The error scaling for Markov-Chain Monte Carlo techniques (MCMC) with $N$samples behaves like $1/\sqrt{N}$. This scaling makes it often very timeintensive to reduce the error of computed observables, in particular forapplications in lattice QCD. It is therefore highly desirable to havealternative methods at hand which show an improved error scaling. One candidatefor such an alternative integration technique is the method of recursivenumerical integration (RNI). The basic idea of this method is to use anefficient low-dimensional quadrature rule (usually of Gaussian type) and applyit iteratively to integrate over high-dimensional observables and Boltzmannweights. We present the application of such an algorithm to the topologicalrotor and the anharmonic oscillator and compare the error scaling to MCMCresults. In particular, we demonstrate that the RNI technique shows an errorscaling in the number of integration points $m$ that is at least exponential.
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机译：具有$ N $ samples的Markov-Chain蒙特卡洛技术（MCMC）的误差缩放行为类似于$ 1 / \ sqrt {N} $。这种缩放通常使减少计算的可观测值的误差非常耗时，尤其是对于格点QCD中的应用。因此，迫切需要具有显示出改进的误差缩放比例的替代方法。这种替代积分技术的一种候选方法是递归数值积分（RNI）方法。该方法的基本思想是使用有效的低维正交规则（通常为高斯型），并迭代地应用其对高维可观量和玻尔兹曼权重进行积分。我们介绍了这种算法在拓扑转子和非谐振荡器上的应用，并将误差缩放比例与MCMC结果进行了比较。特别地，我们证明了RNI技术在积分点$ m $的数量上显示出至少至少指数级的误差。
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