变分蒙特卡罗中优化非线性波函数的加速线性方法
An accelerated linear method for optimizing non-linear wavefunctions in variational Monte Carlo
摘要：
虽然线性方法是优化非线性参数化的波函数在变分蒙特卡罗最强大的算法之一，它遭受的内存瓶颈，由于在每个优化步骤的广义特征值问题的解决，其中的哈密顿和重叠矩阵存储在内存中的事实。在这里，我们证明，通过应用Jacobi-Davidson算法，可以迭代求解广义特征值问题，而不必建立和存储的矩阵问题。由此产生的直接线性方法大大降低了成本，并提高了缩放的算法相对于参数的数量。为了进一步提高具有大量参数的波函数的优化效率，我们使用远离最小值的一阶方法AMSGrad，因为它非常便宜，并且仅在接近优化结束时切换到直接线性方法，其中诸如AMSGrad的方法具有长收敛尾。我们将这种改进的优化器应用于各种波函数与真实和轨道空间Jastrow因子的原子系统，如铍和氖，分子系统，如碳二聚体和铁（II）卟啉，和模型系统，如哈伯德模型和氢链。
英文摘要：
Although the linear method is one of the most robust algorithms for optimizing non-linearly parametrized wavefunctions in variational Monte Carlo, it suffers from a memory bottleneck due to the fact at each optimization step a generalized eigenvalue problem is solved in which the Hamiltonian and overlap matrices are stored in memory. Here we demonstrate that by applying the Jacobi-Davidson algorithm, one can solve the generalized eigenvalue problem iteratively without having to build and store the matrices in question. The resulting direct linear method greatly lowers the cost and improves the scaling of the algorithm with respect to the number of parameters. To further improve the efficiency of optimization for wavefunctions with a large number of parameters, we use the first order method AMSGrad far from the minimum as it is very inexpensive, and only switch to the direct linear method near the end of the optimization where methods such as AMSGrad have long convergence tails. We apply this improved optimizer to various wavefunctions with both real and orbital space Jastrow factors for atomic systems such as Beryllium and Neon, molecular systems such as the Carbon dimer and Iron(II) Porphyrin, and model systems such as the Hubbard model and Hydrogen chains.
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