence, the analysis can be done purely algebraically, independently of an underlying PDE, and the technique is potentially more easily extended to other grid based computations than those directly connected to discrete elliptic PDEs.
In this chapter, we review some recent results on these methods, concerning both self-adjoint and non self-adjoint discrete second order PDEs.
Yvan Notay

On Algebraic Multilevel Preconditioners in Lattice Gauge Theory

Based on a Schur complement approach we develop a parallelizable multi-level preconditioning method for computing quark propagators in lattice gauge theory.
Björn Medeke

Flavor Singlet Operations and Matrix Functionals

Stochastic Estimator Techniques for Disconnected Diagrams

The calculation of physical quantities by lattice QCD simulations requires in some important cases the determination of the inverse of a very large matrix. In this article we describe how stochastic estimator methods can be applied to this problem, and how such techniques can be efficiently implemented.
Stephan Güsken

Noise Methods for Flavor Singlet Quantities

A discussion of methods for reducing the noise variance of flavor singlet quantities (“disconnected diagrams”) in lattice QCD is given. After an introduction, the possible advantage of partitioning the Wilson fermion matrix into disjoint spaces is discussed and a numerical comparison of the variance for three possible partitioning schemes is carried out. The measurement efficiency of lattice operators is examined and shown to be strongly influenced by the Dirac and color partitioning choices. Next, the numerical effects of an automated subtraction algorithm on the noise variance of various disconnected loop matrix elements are examined It is found that there is a dramatic reduction in the variance of the Wilson point-split electromagnetic currents and that this reduction persists at small quark mass.
Walter Wilcox

Novel Numerical Techniques for Full QCD

A Noisy Monte Carlo Algorithm with Fermion Determinant

We propose a Monte Carlo algorithm which accommodates an unbiased stochastic estimates of the fermion determinant and is exact. This is achieved by adopting the Metropolis accept/reject steps for the update of both the dynamical and noise configurations. We demonstrate how this algorithm works even for large noises. We also discuss the Padé - Z2estimates of the fermion determinant as a practical way of estimating the Trln of the fermion matrix.
Keh-Fei Liu

Least-Squares Optimized Polynomials for Fermion Simulations

Least-squares optimized polynomials are discussed which are needed in the two-step multi-bosonic algorithm for Monte Carlo simulations of quantum field theories with fermions. A recurrence scheme for the calculation of necessary coefficients in the recursion and for the evaluation of these polynomials is introduced.
István Montvay

One-Flavour Hybrid Monte Carlo with Wilson Fermions

The Wilson fermion determinant can be written as product of the determinants of two hermitian positive definite matrices. This formulation allows to simulate non-degenerate quark flavors by means of the hybrid Monte Carlo algorithm. A major numerical difficulty is the occurrence of nested inversions. We construct a Uzawa iteration scheme which treats the nested system within one iterative process.
Thomas Lippert


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