2019/09/26 by Eloy Romero, Andreas Stathopoulos, Kostas Orginos · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #Algorithm #Applied mathematics #Computation #Computer science #Deflation #Dirac operator #Eigenvalues and eigenvectors #Estimator #Lattice QCD #Linear system #Mathematical analysis #Mathematical optimization #Mathematics #Matrix Theory and Algorithms #Monte Carlo method #Multigrid method #Operator (biology) #Partial differential equation #Particle physics theoretical and experimental studies #Physics #Quantum Chromodynamics and Particle Interactions #Quantum chromodynamics #Solver #Variance reduction #cs.NA #hep-lat #math.NA
paper · pdf · doi:10.1016/j.jcp.2020.109356
arxiv created 2019/09/26 · openalex publication_date 2020/02/21 · arxiv updated 2020/03/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
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 γ5. 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.