Speaker
Description
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 $\mathrm{tr}( D^{-1} )$, with $D$ the Dirac operator.
The Hutchinson method, which is very frequently used to stochastically estimate the trace of the 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 $\mathcal{O}(1/\sqrt{N})$.
In recent work, we have introduced multigrid multilevel Monte Carlo: having a multigrid hierarchy with operators $A_{\ell}$, $P_{\ell}$ and $R_{\ell}$, for level $\ell$, we can rewrite the trace in the form $\mathrm{tr}(A_{0})^{-1} = \sum_{\ell=0}^{L-1} \mathrm{tr}(A_{\ell}^{-1} - P_{\ell+1}A_{\ell+1}^{-1}R_{\ell+1})+\mathrm{tr}(A_{L}^{-1})$ (this reduced expression is in the special case when $R_{\ell}P_{\ell} = I$). We have seen significant reductions in the variance and the total work with respect to exactly-deflated Hutchinson.
In this talk, 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.
