Multigrid deflation for Lattice QCD

TL;DR

This paper explores efficient methods for computing deflation spaces in Lattice QCD to reduce variance in trace estimation, proposing inexact eigensolvers and multigrid prolongators as cost-effective alternatives to traditional eigenvector computations.

Contribution

It introduces two novel approaches for computing deflation spaces—using inexact eigensolvers and multigrid prolongators—that are more efficient and scalable for large lattice QCD problems.

Findings

Inexact eigensolvers approximate the lower spectrum effectively for ill-conditioned operators.

Multigrid prolongators provide similar variance reduction with lower computational cost.

Proposed methods scale better with problem size than traditional eigenvector approaches.

Abstract

Computing the trace of the inverse of large matrices is typically addressed through statistical methods. Deflating out the lowest eigenvectors or singular vectors of the matrix reduces the variance of the trace estimator. This work summarizes our efforts to reduce the computational cost of computing the deflation space while achieving the desired variance reduction for Lattice QCD applications. Previous efforts computed the lower part of the singular spectrum of the Dirac operator by using an eigensolver preconditioned with a multigrid linear system solver. Despite the improvement in performance in those applications, as the problem size grows the runtime and storage demands of this approach will eventually dominate the stochastic estimation part of the computation. In this work, we propose to compute the deflation space in one of the following two ways. First, by using an inexact eigensolver on the Hermitian, but maximally indefinite, operator $A \gamma_5$. Second, by exploiting the fact that the multigrid prolongator for this operator is rich in components toward the lower part of the singular spectrum. We show experimentally that the inexact eigensolver can approximate the lower part of the spectrum even for ill-conditioned operators. Also, the deflation based on the multigrid prolongator is more efficient to compute and apply, and, despite its limited ability to approximate the fine level spectrum, it obtains similar variance reduction on the trace estimator as deflating with approximate eigenvectors from the fine level operator.