A nested Krylov subspace method for the overlap operator

TL;DR

This paper introduces a nested Krylov subspace method to efficiently compute the overlap Dirac operator, significantly reducing computational complexity while maintaining accuracy for large lattice matrices at various chemical potentials.

Contribution

The novel nested Krylov subspace approach improves the efficiency of computing the overlap operator by further projecting onto smaller subspaces without losing accuracy.

Findings

Method effectively computes the sign function on large matrices

Demonstrates efficiency on both Hermitian and non-Hermitian matrices

Maintains numerical accuracy with reduced computational cost

Abstract

We present a novel method to compute the overlap Dirac operator at zero and nonzero quark chemical potential. To approximate the sign function of large, sparse matrices, standard methods project the operator on a much smaller Krylov subspace, on which the matrix function is computed exactly. However, for large lattices this subspace can still be too large for an efficient calculation of the sign function. The idea of the new method is to nest Krylov subspace approximations by making a further projection on an even smaller subspace, which is then small enough to compute the sign function efficiently, and this without any noticeable loss of numerical accuracy. We demonstrate the efficiency of the method both on Hermitian and non-Hermitian matrices.