Improved Speed and Scaling in Orbital Space Variational Monte Carlo
Improved Speed and Scaling in Orbital Space Variational Monte Carlo
复制标题
改进轨道空间变分蒙特卡罗的速度和缩放比例
DOI：
10.1021/acs.jctc.8b00780
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发表时间：
2018
影响因子：
5.5
通讯作者：
Sharma, Sandeep
中科院分区：
文献类型：
--
作者：
Sabzevari, Iliya;Sharma, Sandeep
在这项工作中，我们介绍了三个算法的改进，以降低成本，提高轨道空间变分蒙特卡罗（VMC）的缩放。首先，我们表明，通过适当筛选的一个和两个电子的积分的哈密顿量，可以提高算法的效率由几个数量级。这种提高的效率带来了额外的好处，即获得每个电子恒定误差的成本从四次方O（N4）下降为系统大小的二次方O（N2）。利用数值结果，我们证明了实际的标度得到的，事实上，O（N1.5）的氢原子链。其次，我们表明，通过使用自适应随机梯度下降算法称为AMSGrad，可以优化波函数能量鲁棒性和有效性。值得注意的是，AMSGrad几乎与简单的随机梯度下降一样便宜，但提供了与随机重新配置算法相当的收敛速度，后者明显更昂贵，并且随着系统大小的缩放更差。第三，我们介绍了使用拒绝自由连续时间蒙特卡罗（CTMC）的抽样决定因素。与前两个改进不同的是，CTMC确实带来了一个开销，即必须在每个Monte Carlo步骤中计算局部能量。然而，由于缩减的缩放算法，该开销在很大程度上被减轻，这确保了计算局部能量的渐近成本等于更新步行者的渐近成本。由此产生的算法允许我们使用包含102 × 105个变分参数的波函数计算160个氢原子链的基态能量，精度为1 mEh/粒子，成本仅为25 CPU h，当分为2个节点24个处理器时，每个处理器仅相当于大约半小时的壁时间。这种低成本加上令人尴尬的并行化的VMC算法和可用的波函数的形式很大的自由度，代表了一个非常有效的方法计算模型和从头算系统的电子结构。
In this work, we introduce three algorithmic improvements to reduce the cost and improve the scaling of orbital space variational Monte Carlo (VMC). First, we show that, by appropriately screening the one- and two-electron integrals of the Hamiltonian, one can improve the efficiency of the algorithm by several orders of magnitude. This improved efficiency comes with the added benefit that the cost of obtaining a constant error per electron scales as the second power of the system sizeO(N2), down from the fourth powerO(N4). Using numerical results, we demonstrate that the practical scaling obtained is, in fact,O(N1.5) for a chain of hydrogen atoms. Second, we show that, by using the adaptive stochastic gradient descent algorithm called AMSGrad, one can optimize the wave function energies robustly and efficiently. Remarkably, AMSGrad is almost as inexpensive as the simple stochastic gradient descent but delivers a convergence rate that is comparable to that of the Stochastic Reconfiguration algorithm, which is significantly more expensive and has a worse scaling with the system size. Third, we introduce the use of the rejection-free continuous time Monte Carlo (CTMC) to sample the determinants. Unlike the first two improvements, CTMC does come at an overhead that the local energy must be calculated at every Monte Carlo step. However, this overhead is mitigated to a large extent because of the reduced scaling algorithm, which ensures that the asymptotic cost of calculating the local energy is equal to that of updating the walker. The resulting algorithm allows us to calculate the ground state energy of a chain of 160 hydrogen atoms using a wave function containing ∼2 × 105variational parameters with an accuracy of 1 mEh/particle at a cost of just 25 CPU h, which when split over 2 nodes of 24 processors each amounts to only about half hour of wall time. This low cost coupled with embarrassing parallelizability of the VMC algorithm and great freedom in the forms of usable wave functions, represents a highly effective method for calculating the electronic structure of model and ab initio systems.