Improved Scaling for Quantum Monte Carlo on Insulators
Improved Scaling for Quantum Monte Carlo on Insulators
复制标题
绝缘体上量子蒙特卡罗的改进缩放
DOI：
10.1137/100805467
复制
发表时间：
2010
期刊：
影响因子：
--
通讯作者：
Jeongnim Kim
中科院分区：
文献类型：
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作者：
K. Ahuja;B. Clark;E. D. Sturler;D. Ceperley;Jeongnim Kim
量子蒙特卡罗（QMC）方法经常用于计算许多体量子系统的性质。许多QMC方法，例如变分蒙特卡罗（VMC）方法，主要成本在于构造斯莱特矩阵序列并计算连续斯莱特矩阵的行列式的比率。最近的工作已经改进了构造绝缘体的斯莱特矩阵的缩放，使得在这些系统中构造斯莱特矩阵的成本现在与粒子数成线性关系，而计算行列式比在粒子数上保持立方。从长远来看，模拟更大的系统的目标，我们改进的VMC方法模拟绝缘子通过使用预处理迭代求解器的行列式比计算的缩放。本文的主要贡献是开发了一种方法来有效地计算斯莱特矩阵的一系列预条件，使迭代求解器迅速收敛。这涉及到廉价的预条件更新，一个有效的重排序策略，和一个廉价的方法来监控不完全LU分解与阈值和旋转（ILUTP）预条件的不稳定性。使用得到的预处理迭代求解器来计算连续斯莱特矩阵的行列式比，将QMC算法的缩放从每次扫描的O（n^3）$减少到大约O（n^2）$，其中$n$是粒子的数量，并且扫描是$n$步的序列，每个步骤尝试移动不同的粒子。我们的实验表明，我们可以实现改进的缩放，而不会增加统计误差。我们的研究结果表明，预处理迭代求解器可以显着降低大（r）系统的VMC的成本。
Quantum Monte Carlo (QMC) methods are often used to calculate properties of many body quantum systems. The main cost of many QMC methods, for example, the variational Monte Carlo (VMC) method, is in constructing a sequence of Slater matrices and computing the ratios of determinants for successive Slater matrices. Recent work has improved the scaling of constructing Slater matrices for insulators so that the cost of constructing Slater matrices in these systems is now linear in the number of particles, whereas computing determinant ratios remains cubic in the number of particles. With the long term aim of simulating much larger systems, we improve the scaling of computing the determinant ratios in the VMC method for simulating insulators by using preconditioned iterative solvers. The main contribution of this paper is the development of a method to efficiently compute for the Slater matrices a sequence of preconditioners that make the iterative solver converge rapidly. This involves cheap preconditioner updates, an effective reordering strategy, and a cheap method to monitor instability of incomplete LU decomposition with threshold and pivoting (ILUTP) preconditioners. Using the resulting preconditioned iterative solvers to compute determinant ratios of consecutive Slater matrices reduces the scaling of QMC algorithms from $O(n^3)$ per sweep to roughly $O(n^2)$, where $n$ is the number of particles, and a sweep is a sequence of $n$ steps, each attempting to move a distinct particle. We demonstrate experimentally that we can achieve the improved scaling without increasing statistical errors. Our results show that preconditioned iterative solvers can dramatically reduce the cost of VMC for large(r) systems.