Numerical Methods for the QCD Overlap Operator:III. Nested Iterations

TL;DR

This paper explores efficient nested iterative methods for computing the overlap operator in lattice QCD, demonstrating that relaxation and preconditioning significantly reduce computational costs while maintaining accuracy.

Contribution

It introduces criteria for inner iteration accuracy, analyzes preconditioning strategies, and shows how relaxation can optimize nested Krylov subspace methods in lattice QCD computations.

Findings

Relaxation of sign function accuracy improves efficiency.

Preconditioning with approximate sign functions reduces computation by up to 4 times.

Projection into a chiral sector offers potential computational benefits.

Abstract

The numerical and computational aspects of chiral fermions in lattice quantum chromodynamics are extremely demanding. In the overlap framework, the computation of the fermion propagator leads to a nested iteration where the matrix vector multiplications in each step of an outer iteration have to be accomplished by an inner iteration; the latter approximates the product of the sign function of the hermitian Wilson fermion matrix with a vector. In this paper we investigate aspects of this nested paradigm. We examine several Krylov subspace methods to be used as an outer iteration for both propagator computations and the Hybrid Monte-Carlo scheme. We establish criteria on the accuracy of the inner iteration which allow to preserve an a priori given precision for the overall computation. It will turn out that the accuracy of the sign function can be relaxed as the outer iteration proceeds. Furthermore, we consider preconditioning strategies, where the preconditioner is built upon an inaccurate approximation to the sign function. Relaxation combined with preconditioning allows for considerable savings in computational efforts up to a factor of 4 as our numerical experiments illustrate. We also discuss the possibility of projecting the squared overlap operator into one chiral sector.