# Deflation methods in lattice QCD.

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The inversion of the Dirac operator is a necessary feature of calculating physical observables within lattice QCD. The calculation of fermionic forces within hybrid Monte Carlo and the formation of quark propagators are two examples where such an inversion is needed. The many discretizations of the Dirac operator pose an algorithmic and computational challenge due to their size and their eigenspectra. As the quark mass approaches its physical value, the low lying eigenspectra of the Dirac operator approaches zero. From this arises the phenomena of critical slowing down, where the number of iterations to obtain an approximate solution for an iterative solver increases as a power law. Deflation and multigrid are two techniques that combat the effects of critical slowing down. We present a deflated multigrid preconditioner of FGMRES for the Wilson-Dirac operator in the lattice Schwinger model. Our method of deflation within the preconditioner demonstrates a remarkable reduction in cost for the inversion of the Wilson-Dirac operator, and also displays very mild scaling with respect to lattice size. The calculation of physical quantities arising from disconnected quark loops is one of the largest challenges in lattice QCD. A direct approach is to calculate the propagator for all lattice sites to all lattice sites. For large lattices, this approach is intractable so stochastic methods are used. The physical signal must be extracted from the noise created by these methods, and thus noise subtraction techniques are mandatory. We present deflation based noise subtraction techniques for the scalar, local vector and non-local vector operators in the quenched approximation at zero quark mass and with the inclusion of dynamical sea quarks at larger than physical pion mass. In both cases, the deflation based methods show dramatic reduction in the variance of these noisy calculations.