  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.
