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In lattice QCD, the trace of the inverse of the discretized Dirac operator appears in the disconnected fermion loop contribution to an observable. As simulation methods get more and more precise, these contributions become increasingly important. Hence, we consider here the problem of computing the trace tr(𝐷 −1 ), with 𝐷 the Dirac operator. The Hutchinson method, which is very frequently used to stochastically estimate the trace of a function of a matrix, approximates the trace as the average over estimates of the form 𝑥 𝐻 𝐷 −1 𝑥, with the entries of the vector 𝑥 following a certain probability distribution. For 𝑁 samples, the accuracy is O (1/ √ 𝑁). In recent work, we have introduced multigrid multilevel Monte Carlo: having a multigrid hierarchy with operators 𝐷 ℓ , 𝑃 ℓ and 𝑅 ℓ , for level ℓ, we can rewrite the trace tr(𝐷 −1 ) via a telescopic sum with difference-levels, written in terms of the aforementioned operators and with a reduced variance. We have seen significant reductions in the variance and the total work with respect to exactly deflated Hutchinson. In this work, we explore the use of exact deflation in combination with the multigrid multilevel Monte Carlo method, and demonstrate how this leads to both algorithmic and computational gains.

Numerical simulations of quantum chromodynamics (QCD) on a lattice require the frequent solution of linear systems of equations with large, sparse and typically ill-conditioned matrices. Algebraic multigrid methods are meanwhile the standard for these difficult solves. Although the linear systems at the coarsest level of the multigrid hierarchy are much smaller than the ones at the finest level, they can be severely ill-conditioned, thus affecting the scalability of the whole solver. In this paper, we investigate different novel ways to enhance the coarsest-level solver and demonstrate their potential using DD-αAMG, one of the publicly available algebraic multigrid solvers for lattice QCD. We do this for two lattice discretizations, namely clover-improved Wilson and twisted mass. For both the combination of two of the investigated enhancements, deflation and polynomial preconditioning, yield significant improvements in the regime of small mass parameters. In the clover-improved Wilson case we observe a significantly improved insensitivity of the solver to conditioning, and for twisted mass we are able to get rid of a somewhat artificial increase of the twisted mass parameter on the coarsest level used so far to make the coarsest level solves converge more rapidly.

The trace of a matrix function f (A), most notably of the matrix inverse, can be estimated stochastically using samples x * f (A)x if the components of the random vectors x obey an appropriate probability distribution. However such a Monte-Carlo sampling suffers from the fact that the accuracy depends quadratically of the samples to use, thus making higher precision estimation very costly. In this paper we suggest and investigate a multilevel Monte-Carlo approach which uses a multigrid hierarchy to stochastically estimate the trace. This results in a substantial reduction of the variance, so that higher precision can be obtained at much less effort. We illustrate this for the trace of the inverse using three different classes of matrices.

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