论文标题
多级蒙特卡洛，用于格子上的量子力学
Multilevel Monte Carlo for quantum mechanics on a lattice
论文作者
论文摘要
随着连续限制的接近，由于连续限制的量量子场理论的蒙特卡洛模拟变得越来越昂贵，因为每个独立样品的成本以较高的反晶格间距的高功率增长。对晶格上的模拟受到关键的放缓，马尔可夫链中自相关的快速增长。这会导致必须生成的晶格配置数量大大增加，以获得统计学上的显着结果。本文讨论了分层抽样方法，以驯服自相关的增长。结合降低多级差异，这大大降低了离散误差上给定公差的模拟成本$ε_{\ text {disc}} $，在统计错误上，$ε_{\ text {disc}} $ $ε_{\ text {dist}} $。对于有订单$α$晶格错误和集成自相关时间的观察到的，它们会像$τ_ {\ Mathrm {int}}} \ propto a^{ - z} $，多级蒙特卡洛（MLMC）降低成本$ \ MATHCAL {O}（ε_{\ text {stat}}^{ - 2}ε_{\ text {disc}}}^{ - （1+z）/α}）$ to $ \ mathcal {o} ε_ {\ text {disc}} \ vert^2+ε_ {\ text {disc}}}^{ - 1/α}）$或$ \ MATHCAL {o}（ε_{ε_{\ text}}}}}}}预计在$ d $尺寸中模拟量子场理论的模拟预计会更高。在两个模型系统上证明了该方法的效率，包括拓扑振荡器，该振荡器受到拓扑电荷冻结的严重减速影响的严重影响。在细晶格上，新方法的数量级比标准杂交蒙特卡洛采样快。对于高分辨率，MLMC也可用于加速拓扑振荡器的簇算法。通过扰动匹配进一步提高了性能，从而确保了多级层次结构上理论的有效耦合。
Monte Carlo simulations of quantum field theories on a lattice become increasingly expensive as the continuum limit is approached since the cost per independent sample grows with a high power of the inverse lattice spacing. Simulations on fine lattices suffer from critical slowdown, the rapid growth of autocorrelations in the Markov chain. This causes a strong increase in the number of lattice configurations that have to be generated to obtain statistically significant results. This paper discusses hierarchical sampling methods to tame the growth in autocorrelations. Combined with multilevel variance reduction, this significantly reduces the computational cost of simulations for given tolerances $ε_{\text{disc}}$ on the discretisation error and $ε_{\text{stat}}$ on the statistical error. For observables with lattice errors of order $α$ and integrated autocorrelation times that grow like $τ_{\mathrm{int}}\propto a^{-z}$, multilevel Monte Carlo (MLMC) reduces the cost from $\mathcal{O}(ε_{\text{stat}}^{-2}ε_{\text{disc}}^{-(1+z)/α})$ to $\mathcal{O}(ε_{\text{stat}}^{-2}\vert\log ε_{\text{disc}} \vert^2+ε_{\text{disc}}^{-1/α})$ or $\mathcal{O}(ε_{\text{stat}}^{-2}+ε_{\text{disc}}^{-1/α})$. Higher gains are expected for simulations of quantum field theories in $D$ dimensions. The efficiency of the approach is demonstrated on two model systems, including a topological oscillator that is badly affected by critical slowdown from topological charge freezing. On fine lattices, the new methods are orders of magnitude faster than standard Hybrid Monte Carlo sampling. For high resolutions, MLMC can be used to accelerate even the cluster algorithm for the topological oscillator. Performance is further improved through perturbative matching which guarantees efficient coupling of theories on the multilevel hierarchy.
