Seed methods for linear equations in lattice qcd problems with multiple right-hand sides

TL;DR

This paper introduces three enhancements to seed methods for solving Hermitian linear systems with multiple right-hand sides in lattice QCD, leading to faster convergence and improved numerical stability.

Contribution

It proposes using the Krylov subspace from the first system for all subsequent systems, solving the first system past convergence, and applying periodic re-orthogonalization to control roundoff errors.

Findings

Significant speed-up in convergence observed.

Effective control of roundoff errors with re-orthogonalization.

Method tested on Wilson fermions near kappa critical.

Abstract

We consider three improvements to seed methods for Hermitian linear systems with multiple right-hand sides: only the Krylov subspace for the first system is used for seeding subsequent right-hand sides, the first right-hand side is solved past convergence, and periodic re-orthogonalization is used in order to control roundoff errors associated with the Conjugate Gradient algorithm. The method is tested for the case of Wilson fermions near kappa critical and a considerable speed up in the convergence is observed.