# A Parallel SSOR Preconditioner for Lattice QCD

## TL;DR

This paper introduces a parallel SSOR preconditioner for lattice QCD that significantly reduces iteration counts and CPU time in simulations, especially on parallel computers like Quadrics QH4.

## Contribution

It proposes a new parallelizable SSOR preconditioning scheme based on local lexicographic ordering, improving efficiency in lattice QCD computations.

## Key findings

- Achieves a factor of 2 reduction in iterations with BiCGstab.
- Realizes a 1.7 times CPU-time reduction on 512-processor Quadrics QH4.
- Effective implementation using the Eisenstat-trick and machine-specific optimizations.

## Abstract

We present a parallelizable SSOR preconditioning scheme for Krylov subspace iterative solvers which proves to be efficient in lattice QCD applications involving Wilson fermions. Our preconditioner is based on a locally lexicographic ordering of the lattice points. In actual hybrid Monte Carlo applications with the bi-conjugate gradient stabilized method BiCGstab, we achieve a gain factor of about 2 in the number of iterations compared to conventional odd-even preconditioning. Whether this translates into similar reductions in run time will depend on the parallel computer in use. We discuss implementation issues using the `Eisenstat-trick' and machine specific advantages of the method for the APE100/Quadrics parallel computer. In a full QCD simulation with Wilson fermions on a 512-processor Quadrics QH4 we find a gain in cpu-time of a factor of 1.7 over odd-even preconditioning for a 24^3 x 40 lattice.