Although the linear method is one of the most robust algorithms for optimizing nonlinearly parametrized wavefunctions in variational Monte Carlo, it suffers from a memory bottleneck due to the fact that 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 wavefunctions with real and orbital space Jastrow factors applied to a symmetry-projected generalized Hartree-Fock reference. Systems addressed include 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.

中文翻译：

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优化变分蒙特卡罗非线性波函数的加速线性方法。

尽管线性方法是用于优化变分蒙特卡洛中非线性参数化波函数的最强大的算法之一，但由于在每个优化步骤中都解决了广义特征值问题，在该问题中汉密尔顿矩阵和重叠矩阵得以解决，因此它存在存储瓶颈存储在内存中。在这里，我们证明了通过应用Jacobi-Davidson算法，可以迭代地解决广义特征值问题，而无需构建和存储所讨论的矩阵。所得的直接线性方法大大降低了成本，并提高了算法在参数数量方面的可伸缩性。为了进一步提高具有大量参数的波函数的优化效率，我们使用一阶方法AMSGrad远离最小值，因为它非常便宜，并且仅在优化接近尾声时才切换到直接线性方法，而AMSGrad等方法的收敛尾巴较长。我们将此改进的优化器应用于具有实空间和轨道空间Jastrow因子的波函数，并将其应用于对称投影的广义Hartree-Fock参考。涉及的系统包括原子系统（例如，铍和氖），分子系统（例如，碳二聚体和卟啉铁）以及模型系统（例如，哈伯德模型和氢链）。我们将此改进的优化器应用于具有实空间和轨道空间Jastrow因子的波函数，并将其应用于对称投影的广义Hartree-Fock参考。涉及的系统包括原子系统（例如，铍和氖），分子系统（例如，碳二聚体和卟啉铁）以及模型系统（例如，哈伯德模型和氢链）。我们将此改进的优化器应用于具有实空间和轨道空间Jastrow因子的波函数，并将其应用于对称投影的广义Hartree-Fock参考。涉及的系统包括原子系统（例如，铍和氖），分子系统（例如，碳二聚体和卟啉铁）以及模型系统（例如，哈伯德模型和氢链）。