The solution of the Poisson equation is a crucial step in electronic structure calculations, yielding the electrostatic potential—a key component of the quantum mechanical Hamiltonian. In recent decades, theoretical advances and increases in computer performance have made it possible to simulate the electronic structure of extended systems in complex environments. This requires the solution of more complicated variants of the Poisson equation, featuring nonhomogeneous dielectric permittivities, ionic concentrations with nonlinear dependencies, and diverse boundary conditions. The analytic solutions generally used to solve the Poisson equation in vacuum (or with homogeneous permittivity) are not applicable in these circumstances, and numerical methods must be used. In this work, we present DL_MG, a flexible, scalable, and accurate solver library, developed specifically to tackle the challenges of solving the Poisson equation in modern large-scale electronic structure calculations on parallel computers. Our solver is based on the multigrid approach and uses an iterative high-order defect correction method to improve the accuracy of solutions. Using two chemically relevant model systems, we tested the accuracy and computational performance of DL_MG when solving the generalized Poisson and Poisson–Boltzmann equations, demonstrating excellent agreement with analytic solutions and efficient scaling to ∼10⁹ unknowns and 100s of CPU cores. We also applied DL_MG in actual large-scale electronic structure calculations, using the ONETEP linear-scaling electronic structure package to study a 2615 atom protein–ligand complex with routinely available computational resources. In these calculations, the overall execution time with DL_MG was not significantly greater than the time required for calculations using a conventional FFT-based solver.

中文翻译：

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DL_MG：并行多重网格Poisson和Poisson-Boltzmann求解器，用于真空和溶液中的电子结构计算

泊松方程的解是电子结构计算中的关键步骤，产生静电势，这是量子力学哈密顿量的关键组成部分。近几十年来，理论上的进步和计算机性能的提高使得在复杂环境中模拟扩展系统的电子结构成为可能。这就需要求解更复杂的泊松方程，具有不均匀的介电常数，具有非线性依赖性的离子浓度以及多样的边界条件。通常用于在真空中（或具有均匀介电常数）求解泊松方程的解析解在这些情况下不适用，必须使用数值方法。在这项工作中，我们展示了DL_MG，它是一个灵活，可扩展且准确的求解器库，专门开发用于解决在并行计算机上现代大规模电子结构计算中求解泊松方程的挑战。我们的求解器基于多网格方法，并使用迭代的高阶缺陷校正方法来提高求解的准确性。使用两个化学相关的模型系统，我们在求解广义Poisson和Poisson-Boltzmann方程时测试了DL_MG的准确性和计算性能，证明了与解析解决方案的出色一致性以及有效缩放至约10 我们的求解器基于多网格方法，并使用迭代的高阶缺陷校正方法来提高求解的准确性。使用两个化学相关的模型系统，我们在求解广义Poisson和Poisson-Boltzmann方程时测试了DL_MG的准确性和计算性能，证明了与解析解决方案的出色一致性以及有效缩放至约10 我们的求解器基于多网格方法，并使用迭代的高阶缺陷校正方法来提高求解的准确性。使用两个化学相关的模型系统，我们在求解广义Poisson和Poisson-Boltzmann方程时测试了DL_MG的准确性和计算性能，证明了与解析解决方案的出色一致性以及有效缩放至约10^9个未知数和100个CPU内核。我们还使用DL_MG在实际的大规模电子结构计算中，使用ONETEP线性缩放电子结构包来研究具有常规计算资源的2615原子蛋白质-配体复合物。在这些计算中，使用DL_MG的总执行时间不会明显大于使用常规基于FFT的求解器进行计算所需的时间。