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

TL;DR

This paper presents an accelerated conjugate gradient algorithm for computing low-lying eigenvalues of the Dirac operator in SU(2) lattice QCD, demonstrating significant speedups and stability improvements.

Contribution

The study introduces a combined conjugate gradient and exact diagonalization method that accelerates eigenvalue computations for the Dirac operator in lattice QCD.

Findings

Achieves 4-8 times faster eigenvalue computation

Demonstrates numerical stability and parallelizability

Effective on lattices from 4^4 to 16^4 sizes

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.