论文标题
缩气的多人多层蒙特卡洛
Deflated Multigrid Multilevel Monte Carlo
论文作者
论文摘要
在晶格QCD中，离散的狄拉克运算符的倒数轨迹出现在断开的fermion回路对可观察的贡献中。随着仿真方法变得越来越精确，这些贡献变得越来越重要。因此，我们在这里考虑计算跟踪$ \ mathrm {tr}（d^{ - 1}）$的问题，以及$ d $ the dirac Operator。 Hutchinson方法非常经常用于随机估计矩阵函数的轨迹，将轨迹近似为$ x^{h} d^{ - 1} x $的平均值，并带有vector $ x $的条目，并具有一定的概率分布。对于$ n $样本，准确性为$ \ MATHCAL {O}（1/\ sqrt {n}）$。在最近的工作中，我们介绍了Multigrid Multilevel Monte Carlo：与运营商$ d _ {\ ell} $，$ p _ {\ ell} $和$ r _ {\ ell} $具有多个层次结构差异级别，根据上述操作员和差异降低。我们已经看到，方差和相对于精确放气的Hutchinson的总工作量显着降低。在这项工作中，我们探讨了与多族多层蒙特卡洛法结合使用精确放气的使用，并演示了这如何导致算法和计算增长。
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 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 $\mathcal{O}(1/\sqrt{N})$. In recent work, we have introduced multigrid multilevel Monte Carlo: having a multigrid hierarchy with operators $D_{\ell}$, $P_{\ell}$ and $R_{\ell}$, for level $\ell$, we can rewrite the trace $\mathrm{tr}(D^{-1})$ 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.
