Error Bounds for the Sign Function

TL;DR

This paper develops efficient methods to estimate and bound errors in approximating the sign function of a matrix using Lanczos iterations, crucial for accurate overlap operator computations in lattice QCD.

Contribution

It introduces a technique to recover secondary Lanczos processes from primary ones, enabling reliable error bounds for sign function approximations in lattice QCD simulations.

Findings

Secondary Lanczos process can be cheaply recovered from primary process.

Error bounds improve the reliability of sign function approximations.

Method enhances accuracy and efficiency in overlap operator computations.

Abstract

The Overlap operator fulfills the Ginsparg-Wilson relation exactly and therefore represents an optimal discretization of the QCD Dirac operator with respect to chiral symmetry. When computing propagators or in HMC simulations, where one has to invert the overlap operator using some iterative solver, one has to approxomate the action of the sign function of the (symmetrized) Wilson fermion matrix Q on a vector b in each iteration. This is usually done iteratively using a "primary" Lanczos iteration. In this process, it is very important to have good stopping criteria which allow to reliably assess the quality of the approximation to the action of the sign function computed so far. In this work we show how to cheaply recover a secondary Lanczos process, starting at an arbitrary Lanczos vector of the primary process and how to use this secondary process to efficiently obtain computable error estimates and error bounds for the Lanczos approximations to sign(Q)b, where the sign function is approximated by the Zolotarev rational approximation.