Electrostatic correlations and fluctuations in ionic systems can be described within an extended Poisson-Boltzmann theory using a Gaussian variational form. The resulting equations are challenging to solve because they require the solution of a non-linear partial differential equation for the pair correlation function. This has limited existing studies to simple approximations or to one-dimensional geometries. In this paper we show that the numerical solution of the equations is greatly simplified by the use of selective inversion of a finite difference operator which occurs in the theory. This selective inversion preserves the sparse structure of the problem and leads to substantial savings in computer effort. In one and two dimensions further simplifications are made by using a mixture of selective inversion and Fourier techniques.
A Self-Adjusting FEM-BEM Coupling Scheme for the Nonlinear Poisson-Boltzmann Equation
Universidad Técnica Federico Santa María（弗雷德里科·圣塔玛丽亚技术大学）；Kingston University London（金斯顿大学伦敦分校）；Centro Científico Tecnológico de Valparaíso (CCTVal)（瓦尔帕莱索科技中心）
Linear-Scaling Selected Inversion based on Hierarchical Interpolative Factorization for Self Green's Function for Modified Poisson-Boltzmann Equation in Two Dimensions
Implicit temporal discretization and exact energy conservation for particle methods applied to the Poisson-Boltzmann equation
本文报道了一种新的多尺度方法，用于研究作用在短时间尺度上的短程力和作用在慢得多的尺度上的长程力之间有很大分离的系统。我们考虑泊松-玻尔兹曼方程的情况下，使用玻尔兹曼公式（即，我们假设介质是在准局部热平衡）描述的长程力。我们开发了一种新的方法，其中字段和粒子信息（介导的方程为他们的时刻）解决自洽。新方法是隐式的，数值稳定，提供精确的能量守恒。我们测试了不同的实现方式，所有这些都能实现精确的节能。新方法需要求解大量的非线性方程组。我们考虑了三种解决方案策略：Jacobian Free Newton Krylov，一种称为场隐藏的替代方案，基于隐藏部分残差计算并将其替换为直接解决方案和考虑简化的基于单个粒子的Jacobian的直接Newton Schwarz求解器。域隐藏策略被证明是最有效的方法。
A Domain Decomposition Method for the Poisson-Boltzmann Solvation Models
Sorbonne Universités, UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, and Institut des Sciences du Calcul et des Données（索邦大学，巴黎第六大学，UMR 7598，雅克-路易·利翁斯实验室，计算与数据科学研究所）；Center for Computational Engineering Science, RWTH Aachen University（亚琛工业大学计算工程科学中心）；Computational Biomedicine, Institute for Advanced Simulation IAS-5 and Institute of Neuroscience and Medicine INM-9, Forschungszentrum Jülich（朱利希研究中心计算生物医学，先进模拟研究所IAS-5，神经科学与医学研究所INM-9）；Sorbonne Universités, UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, and Institut Universitaire de France（索邦大学，巴黎第六大学，UMR 7598，雅克-路易·利翁斯实验室，法国高等研究院）；Division of Applied Mathematics, Brown University（布朗大学应用数学系）
