Computing and deflating eigenvalues while solving multiple right hand side linear systems in Quantum Chromodynamics

TL;DR

This paper introduces an efficient algorithm that computes eigenvalues during linear system solutions in lattice QCD, enabling eigenvector deflation to significantly accelerate convergence across multiple right-hand sides.

Contribution

The authors develop a novel algorithm that updates eigenvectors during CG solves without affecting the linear system process, improving efficiency in lattice QCD applications.

Findings

Eigenvectors converge at rates similar to unrestarted Lanczos.

Deflating eigenvectors accelerates convergence for subsequent systems.

Achieves speedups of up to 8 times over standard CG for light quark masses.

Abstract

We present a new algorithm that computes eigenvalues and eigenvectors of a Hermitian positive definite matrix while solving a linear system of equations with Conjugate Gradient (CG). Traditionally, all the CG iteration vectors could be saved and recombined through the eigenvectors of the tridiagonal projection matrix, which is equivalent theoretically to unrestarted Lanczos. Our algorithm capitalizes on the iteration vectors produced by CG to update only a small window of vectors that approximate the eigenvectors. While this window is restarted in a locally optimal way, the CG algorithm for the linear system is unaffected. Yet, in all our experiments, this small window converges to the required eigenvectors at a rate identical to unrestarted Lanczos. After the solution of the linear system, eigenvectors that have not accurately converged can be improved in an incremental fashion by solving additional linear systems. In this case, eigenvectors identified in earlier systems can be used to deflate, and thus accelerate, the convergence of subsequent systems. We have used this algorithm with excellent results in lattice QCD applications, where hundreds of right hand sides may be needed. Specifically, about 70 eigenvectors are obtained to full accuracy after solving 24 right hand sides. Deflating these from the large number of subsequent right hand sides removes the dreaded critical slowdown, where the conditioning of the matrix increases as the quark mass reaches a critical value. Our experiments show almost a constant number of iterations for our method, regardless of quark mass, and speedups of 8 over original CG for light quark masses.